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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nNanoscale confinement of a polymer strongly affects its conformational and dynamical \nproperties.  Recent advances in nanofabrication techniques have facilitated the creation of \nlab-on-a-chip nanofluidic devices that are ideal for studying and characterizing \nsuch physical effects.  In recent years, nanofluidics experiments employing optical \nimaging techniques to study biopolymers such as DNA have been instrumental in testing and refining \ndecades-old theories of confined polymers. A notable example is confinement of DNA in \nnanochannels.\\cite{dai2016polymer,reisner2012dna} A thorough understanding of the \nfundamental physics of such systems is vital for various applications that require \nstretching of DNA in channels, including DNA sorting,\\cite{dorfman2012beyond} \nDNA denaturation mapping,\\cite{reisner2010single,marie2013integrated} and genome mapping.%\n\\cite{lam2012genome,hastie2013rapid,dorfman2013fluid,muller2017optical} \nA number of other studies have examined the confinement effects on DNA in embedded nanotopography\ndevices composed of a nanoslit with nanogrooves or nanopits etched into one surface\ndeeper than the surrounding slit. While the pits and grooves generally promote entropic trapping \nof the DNA, some portion of the contour of the molecule can occupy the narrow region\nof the slit outside these structures. This enables a novel means for conformational manipulation \nof single polymers. For example, recent experiments by Reisner and coworkers have observed \nand characterized `digitized' or `tetris'-like conformations for polymers that share their \ncontour between multiple adjacent nanopits.\\cite{klotz2015correlated,klotz2015measuring} \nEmbedded nanotopography devices can also serve as useful models to characterize single-molecule \ntransport across free energy landscapes.\\cite{mikkelsen2011pressure,kim2017giant,%\nsmith2017photothermal,klotz2016waves,del2009pressure,klotz2012diffusion,ruggeri2017lattice}\n\nOne notable study using nanofluidics techniques to study the effects of nanoconfinement on polymers \nwas reported recently by Capaldi {\\it et al.}\\cite{capaldi2018probing} Their experiment \nemployed pneumatic pressure to deflect a thin nitride lid into a nanoslit containing a solution\nof fluorescently stained $\\lambda$-DNA chains, forcing the molecules into an array of nanocavities \nembedded in one surface of the slit. Each cavity had a square cross section of side length 2~$\\mu$m,\nwas 200~nm deep, and was able to trap up to two $\\lambda$-DNA chains per cavity.  \nDifferential staining of the chains enabled monitoring of \nindividual chain conformation, the degree of partitioning or mixing of the chains, and coupled\ndiffusion of the centre-of-mass chain positions. Comparing the results to those for cavities\nwith a single trapped DNA chain, the drastic impact of the presence of a second chain\non the conformation and dynamics was quantified. Similar, though less pronounced, effects\nwere observed for a cavity-confined system containing a single $\\lambda$-DNA chain together \nwith a small plasmid. \n\nNumerous theoretical and simulation studies have examined the mixing\/partitioning behaviour \nof two polymers confined to nanoscale cavities and channels in recent years.\\cite{jun2006entropy,%\nteraoka2004computer,jun2007confined,arnold2007time,jacobsen2010demixing,jung2010overlapping,%\njung2012ring,jung2012intrachain,liu2012segregation,dorier2013modelling,racko2013segregation,%\nshin2014mixing,minina2014induction,minina2015entropic,chen2015polymer,polson2014polymer,%\ndu2018polymer,polson2018segregation,nowicki2019segregation,nowicki2019electrostatic,polson2021free} \nUnder sufficiently strong confinement, polymers tend to segregate due to entropic repulsion between\nthe chains.  It has been suggested that this effect may contribute to the driving force for chromosome\nsegregation in self-replicating bacteria,\\cite{jun2006entropy,jun2010entropy,diventura2013chromosome,%\nyoungren2014multifork,mannik2016role} and recent experimental studies have reported results consistent\nwith this hypothesis.\\cite{diventura2013chromosome,mannik2016role,cass2016escherichia,wu2020geometric,%\nelnajjar2020chromosome,japaridze2020direct} \nUnfortunately, {\\it in vivo} experiments on replicating bacteria do not provide a straightforward\nmeans to quantify the degree of entropic repulsion. By contrast, {\\it in vitro} nanofluidics experiments \nsuch as that by Capaldi {\\it et al.}, which consider much simpler systems, are much better\nsuited for direct comparison with the predictions of theory and simulation.\n\nIn this study, we use Brownian dynamics (BD) and Monte Carlo (MC) simulations to study the organization,\nconformational behaviour, and equilibrium dynamics of a system of two polymers under confinement\nin a box-like cavity. Ideally, the molecular model should incorporate correct length scale ratios \nfor the width, contour length, and persistence length of $\\lambda$-DNA. However, this choice leads \nto simulations that are far too time consuming to be of practical benefit, especially in the case \nof dynamics.  Consequently, we employ instead a simple coarse-grained molecular model, in which the \npolymers are described as relatively short flexible chains of spherical Lennard-Jones beads. One goal \nof this study is to determine whether the general trends observed in the study of Capaldi {\\it et al.} \ncan be accounted for using such a simplistic model. The simulations also provide a means to\ntest the validity of the interpretation proposed by Capaldi {\\it et al.} for the observed dynamical\nbehaviour. In addition, we examine effects of varying system parameters not considered in the \nexperiments. Most notably, we study the effects of varying the confining box dimensions on the \npolymer dynamics and organization. For sufficiently small cavities, we find that the polymers tend \nto segregate to opposite sides of the box and that the rates of polymer diffusion and internal \nmotion are both strongly affected by interpolymer crowding. These observations are qualitatively\nconsistent with those of the experimental study, demonstrating the utility of the very simplistic\nmodel employed in the simulations.  The observed behaviour in this model system may also be of value in\ninterpreting results of future experiments.\n\nThe remainder of this article is organized as follows. Section~\\ref{sec:model} presents a\nbrief description of the model used in the simulations, following which\nSection~\\ref{sec:methods} gives an outline of the methodology employed together with\nthe relevant details of the simulations. Section~\\ref{sec:results} presents the simulation\nresults for the various systems we have examined, and Section~\\ref{sec:experiment} describes\nthe relevance of the simulation results to experiment.  Finally, Section~\\ref{sec:conclusions}\nsummarizes the main conclusions of this work.\n\n\n\\section{Model}\n\\label{sec:model}\n\nWe examine systems of either one or two polymer chains confined to a box-like cavity.\nEach polymer is a flexible linear chain of $N$ spherical monomers. Polymer lengths are in\nthe range $N$=40--80 for BD simulations and 40--300 for MC simulations. {\nFor the two-polymer systems, the lengths of the two polymers are equal.}\nNon-bonded interactions are given by the repulsive Lennard-Jones potential,\n\\begin{eqnarray}\nu_{\\rm nb}(r) =\n\\begin{cases}\n u_{\\rm LJ}(r) - u_{\\rm LJ}(r_{\\rm c}),  & r \\leq r_{\\rm c} \\\\\n0,   & r \\geq r_{\\rm c}\n\\end{cases}\n\\label{eq:LJ}\n\\end{eqnarray}\nwhere $r$ is the distance between the monomer centres, $r_{\\rm c} \\equiv 2^{1\/6}\\sigma$,\nand where $u_{\\rm LJ}(r)$ is the standard Lennard-Jones 6-12 potential,\n\\begin{equation}\nu_{\\rm LJ}(r) = 4\\epsilon\\left[\\left(\\frac{\\sigma}{r}\\right)^{12}-\n\\left(\\frac{\\sigma}{r}\\right)^6\\right].\n\\end{equation}\nBonded monomers interact with a combination of the potential in Eq.~(\\ref{eq:LJ})\nand the finite extensible nonlinear elastic (FENE) potential,\n\\begin{equation}\nu_{\\rm FENE}(r)=-{\\textstyle\\frac{1}{2}} kr_0^2 \\ln(1-(r\/r_0)^2), \n\\label{eq:FENE}\n\\end{equation}\nwhere we choose $k\\sigma^2\/\\epsilon=30$ and where $r_0=1.5\\sigma$.\n\nThe polymers are enclosed in a rectangular box with a square cross section in the $x-y$ \nplane of side length $L$ and a height $h$ in the $z$ direction. The box dimensions are \ndefined such that $L$ is the range of positions along the $x$ and $y$ axes accessible to \nthe centres of the monomers, and $h$ is the corresponding range along $z$. To impose\nthis condition, each monomer interacts with each wall through Eq.~(\\ref{eq:LJ}), where \n$r+\\sigma$ is the distance of the monomer to the nearest point on the wall.  Most \ncalculations used $h$=4, a value that is low enough to compress the polymer along the\n$z$ direction, as was the case in the experiments of Ref.~\\onlinecite{capaldi2018probing}.  \nWe use a wide range of values for the box width. \n\n\\section{Methods}\n\\label{sec:methods}\n\nWe use two different simulation methods to study the confined-polymer system. BD simulations \nare used to monitor the dynamics of centre of mass motion as well as the internal dynamics \nof each chain. {Since the focus is on characterizing dynamics at longer time \nscales, results obtained using the BD method are not expected to differ significantly from \nthose obtained using the more computationally costly Langevin dynamics method.} \nMC simulations employing the standard Metropolis method are used to measure \nprobability distributions associated with polymer position, as well as to characterize the \nconformational statistics.  {Although in principle BD simulations could also be \nused for these measurements, they were far too computationally costly for larger $N$ to\nobtain statistically sound results in a reasonable time. By contrast, this presented\nno problem for the much more efficient MC simulations. For convenience, we chose to\nuse MC simulations for the static quantities for all $N$.}  Both methods employ the molecular \nmodel described in Section~\\ref{sec:model}. A brief description of each method follows below.\n\n\\subsection{Monte Carlo simulations}\n\\label{subsec:mc}\n\nWe use a standard MC simulation method in which polymer configurations are generated using \ntrial moves that are accepted or rejected based on the Metropolis MC criterion. The trial\nmoves consist of a combination of single-monomer crankshaft rotations, reptation \nmoves and whole-polymer displacements. The type of each trial move is randomly selected.\nA single MC cycle consists of $2N$ trial moves,  each consisting\nof of $N-1$ crankshaft moves, $N-1$ reptation moves, and two whole-polymer translations. \nFor each crankshaft move, a randomly selected monomer was rotated about an \naxis connecting adjacent monomers through a random angle drawn from uniform distribution in \nthe range $[-\\Delta \\phi_{\\rm max}, +\\Delta\\phi_{\\rm max}]$. In the case of end monomers, \nrotation was about the second bond from the end.  Whole-polymer translation \nwas achieved by moving all monomers of a randomly selected polymer through a displacement drawn \nfrom a uniform distribution in the range $[-\\Delta_{\\rm max},+\\Delta_{\\rm max}]$ for each \ncoordinate. The parameters $\\Delta \\phi_{\\rm max}$ and $\\Delta_{\\rm max}$ were chosen to achieve \nan acceptance ratio of approximately 50\\%. For each reptation move, the polymer and the\nreptation direction were both randomly selected.\n\n{For each system size, as defined by $N$ and $L$, we carried out numerous simulations on\nan array of processors, each using a different sequence of random numbers, to acquire a collection \nof statistically uncorrelated results. This collection of results was then averaged. Dividing the \ncalculation into such independent runs essentially parallelizes the simulation and dramatically\nincreases the computational efficiency.} Each simulation consisted of an equilibration period of \ntypically $10^6$ MC cycles followed by a production run of $10^8$ MC cycles. The number of these \nsimulations ranged from 50 for $N$=40 to 1000 for $N$=300, corresponding to total simulation times \nof 540 CPU-hours for $N$=40 to 15000 CPU-hours for $N$=300, respectively.\n\n\\subsection{Brownian dynamics simulations}\n\\label{subsec:bd}\n\nThe BD simulations used to study the polymer dynamics employ standard methods.\nThe coordinates of the {\\it i}th particle are advanced through a time $\\Delta t$ according to \nthe algorithm:\n\\begin{eqnarray}\nx_i(t+\\Delta t) & = & x_i(t) + \\frac{f_{i,x}}{\\gamma_0} + \\sqrt{2 k_{\\rm B}T \\Delta t\/\\gamma_0} \\Delta w,\n\\label{eq:BDeq}\n\\end{eqnarray}\nand likewise for $y_i$ and $z_i$.  Here, $f_{i,x}$ is the $x$-component of the conservative \nforce on particle $i$, and $\\gamma_0$ is the friction coefficient of each monomer. The conservative \nforce is calculated as $f_{i,\\alpha}=-\\nabla_{i,\\alpha} U_{\\rm tot}$, where $\\nabla_{i,\\alpha}$ \nis the $\\alpha$-component of the gradient with respect to the coordinates of the $i$th particle \nof the total potential energy of the system, $U_{\\rm tot}$. In addition, $\\Delta w$ is a random \nquantity drawn from a Gaussian of unit variance.  \n\nAll simulations were carried out at a temperature $T = \\epsilon\/k_{\\rm B}$, where $k_{\\rm B}$\nis Boltzmann's constant.  The time step used in Eq.~(\\ref{eq:BDeq}) was \n$\\Delta t = 0.0001\\tau_{\\rm BD}$, where $\\tau_{\\rm BD}\\equiv\\gamma_0\\sigma^2\/\\epsilon$. The \nrun time of each simulation was typically $10^5\\tau_{\\rm BD}$, following an equilibration \nperiod of typically $10^4\\tau_{\\rm BD}$. For each system size, as defined by $N$ and $L$, the \nresults of numerous simulations were averaged and used to estimate uncertainties. This ranged from \n750 simulations for $N$=40 to 1000 simulations for $N$=80, corresponding to total simulation times \nof roughly 10000 CPU-hours and 30000 CPU-hours, respectively.\n\n\\subsection{Measured quantities}\n\\label{subsec:measurements}\n\nTo track the polymer centre-of-mass motion, we use the mean-square displacement, \n\\begin{eqnarray}\n{\\rm MSD}(t) = \\left\\langle \\left( x_i(t) - x_i(0) \\right)^2 \\right\\rangle.\n\\label{eq:MSDdef}\n\\end{eqnarray}\nwhere $x_i$ is the centre-of-mass $x$ coordinate for the $i$th polymer. The angular brackets \ndenote an average over sequences of configurations generated in separate simulations, as \nwell as over the time origin for each simulation. In addition, an average was carried out \nover both polymer positions in the 2-polymer system, as well as over the $y$ coordinates \nof the centres of mass. The latter average is valid since the box width in the $x$ and $y$ \ndimensions is equal. \n\nA related measure of polymer motion is the position autocorrelation function,\n\\begin{eqnarray}\nC_{\\rm auto}(t) \\equiv \\left\\langle x_i(t)x_i(0)\\right\\rangle.\n\\label{eq:Cauto}\n\\end{eqnarray}\nNote that since the box centre lies at $x=0$, then $\\langle x_i\\rangle=0$, and\nthus $\\langle  x_i(t)x_i(0)\\rangle = \\langle x_i^2\\rangle - \\frac{1}{2}{\\rm MSD}(t)$.\nCorrelations between the centre-of-mass kinetics of the two polymers is quantified\nby the cross-correlation function,\n\\begin{eqnarray}\nC_{\\rm cross}(t) \\equiv -\\left\\langle x_1(t)x_2(0)\\right\\rangle,\n\\label{eq:Ccross}\n\\end{eqnarray}\nwhere the subscripts denote different polymers. Since the signs of $x_1$ and $x_2$ tend \nto be opposite (i.e. the polymers tend to be situated on opposite sides of the box), the \nnegative sign leads to the property, $C_{\\rm cross}(t) \\geq 0$.\n\nTo examine internal motion of the polymers, we use Rouse coordinates, defined as\n\\begin{eqnarray}\n{\\bf R}_p \\equiv \\frac{1}{N}\\sum_{n=1}^N {\\bf r}_n \\cos\\left(\\frac{p(n-{\\textstyle\\frac{1}{2}})\\pi}{N}\\right),\n\\end{eqnarray}\nwhere ${\\bf r}_n$ is the position of particle $n$, and $p=1,2,3...$. These are used to calculate the\ncorrelation functions\n\\begin{eqnarray}\nC_p(t) & = & \\left\\langle {\\bf R}_p(t+t_0)\\cdot {\\bf R}_p(t_0)\\right\\rangle_{xy},\n\\end{eqnarray}\nwhere the subscript indicates that only transverse components of the coordinates are used in\nthe calculation of the average. In most cases we find that the correlation function decays \nexponentially such that $C_p\\propto e^{-t\/\\tau_p}$, where $\\tau_p$ is the correlation \ntime for the $p$th mode. Typically, we observe a small transient at short times, which is excluded\nfrom the fit. {The consistently exponential form of $C_p(t)$ for both one- and \ntwo-polymer systems with excluded volume under even strong confinement conditions is a somewhat\nsurprising but useful property. For example,} we find that the $p=1$ mode provides a more \nconvenient probe of large-scale conformational changes than, e.g., the end-to-end displacement \nsince the correlation function of the latter tends not to be exponential under the conditions \nexamined here.\n\nAs a measurement of the polymer shape isometry, we use the 2-D version of asphericity,\n$A_2$, defined as\n\\begin{eqnarray}\nA_2 = \\frac{\\left\\langle R_1^2\\right\\rangle - \\left\\langle R_2^2\\right\\rangle } \n         {\\left\\langle R_1^2\\right\\rangle + \\left\\langle R_2^2\\right\\rangle },\n\\label{eq:asph}\n\\end{eqnarray}\nwhere the angular brackets denote an average over sampled configurations.\nThe quantities $R_1^2$ and $R_2^2$ ($\\leq R_1^2$) are eigenvalues of the \n2-D gyration matrix, whose elements are defined\n\\begin{eqnarray}\nS_{\\alpha\\beta} & = & \\frac{1}{N} \\sum_{i=1}^{N} \n\\left(r_{\\alpha,i} - r_{\\alpha,{\\rm cm}}\\right)\n\\left(r_{\\beta,i}  - r_{\\beta,{\\rm cm}}\\right), \\\\\n&&\\nonumber\n\\end{eqnarray}\nwhere $r_{\\alpha,i}$ is the $\\alpha$-coordinate ($\\alpha=x, y$) of particle $i$ and \n$r_{\\alpha,{\\rm cm}}$ is the instantaneous $\\alpha$-coordinate of the centre-of-mass.  The \nquantities $R_1$ and $R_2$ ($\\leq R_1$) can be viewed as the semi-major and \nsemi-minor axes of an equivalent ellipse that very roughly approximates the shape\nof the polymer in the $x-y$ plane. Note the two limiting cases for the asphericity: \n$A_2=0$ corresponds to a circular disk, and $A_2=1$ corresponds to an infinitesimally thin\nneedle. Note as well that $R_{{\\rm g},xy}^2 = R_1^2+R_2^2$ is the instantaneous\nsquare radius of gyration in the $x-y$ plane.\nAs a measure of inter-polymer overlap, we define the overlap parameter $\\chi_{\\rm ov}\\equiv \nN_{\\rm ov }\/N$, where $N_{\\rm ov}$ is the average number of monomers per polymer that lie \ninside the overlap area of the two equivalent ellipses defined above. \n\n\nIn the results presented below, distances are measured in units of $\\sigma$, energy\nis measured in units of $\\epsilon$ ($=k_{\\rm B}T$), and time is measured in units of \n$\\gamma_0\\sigma^2\/\\epsilon$.\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Polymer organization and conformational statistics}\n\\label{subsec:statics}\n\nWe consider first the effects of confinement and crowding on the organization of the polymers \nin the cavity.  Figure~\\ref{fig:prob2d_N60} shows 2-D probability distributions that characterize \nthe position of the polymers inside the the box-like cavity in the transverse plane. Results\nare shown for wide range of box widths.  {The lowest value of $L$ was chosen to \nmaintain the condition $L>h$, while the highest value of $L$ (as explained below) corresponds\nto a system for which confinement and inter-polymer crowding effects are negligible.}\n{Row (a)} shows the probability of the centre\nof mass of a polymer at any position in the $x-y$ plane for the case where just a single\npolymer is confined to the cavity. {Row (b)} shows the same probability distribution\nfor the case where two identical polymers are confined to the box.  We label these distributions\n${\\cal P}_1(x,y)$, where the subscript denotes the single-polymer aspect of the distribution\n(and {\\it not} the number of confined polymers). {Row (c)} shows probability distributions \nfor the difference in the centre-of-mass coordinates, $\\delta x$ and $\\delta y$, of the polymers \nin the two-polymer system. We label these distributions ${\\cal P}_2(\\delta x,\\delta y)$, where \nthe subscript denotes the fact that this is a 2-polymer property. (Note that this quantity\nis a physically meaningful descriptor only for the 2-polymer system.)  Together, the two complementary \ndistributions provide a clear picture of the effect of varying box width $L$ on the polymer \nposition. {As an aid to interpret these 2D probability maps, Figs.~\\ref{fig:prob2d_N60}(d), \n(e) and (f) show probability cross sections of ${\\cal P}_1(x,y)$ and ${\\cal P}_2(\\delta x,\\delta y)$ \nfor some of the two-polymer systems.}\n\nA useful relative measure of the box width is the ratio $L\/R_{{\\rm g},xy}^*$, where \n$R_{{\\rm g},xy}^*$ is the transverse root-mean-square radius of gyration for a polymer in a \nslit, i.e., $L$=$\\infty$.  A box with $L\/R_{{\\rm g},xy}^*\\gg 1$ is large in the sense that \nthe polymers are unlikely to interact with either the walls or with each other. As the box \nwidth approaches the regime where $L\/R_{{\\rm g},xy}^*$ is of the order of unity, the combined \neffects of the polymer-polymer and polymer-wall interactions are expected to strongly affect \nthe organization of the polymers inside the box as well as their conformational behaviour.  \nNote that the system in Fig.~\\ref{fig:prob2d_N60} is characterized by $N$=60 and $h$=4, for \nwhich $R_{{\\rm g},xy}^*=5.710\\pm 0.001$.\n\n\\begin{figure*}[!ht]\n\\begin{center}\n\\vspace*{0.2in}\n\\includegraphics[angle=0,width=0.95\\textwidth]{fig1.pdf}\n\\end{center}\n\\caption{Probability distributions for a system of one and two polymers of length $N$=60 confined to a \nbox of height $h$=4. Results for various box widths are shown. Row (a) shows results for the distribution\n${\\cal P}_1(x,y)$ for a {\\it single-polymer system}. Row (b) shows distribution ${\\cal P}_1(x,y)$ for\na {\\it two-polymer system}. Row (c) shows ${\\cal P}_2(\\delta x,\\delta y)$, the probability\ndistributions for the {\\it difference} in the centre-of-mass coordinates, $\\delta x$ and $\\delta y$\nof the polymers in the two-polymer system. {In (a), (b) and (c) the axes are scaled by \nthe box width, $L$, and the colour intensity maximum corresponds to the maximum value of the probability \nfor each graph.  (d) Probability distribution cross-sections ${\\cal P}_1(x|y)$ for the two-polymer system \nwith $L$=22 for various values of $y$. (e) As in (d), except for $y$=0 (which bisects the box) for various \n$L$.  (f) Probability distribution cross-sections ${\\cal P}_2(\\delta x|\\delta y)$ for $\\delta y$=0 and \nfor various $L$.}}\n\\label{fig:prob2d_N60}\n\\end{figure*}\n\nLet us consider first the behaviour of the 1-polymer system. At the largest box size of $L=78$,\nthe probability distribution ${\\cal P}_1(x,y)$ is fairly flat over the $x-y$ plane, with\nthe exception of an entropy-induced depletion layer along the lateral walls. As $L$ is reduced,\nrelative width of the depletion zone grows, and the polymer centre-of-mass distribution \nincreasingly narrows to the box centre. Concomitantly, the distribution transforms from a\nsquare shape (corresponding to the shape of the box) to a circular shape.\n\nA comparison of the ${\\cal P}_1(x,y)$ distributions for the 1-polymer system with those for\nthe 2-polymer systems shows the increasingly pronounced effect of crowding as $L$ decreases.\nFor the largest box size of $L=78$ ($L\/R_{{\\rm g},xy}^*=13.7$), ${\\cal P}_1(x,y)$ is essentially\nidentical to that for the 1-polymer system: flat over the $x-y$ plane, with an entropy-induced \ndepletion layer along the lateral walls. This is precisely the behaviour expected in the dilute \nlimit where interpolymer interactions are infrequent. The corresponding distribution \n${\\cal P}_2(\\delta x,\\delta y)$ exhibits a high-probability ring with a probability hole in the \nmiddle, as well as very low probability for large inter-polymer displacements. The depletion hole \ncorresponds simply to the tendency of the polymers to avoid transverse overlap, as such configurations \nwould reduce the conformational entropy. Likewise, very large separations require the polymers to \npress against the walls of the box, which also has an entropy-reducing effect. Within the \nhigh-probability ring, there are slight enhancements at four symmetrically distributed locations, \ntwo at $\\delta x=0$ and two at $\\delta y=0$. \n\nAs $L$ becomes smaller, ${\\cal P}_1(x,y)$ changes significantly. At $L$=38 ($L\/R_{{\\rm g},xy}^*$=6.7),\nsmall peaks appear near the corners of the distribution. Thus, the polymers increasingly tend to \nbe situated near the corners of the box. No such feature is present for the 1-polymer system,\nindicating that it is a consequence of interpolymer crowding.  At a smaller box size \n$L$=22 ($L\/R_{{\\rm g},xy}^*$=3.9), the distribution has a ring-like structure, with a significant\ndepletion hole in the centre of the box. Slight enhancements are evident at four symmetrically\nrelated positions. The narrowness of the ring structure for the corresponding\n${\\cal P}_2(\\delta x,\\delta y)$ indicates that when one polymer centre lies at $(x,y)$ the other\nwill tend to lie at an inverted position of $(-x,-y)$. Thus, each polymer centre is expected to \ndiffuse around the centre of the box with the other polymer moving in a correlated manner in\nthis inverted position.  Note the pronounced qualitative difference in ${\\cal P}_1(x,y)$ between\nthe one- and two-polymer systems.\n\nAs the box size decreases to $L$=13 ($L\/R_{{\\rm g},xy}^*$=2.3) the ring structure for the \ntwo distributions gives way to strong probability enhancements at four symmetrically related \npositions. For ${\\cal P}_1$, two are located along $x$=0 and the other two are at $y$=0. \nA similar structure is evident for ${\\cal P}_2$.\nThis behaviour indicates that the polymers tend to occupy quasi-discrete locations \nin opposite halves of the box divided by boundaries at $x$=0 or $y$=0. Such behaviour is a consequence \nof the significant interaction between the polymers (i.e., crowding) and between each polymer and \nthe confining walls (i.e., confinement) for this small box size. As will be shown below, these \ninteractions also strongly deform the polymer conformation leading to significant changes \nin its average size and shape.  At the smallest examined box size of $L=7$ ($L\/R_{{\\rm g},xy}^*$=1.23), \na new trend emerges: an enhancement of the probability at positions near the centre of the box \nand for very small inter-polymer displacements (i.e. near $\\delta x = \\delta y = 0$).  Note that \nthese broad probability peaks for ${\\cal P}_1(x,y)$ and ${\\cal P}_2(\\delta x,\\delta y)$ each coexist \nwith the remaining four peaks associated with the the quasi-discrete states described above.\nAgain, we note the qualitatively different behaviour in ${\\cal P}_1(x,y)$ for the 1- and 2-polymer\nsystems for these small box sizes.\n\nAlthough the distributions in Fig.~\\ref{fig:prob2d_N60} were calculated for $N$=60 and $h$=4,\nthe qualitative trends are unaffected by other choices of $N$ and $h$ (as long as $h$ is\nsufficiently small to compress the polymer along the $z$ direction). Distributions for a polymer\nlength of $N$=300 and box heights of $h$=4, 6, and 8 presented in Figs.~1 and 2 of the ESI indeed \nshow the same trends as those in Fig.~~\\ref{fig:prob2d_N60} above.\\dag\\footnotetext{\\dag~Electronic \nSupplementary Information (ESI) available: [details of any supplementary information available \nshould be included here]. See DOI: 10.1039\/b000000x\/}\n\nThe tendency of the polymer positions to become inversely correlated with respect to the box\ncentre for highly confined systems is also illustrated in Fig.~\\ref{fig:corr_LJ}. \n{The mean-square centre-of-mass position of each polymer, $\\langle x_1^2\\rangle$ \n($=\\langle x_2^2\\rangle$),\nand the position cross-correlation, $-\\langle x_1 x_2\\rangle$, both decrease monotonically\nwith decreasing $L$.} However, the ratio, $-\\langle x_1x_2\\rangle\/\\langle x_1^2\\rangle$, shown \nin the inset increases as the polymers become more confined. This ratio is a measure of the\ndegree of inverse correlation of the polymer positions. Note that the ratio is independent\nof $N$ when plotted against the scaled box length, $L\/R_{{\\rm g},xy}^*$. We expect qualitatively \nsimilar behaviour for more a realistic polymer model (e.g., one providing a better description of \n$\\lambda$ DNA), except with a much lower degree of correlation. The convergence of the ratio to \nunity here likely arises from packing effects due to the high volume fraction at low $L$.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\vspace*{0.2in}\n\\includegraphics[width=0.4\\textwidth]{fig2.pdf}\n\\end{center}\n\\caption{Variation of $\\langle x_1^2\\rangle^*\\equiv \\langle x_1^2\\rangle\/(R_{{\\rm g},xy}^*)^2$ \n(closed symbols) and $-\\langle x_1 x_2\\rangle^*\\equiv -\\langle x_1 x_2\\rangle\/(R_{{\\rm g},xy}^*)^2$ \n(open symbols) with scaled box width, $L\/R_{{\\rm g},xy}^*$. Results are shown for $N$=40, 60, 80, \nand 300. The solid and dashed curves are guides for the eye. The inset shows the ratio of the two \nquantities for each chain length.}\n\\label{fig:corr_LJ}\n\\end{figure}\n\nNow let us examine conformational behaviour of each individual polymer.  Figure~\\ref{fig:size_shape} \nillustrates the effect of lateral confinement on the mean size and shape of the polymer. \nFigure~\\ref{fig:size_shape}(a) shows the variation of the transverse radius of gyration, \n$R_{{\\rm g},xy}$, with respect to the box width. Results are shown for both 1- and 2-polymer \nsystems for comparison. {For sufficiently large $L$, where polymer-wall and \npolymer-polymer interactions are infrequent} the transverse size of each polymer \nis close to the value for a slit For $L\/R_{{\\rm g},xy}^*\\lesssim 5$, \nthe size decreases rapidly with decreasing $L$. This decrease is comparable for both 1- and 2-polymer\nsystems, indicating that it is driven primarily by the interactions with the lateral confining walls\nand less so by inter-polymer crowding. The relative difference between the two sets of results, \n$\\Delta R \\equiv (R_{{\\rm g},xy}(1~{\\rm pol})-R_{{\\rm g},xy}(2~{\\rm pol}))\/R_{{\\rm g},xy}^*$, is \nshown in the inset of Fig.~\\ref{fig:size_shape}(a). The difference peaks at $L\\approx \n4R_{{\\rm g},xy}^*$, below which it decreases rapidly. Thus, at very small $L$, the \ninter-polymer crowding no longer drives compression of the polymer in the $x-y$ plane. This likely\narises from the increased tendency of the polymers to overlap with each other at the centre\nof the box in this regime.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\vspace*{0.2in}\n\\includegraphics[width=0.45\\textwidth]{fig3.pdf}\n\\end{center}\n\\caption{(a) Scaled transverse radius of gyration, $R_{{\\rm g},xy}\/R_{{\\rm g},xy}^*$,\nvs. scaled box width, $L\/R_{{\\rm g},xy}^*$. Results are shown for box height $h$=4 and for both\n1-polymer and 2-polymer systems for $N$=40, 60 and 300. The horizontal dashed line shows the value for \n$L$=$\\infty$.  The inset shows the relative difference $\\Delta R$ (defined in the text) between the \ndata for the 1- and 2-polymer systems. (b) Asphericity, $A_2$, vs. scaled box width for the same systems \nas in panel (a).  The horizontal dashed line is the value measured for a slit for $N=60$. The inset \nshows the difference $\\Delta A_2$ between the 1- and 2-polymer results.}\n\\label{fig:size_shape}\n\\end{figure}\n\nFig.~\\ref{fig:size_shape}(b) shows the variation of the asphericity $A_2$, defined in \nEq.~(\\ref{eq:asph}) with respect to box width. As in Fig.~\\ref{fig:size_shape}(a), the results\nare comparable to those for a slit when $L$ is sufficiently large. As $L$ decreases, increased \npolymer-wall and polymer-polymer interactions result in a reduction in $A_2$. This implies that \neach polymer becomes less elongated and more disc-like as crowding increases.  The difference \nbetween the 1- and 2-polymer results is shown in the inset. While qualitatively similar to the data \nin the Fig.~\\ref{fig:overlap}(a) inset for the polymer size, the difference for $\\Delta A_2$ peaks \nat a much smaller box size of $L\\approx 1.6R_{{\\rm g},xy}^*$. The difference is appreciable.\nFor example, at $L$=7 we note $\\Delta A_2=0.13$, which is 21\\% of the asphericity for slit confinement.\nIn addition, maximum $\\Delta A_2$ occurs at greatest confinement, precisely where the size difference \nis negligible and where Fig.~\\ref{fig:prob2d_N60} indicates that the polymers in the 2-polymer\nsystem have a tendency to overlap in the middle of opposite halves of the box. The concomitant \ninter-polymer repulsion evidently leads to a larger asphericity than for the case of a single \npolymer that interacts solely with the walls.\n\nGiven the effects of inter-polymer repulsion on polymer size and shape for small and medium\nbox widths, it is instructive to quantify the degree of polymer overlap. As described in \nSection~\\ref{subsec:measurements}, {we define the overlap parameter}\n$\\chi_{\\rm ov }\\equiv N_{\\rm ov}\/N$, where $N_{\\rm ov}$ is the mean number of monomers inside\noverlapping equivalent ellipses for the polymers.\nAs a reference, we also show results for an {\\it artificial} system of two \n{\\it non}-interacting polymers, i.e., a model system in which overlap between pairs of monomers \non different polymers are ignored (though non-bonded {\\it intra}-polymer interactions are present).\nWe refer to this as a ``non-interacting'' (NI) system, and the real system as an ``interacting''\n(I) system.\n\nFigure~\\ref{fig:overlap}(a) shows the variation of $\\chi_{\\rm ov }$ with box size for three \ndifferent polymer lengths.  The degree of overlap is very small for large $L$ \nand increases significantly with increasing confinement. This is true for both interacting \nand non-interacting polymers. The values for the non-interacting system are larger than those \nfor the interacting system. This arises simply from the entropic repulsion between polymers \ncaused by the inter-polymer interactions. In the absence of such interactions the polymers \nhave a greater tendency to overlap in the $x-y$ plane. For interacting polymers, the figure \nshows a transition in the rate of change of overlap with box width at $L\/R_{{\\rm g},xy}^*\\approx 5$.\nFor $L\/R_{{\\rm g},xy}^*\\lesssim 5$ the decrease in $\\chi_{\\rm ov}$ with box width is exponential\ncharacterized by a decay constant of $\\approx 1.0$, while for $L\/R_{{\\rm g},xy}^*\\gtrsim 5$ \nthe exponential decay constant is $\\approx 6.8$. Thus, the degree of overlap increases rapidly\nwith increasing confinement in the regime for $L\/R_{{\\rm g},xy}^*\\lesssim 5$. This inference \nis corroborated by the results in Fig.~\\ref{fig:overlap}(b), which shows the ratio of the 1- \nand 2-polymer values for $\\chi_{\\rm ov }$. For large box widths of $L\/R_{{\\rm g},xy}^*\\gtrsim 5$, \neach ratio is small and constant. For smaller box widths of $L\/R_{{\\rm g},xy}^*\\lesssim 5$ the \nratio rapidly increases for smaller box widths. Together, the results of Fig.~\\ref{fig:overlap}(a) \nand (b) indicate that the entropic repulsion preventing overlap of interacting polymers is being \noverridden by the even stronger repulsion from the walls, contact with which becomes increasingly \nunavoidable at smaller $L$. This region of enhanced overlap coincides with that of the dramatic \nreduction in size and asphericity of the polymers. {Finally,} in Fig.~3 of the ESI, \n{results obtained using an alternative as a measure of overlap} show the same \nqualitative trend as in Fig.~\\ref{fig:overlap}(b).\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\vspace*{0.2in}\n\\includegraphics[width=0.4\\textwidth]{fig4.pdf}\n\\end{center}\n\\caption{(a) Variation of the overlap parameter, $\\chi_{\\rm ov}$, with scaled box width, \n$L\/R_{{\\rm g},xy}^*$. Results are shown for $N$=40, 60, and 300 for both interacting (I)\nand noninteracting (NI) polymer systems. The definitions of these systems is described in\nthe text. The dashed and solid lines are fits to the interacting-system data in the\nregimes $L\/R_{{\\rm g},xy}^*<5$ and $L\/R_{{\\rm g},xy}^*>10$, respectively. (b) Ratio of \nthe overlap parameter for the {NI and I} cases as a function of box width.} \n\\label{fig:overlap}\n\\end{figure}\n\nTo summarize, the equilibrium statistics of the confined-polymer system exhibits behaviour\nthat depends strongly on the lateral width, $L$, of the confining box. At $L\/R_{{\\rm g},xy}^*\\gtrsim 5$ \nthe polymers rarely interact with each other or with the wall, and each chain behaves in most ways \ncomparably to a single slit-confined polymer. For smaller boxes with \n$L\/R_{{\\rm g},xy}^*\\lesssim 5$, the effects of confinement and inter-polymer crowding become \nappreciable. As $L$ decreases in this regime, the centres of mass of the two become increasingly \nlocalized at positions in opposite halves of the box, except for very small boxes where there\nis a simultaneous tendency for them to overlap the the box centre. In addition, there is an\nincrease in the overlap in the lateral distributions of monomers, which is driven by the\nincrease in confinement and, counter-intuitively, enhanced by inter-polymer repulsion.\nWe also find that increasing lateral confinement decreases the average size\nof the chains, an effect that is enhanced by inter-polymer crowding. The shape anisometry\n(``asphericity'') also decreases with decreasing $L$ in this regime, though this effect\nis slightly offset by that caused by inter-chain crowding.\n\n\n\\subsection{Polymer dynamics}\n\\label{subsec:dynamics}\n\nWe now examine the equilibrium dynamics of the confined-polymer system. As in Section~\\ref{subsec:statics} \nwe choose a single box height of $h$=4 and consider the effects of varying the box width. The trends \nin the dynamics can then be explained in the context of those for the equilibrium statistics described \nin the previous section.  Both the diffusion of the centre of mass and the internal dynamics \nare characterized.  As in Section~\\ref{subsec:statics} it is useful to consider a 1-polymer system for \ncomparison with the 2-polymer case in order to elucidate the effects of inter-polymer repulsion on \nthe system behaviour.\n\nWe consider first the mean-square displacement (MSD) of the polymer centre of mass, defined\nin Eq.~(\\ref{eq:MSDdef}).\nFigure~\\ref{fig:msd}(a) shows the time dependence of the MSD for a 2-polymer system with $N$=60, \n$L$=13. Initially, the curve rises rapidly with time, after which it levels off to a \nconstant value of approximately 7.8. The leveling off of the MSD is a straightforward consequence \nof the confinement of the polymer in the lateral directions. The basic features of the MSD shown \nin the figure are generic to the results for all $L$.  Generally, the limiting value of the MSD \nat long time and the characteristic time, $\\tau$ (defined below), required to reach this plateau \nboth increase with increasing box size.  Increasing the polymer length reduces the long-time \nvalue of the MSD and increases $\\tau$.\n \n\\begin{figure}[!ht]\n\\begin{center}\n\\vspace*{0.2in}\n\\includegraphics[width=0.45\\textwidth]{fig5.pdf}\n\\end{center}\n\\caption{(a) MSD versus time for a polymer of length $N$=60 in a box of width \n$L$=13 and height $h$=4. (b) $\\xi$ versus time for the system in (a), where \n$\\xi\\equiv C_0 - {\\rm MSD}(t)$, and where $C_0$ is the mean value of the MSD for $t>20000$.  \nThe red curve is a fit to an exponential using data in the range of $t=1000-6000$.\nThe inset is a closeup of the data at small $t$ illustrating the deviation in the\nexponential fit in this regime.}\n\\label{fig:msd}\n\\end{figure}\n\nA quantitative analysis of the MSD is aided by employing a theoretical model used in\nRef.~\\onlinecite{capaldi2018probing} to analyze comparable experimental data. In this \ndescription, the centre-of-mass motion of a single DNA molecule confined to a box-like cavity is\nmodeled as free Brownian diffusion of a particle subject to an infinite square-well \npotential.\\cite{kusumi1993confined} Interactions between polymer chains are effectively\nignored, implying that the model should only be quantitatively accurate for sufficiently wide boxes.\nSince the cavity is symmetric in the $x$ and $y$ dimensions, the MSD is the same along \nthese axes and can be averaged.  Solving the diffusion equation for a single particle \nin a square box of side length $L_{\\rm e}$ yields\\cite{kusumi1993confined}\n\\begin{eqnarray}\n{\\rm MSD}(t) = C_0 - C_1 \\sum_{n=1,3,5,...}^\\infty \n\\frac{1}{n^4}\\exp\\left[-D\\left(\\frac{n\\pi}{L_{\\rm e}}\\right)^2 t \\right],~~\n\\label{eq:MSD}\n\\end{eqnarray}\nwhere $D$ is the diffusion coefficient of the particle, and where $C_0\\equiv L_{\\rm e}^2\/6$\nand $C_1\\equiv 16L_{\\rm e}^2\/\\pi^4$. Defining the quantity $\\xi(t)$ as\n \\begin{eqnarray}\n\\xi(t) \\equiv C_0 - {\\rm MSD}(t),\n\\label{eq:xi}\n\\end{eqnarray}\nand noting that all of the terms with $n\\geq 3$ and are negligible in comparison to the $n=1$ \nterm for sufficiently long $t$, it follows that\n\\begin{eqnarray}\n\\xi(t) \\approx C_1 \\exp\\left(-t\/\\tau)\\right)\n\\label{eq:xi2}\n\\end{eqnarray}\nin this long-time limit, where $\\tau\\equiv DL_{\\rm e}^2\/\\pi^2$.\n\nTo apply these results to the mean-square displacement of the centre of mass of a polymer \ndiffusing in two dimensional square box, a reasonable choice of the effective box length\nis $L_{\\rm e}=L-2R_{{\\rm g},xy}^*$, where $L$ is the true side length of the confining box\nand $R_{{\\rm g},xy}^*$ is the radius of gyration of the polymer measured in the $x-y$ plane\nfor a slit geometry. Thus, the polymer is modeled as a hard 2-D disk of radius $R_{{\\rm g},xy}^*$, \nand shape deformations associated with pressing the polymer against a side wall are \nneglected.\\cite{capaldi2018probing} In addition, noting that Rouse dynamics are obeyed for \nthe simulation model, it follows that $D=k_{\\rm B}T\/\\gamma$. Since the friction efficient \nsatisfies $\\gamma=N\\gamma_0$, where $\\gamma_0$ is the friction per monomer, and noting that \n$k_{\\rm B}T\/\\gamma_0=1$, it follows that $D=1\/N$. Consequently, the time constant in \nEq.~(\\ref{eq:xi2}) satisfies:\n\\begin{eqnarray}\n\\tau\/N = L_{\\rm e}^2\/\\pi^2.\n\\label{eq:tauN}\n\\end{eqnarray}\n\nFigure~\\ref{fig:msd}(b) shows the time dependence of the quantity $\\xi$, calculated using the \ndata in Fig.~\\ref{fig:msd}(a).  {Consistent with} the theoretical model, we find \nthat $\\xi$ decreases exponentially with time. Only a small deviation from this behaviour is \nobserved at low values of $t$, as illustrated in the inset of the figure. This arises from \nthe non-negligible contribution to the MSD from the higher-$n$ terms in Eq.~(\\ref{eq:MSD}).\n\nFigure~\\ref{fig:tau.N.Le}(a) shows the variation of the scaled time constant, $\\tau\/N$, with\nrespect to the scaled box length, $L\/R_{{\\rm g},xy}^*$, while Fig.~\\ref{fig:tau.N.Le}(b) shows\n$\\tau\/N$ vs. the effective box length, $L_{\\rm e}$. Here, $\\tau$ is extracted from the fit of \n$\\xi$ to Eq.~(\\ref{eq:xi2}), and the effective box with is obtained from $L_{\\rm e} = \\sqrt{6C_0}$,\nwhere the quantity $C_0$, defined in Eq.~(\\ref{eq:MSD}), is estimated from the MSD in the \nplateau region.  Results are shown for both 1- and 2-polymer systems for polymer lengths of $N$=40, \n60 and 80. In Fig.~\\ref{fig:tau.N.Le}(b) the theoretical prediction of Eq.~(\\ref{eq:tauN}) \nis shown as a dashed curve. As expected, the data for both 1- and 2-polymer systems both \nconverge to the theoretical curve in the limit of large box length. In this regime, interactions\nbetween the two polymers are infrequent and therefore do not change the dynamical behaviour\nof either polymer. As $L_{\\rm e}$ decreases, such interactions become more frequent, leading\nto a reduction in the rate of diffusion of the polymers. This is characterized by an increase\nin the time constant relative to both the predictions of the theoretical model and the 1-polymer\ntime constant. Surprisingly, the prediction of Eq.~(\\ref{eq:xi2}) remains very accurate\nfor the 1-polymer system even to very small box sizes where $L_{\\rm eff}\\approx R_{\\rm g,xy}^*$.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\vspace*{0.2in}\n\\includegraphics[width=0.45\\textwidth]{fig6.pdf}\n\\end{center}\n\\caption{(a) Variation of $\\tau\/N$ with respect to $L\/R_{{\\rm g},xy}^*$. The time constant $\\tau$ \nis  extracted from fits to $\\xi(t)$ at sufficiently long times, where $\\xi(t)\\equiv C_0-{\\rm MSD}(t)$, \nand where $C_0 = \\lim_{t\\rightarrow\\infty} {\\rm MSD}(t)$. Results are shown for systems of a single\npolymer (open symbols) and for two polymers (closed symbols) and for polymer lengths of\n$N$=40, 60 and 80. The solid and dashed lines are guides for the eye for 2-polymer and 1-polymer\ndata sets, respectively.  (b) $\\tau\/N$ vs. $L_{\\rm e}$, where the effective box width is defined\n$L_{\\rm e}=\\sqrt{6C_0}$, as discussed in the text.  The dashed line shows the prediction of \nEq.~(\\ref{eq:tauN}), which is expected to be valid at $L_{\\rm e}\/R_{{\\rm g},xy}^*\\gg 1$.  \n{The inset shows the variation of $\\zeta\\equiv (L-L_{\\rm e})\/2R_{{\\rm g},xy}^*$ with \n$L\/R_{{\\rm g},xy}^*$.}  }\n\\label{fig:tau.N.Le}\n\\end{figure}\n\nAs an additional test of the approximations employed in the theoretical model, we consider \nthe quantity $\\zeta \\equiv (L-L_{\\rm e})\/2R_{{\\rm g},xy}^*$.  As noted above, the effective \nbox width is expected to be $L_{\\rm e} \\approx L-2R_{{\\rm g},xy}^*$ if we neglect the deformations \nin the size and shape of the polymer. In this picture, $\\zeta$ is a constant of \norder unity.  {The inset of Fig.~\\ref{fig:tau.N.Le}(b) shows the measured variation of\n$\\zeta$ with $L\/R_{{\\rm g},xy}^*$ for both 1- and 2-polymer systems.} For the single-polymer\nsystem, $\\zeta$ is indeed constant and close to unity for $L\/R_{{\\rm g},xy}^* \\gtrsim 5$. \nHowever, for $L\/R_{{\\rm g},xy}^* \\lesssim 5$, $\\zeta$ decreases slightly with decreasing box width.\nIn this regime, there is no extended area in the $x-y$ plane over which the polymer centre of mass \ncan move without interacting with the walls. Thus, size and shape deformations are significant \nand neglecting them leads to the observed deviation from the prediction.\n\nA similar trend is observed for the 2-polymer system. However, in this case the transition\noccurs at a larger box width of $L\/R_{{\\rm g},xy}^*\\approx 10$. The additional crowding\ncaused by the presence of the second polymer leads to increased interaction with the lateral\nwalls as well as shape deformations at box sizes where such effects are not as prominent\nin the 1-polymer system.  Note that the 2-polymer $\\zeta$ begins to decrease significantly \nwith decreasing $L$ at $L\/R_{{\\rm g},xy}^*\\approx 5$, which corresponds to $L\\approx 29$ and\n$L_{\\rm e}\\approx 17$. This is precisely where the measured $\\tau\/N$ for the 2-polymer system \nbegins to deviate from the dilute-limit approximation. \n\nNext, we examine the time dependence of correlations in the centre-of-mass positions of\nthe polymers. We use the autocorrelation function, $C_{\\rm auto}(t)$, and cross-correlation\nfunction, $C_{\\rm cross}(t)$, defined in Eqs.~(\\ref{eq:Cauto}) and (\\ref{eq:Ccross}), respectively. \nFigure~\\ref{fig:corr} shows both functions for box widths of $L$=13, 22, and 33. Both correlation \nfunctions tend decay exponentially at large $t$. Note that in each case the decay to zero at long \n$t$ is a consequence of choosing $x=y=0$ at the box centre.  The individual functions diverge with \ndecreasing $t$. This divergence is greater for larger box sizes.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\vspace*{0.2in}\n\\includegraphics[width=0.4\\textwidth]{fig7.pdf}\n\\end{center}\n\\caption{Autocorrelation function, $C_{\\rm auto}\\equiv\\left\\langle x_1(t)x_1(0)\\right\\rangle$, and \ncross-correlation function, $C_{\\rm cross}\\equiv -\\left\\langle x_1(t)x_2(0)\\right\\rangle$, for the \ncentre-of-mass positions for two $N$=60 polymers.  Results are shown for $L$=13, 22, and 33. The inset \nshows the same data plotted on a semi-log scale. }\n\\label{fig:corr}\n\\end{figure}\n\nThe system behaviour at $t=0$ has a simple explanation. Note that $C_{\\rm auto}(0)=\\langle x_1^2 \\rangle$\nand $C_{\\rm cross}(0)=-\\langle x_1 x_2\\rangle$. As previously noted for the results in Fig.~\\ref{fig:corr_LJ},\n$\\langle x_1^2 \\rangle$ approaches $-\\langle x_1x_2 \\rangle$ for small box sizes since the polymer positions \nin the $x-y$ plane are highly anti-correlated in this regime (i.e. if one polymer lies at $(x,y)$ the other \nis likely to be near $(-x,-y)$).  For larger box sizes the degree of anticorrelation is smaller, leading \nthe observed wider divergence between the correlation functions for larger $L$. \n\nTo better understand the origins of the observed time dependence of the functions, we employ a \nsimple model to describe the box-size regime where the polymers are pushed out from the centre of \nthe box (see, e.g., the results for $L$=13 and $L$=22 in Fig.~\\ref{fig:prob2d_N60}). Here, the \npolymers are pictured as two point particles that each occupy one of four discrete sites on the \ncorners of a square and whose interactions mimic the effects of entropic repulsion. The model can \nbe used in MC dynamics simulations to measure the two position correlation functions. The model \nis fully described in the appendix, and the calculated correlation functions are shown \nin Fig.~\\ref{fig:corr_discrete_model}. The convergence to exponential decay at large $t$ and the \ndivergence at low $t$ are both present. Thus, a simple model incorporating the basic features of \nthe probability distributions of Fig.~\\ref{fig:prob2d_N60} and a simple description of entropic \nrepulsion between the polymers is capable of accounting for the general behaviour of the correlation \nfunctions. \n\nLet us now examine the effects of confinement on the internal motion of the polymers.\nA useful means to characterize such motion is the autocorrelation function of the\nRouse coordinates of the polymer. As noted in Section~\\ref{sec:methods}, these functions\ntend to be exponential over at least one decade of decay (except at very short times)\nfor both 1- and 2-polymer systems.  \n\nFigure~\\ref{fig:rouse}(a) shows the variation of the Rouse mode correlation times for the \n$p$=1 mode with respect to box width. Note that $\\tau_1$ describes rates of conformational \nmotion on large length scales.  For sufficiently wide boxes $\\tau_1$ approaches \nthe value for slit confinement for both 1- and 2-polymer systems. Thus, the rate\nof conformational change is independent of box width when the polymers rarely encounter\nthe confining walls.  However, as $L$ decreases, the behaviours of the two systems diverge.\nFor the 2-polymer system the $\\tau_1$ exhibits a peak at at $L\/R_{{\\rm g},xy}^*\\approx 4$, \nfollowed by a sharp decrease at lower $L$.  By contrast, the 1-polymer system has no maximum \nin this region, and instead $\\tau_1$ exhibits solely a sharp decrease at \n$L\/R_{{\\rm g},xy}^*\\lesssim 5$.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\vspace*{0.2in}\n\\includegraphics[width=0.45\\textwidth]{fig8.pdf}\n\\end{center}\n\\caption{(a) Variation of scaled Rouse mode correlation time $\\tau_1\/\\tau_{\\rm slit}$ with box \nwidth, $L$, where $\\tau_{\\rm slit}$ is the value of $\\tau_1$ for slit confinement. Results are shown\nfor $N$=60 for 1- and 2-polymer systems. The dashed line corresponds to $\\tau_1=\\tau_{\\rm slit}$.  \n(b) Variation of $\\tau_p$ with mode index $p$ for a 2-polymer system with $N$=60. Results are shown \nfor various box widths. The black curve shows a fit to the results for slit-confinement ($L=\\infty$) \nto a power law.  The other curves are guides for the eye for $L$=6, 8, and 13. }\n\\label{fig:rouse}\n\\end{figure}\n\nThe effect of confinement on Rouse mode relaxation has been previously examined \nanalytically using a Gaussian chain subject to a harmonic confining potential.\\cite{denissov2002segment}\nIn each dimension subject to this confinement, it was shown that $C_p(t)$ decays exponentially\nwith a relaxation time\n\\begin{eqnarray}\n1\/\\tau_p = 1\/\\tau_p^{(0)} + 1\/\\tau_{\\rm e},\n\\label{eq:taupinv}\n\\end{eqnarray}\nwhere $\\tau_p^{(0)}\\propto (N\/p)^2$ is the relaxation time for an unconfined chain,\nand $\\tau_{\\rm e}$ is a constant proportional to $\\tilde{d}^2$, where $\\tilde{d}$ is the\neffective confinement width. Thus, in this model $\\tau_p$ decreases as the confinement\nincreases, qualitatively consistent with the present results for $\\tau_1$ for a single \nself-avoiding chain laterally confined between hard walls. The presence of a second \npolymer chain effects an increase in $\\tau_1$ over the range of $L$ where the two polymers\nare forced to be in contact with each other, including lower values of $L$ where \n$\\tau_1<\\tau_{\\rm slit}$. This is likely an effect of polymer crowding impeding conformational \nmotion on large length scales. The competition between the effects of confinement, which\ntends to decrease $\\tau_1$, and crowding, which tends to increase $\\tau_1$, leads to\nthe local maximum. \n\nFigure~\\ref{fig:rouse}(b) shows the variation of the Rouse mode correlation times, $\\tau_p$,\nas a function of mode index $p$ for the 2-polymer system with $N$=60. Results are shown for various\nbox widths for mode indices in the range $p=1-7$. In all cases $\\tau_p$ decreases monotonically\nwith $p$.  For confinement to a slit, i.e. $L=\\infty$, the results exhibit power-law behaviour\nwith $\\tau_p \\propto p^{-2.66}$. This is close to the expected Rouse scaling for a 2-D polymer\nin good solvent conditions: $\\tau_p \\propto p^{-(1+2\\nu)} = p^{-2.5}$, where $\\nu=0.75$ is the Flory\nscaling exponent for two dimensions.\\cite{deGennes_book} The discrepancy is likely a result\nof finite-size effects. The relaxation times $\\tau_p$ are independent of $L$ for $L \\gtrsim 20$.\nFor $L\\lesssim 20$ ($L\/R_{{\\rm g},xy}^*\\lesssim 3.5$), $\\tau_p$ increases with decreasing box width \nfor all $p$ except for the case of $p=1$ at very small $L$, as noted above. Thus, in the confinement \nregime where 2-polymer crowding effects become noticeable, the conformational dynamics appear to \nbe slowed over all length scales. Finally, we speculate that $\\tau_p<\\tau_p^{\\rm (slit)}$ for \n$p\\geq 2$ for smaller box widths than those examined in our simulations. This follows from the \nfact that Eq.~(\\ref{eq:taupinv}) suggests that as $p$ increases the second term in \nEq.~(\\ref{eq:taupinv}) can remain appreciable relative to the first term if the confinement \ndimension $\\tilde{d}$ (analogous to $L$) is reduced. \n\nTo summarize, we find that the dynamics of the confined-polymer system are significantly affected \nby size of the confining cavity for sufficiently small $L$.  The centre-of-mass motion and the \nconformational dynamics are both influenced by a combination of confinement effects that are \nalso present for a single polymer, as well as crowding effects arising from interactions between\nthe two polymers. These effects become pronounced precisely in the regime where the centre-of-mass \nprobability distributions, the average size and shape of individual polymers, and the degree of \npolymer overlap are also strongly altered.\n\n\\section{Relevance to experiment}\n\\label{sec:experiment}\n\nAs noted in the introduction, this study is principally motivated by the recent work of Capaldi \n{\\it et al.},\\cite{capaldi2018probing} who used optical imaging methods to probe the organization \nand dynamics of DNA molecules trapped in nanofluidic cavities. Consequently, it is of value to \nexamine the relevance of the results of the present study to those experiments. \n\nIt is important to first note how the choice of model limits direct quantitative comparison. \nIn their experiments using fluorescently stained $\\lambda$ DNA, Capaldi {\\it et al.} used a solution \ncontaining 10 mM tris at pH 8 with 2\\% BME, corresponding to an ionic strength of about $I=$12~mM. \nUsing the empirical relation between the Kuhn length, $l_{\\rm K}$, and ionic strength from \nDobrynin\\cite{dobrynin2006effect} we estimate $l_{\\rm K}\\approx 100$~nm. Likewise,\nusing the theory of Stigter\\cite{stigter1977interactions} for the relation between the $I$ and\nthe effective chain width, $w$, we estimate $w\\approx 10$~nm, and thus the monomer anisotropy ratio\nis $l_{\\rm K}\/w\\approx 10$. For the staining ratio of 10:1 (bp:fluorophore) employed, the Kuhn \nlength is not expected to change.\\cite{kundukad2014effect} On the other hand, the contour length \nis expected to increase due to unwinding of the double helix. From Fig.~7 of \nRef.~\\onlinecite{kundukad2014effect} we estimate that a 10:1 YOYO-1 staining ratio leads to\na contour length increase of about 15\\%. As unstained $\\lambda$ DNA has a contour length\nof $L_{\\rm c}=16.5$~$\\mu$m, the value for stained DNA is expected to be $L_{\\rm c}\\approx 19$~$\\mu$m.\nThus, we estimate a ratio of $L_{\\rm c}\/l_{\\rm K}\\approx 190$.  The simulation model employed a \nflexible chain of spherical beads. If the bead width (which is approximately the mean bond \nlength) represents one Kuhn length, then the model polymers should have a length of $N$=190 to \nachieve the correct $L_{\\rm c}\/l_{\\rm K}$ ratio. This is larger by a factor of 2.4--5 than the \npolymer lengths ($N$=40--80) used in most simulations, with the exception of some calculations \nfor $N$=300 carried out in MC simulations. More problematic is the ratio of $l_{\\rm K}\/w=1$ \nin the simulation model.  Finally, the ratio of the bulk radius gyration ($\\approx \n700$~nm\\cite{lin2011partial}) to the nanofluidic cavity depth (200~nm) is $R_{\\rm g}\/h\\approx \n3.5$. By contrast, the ratio for the simulation model using $N$=60 and $h$=4 is $R_{\\rm g}\/h\\approx \n1.35$. Thus, confinement in the narrow dimension of the cavity deforms $\\lambda$-DNA significantly \nmore than is the case in the model system. Increasing the ratio by decreasing $h$ in the simulation \nleads to small ratios of $h\/w$ that result in unacceptable artifacts.  Choosing model parameter \nvalues to better match the other length scale ratios to those in the experiments\nleads to simulations that require infeasibly long simulation times, especially in the case of \nBrownian dynamics simulations. Consequently, we must use a model system for which we can\nexpect only qualitative or, at best, semi-quantitative agreement between experiment and simulation.\n\nThe simulations examined cavities with a wide range of width values. By contrast the nanocavities\nemployed in the experiments had a single fixed width of 2~$\\mu$m. The ratio of the in-plane \nradius of gyration (0.91~$\\mu$m) for a slit confined $\\lambda$-DNA molecule and box length \nwas $L\/R_{{\\rm g},xy}^*\\approx 2.2$. Imposing this ratio on the simulation model with $N$=60 \n(for which $R_{{\\rm g},xy}^*\\approx 5.7$) implies a box width of $L\\approx 13$.  \nFigure~\\ref{fig:prob2d_N60} shows that a single molecule confined to a box of this \nwidth is expected to have its centre of mass be strongly localized to the centre of the box. \nThis is qualitatively consistent with the experimental results presented in Fig.~4(a) and 4(c) \nof Ref.~\\onlinecite{capaldi2018probing}. The wider, more square-like distribution for \nthe larger box sizes in Fig.~\\ref{fig:prob2d_N60}(a) better resemble the\nmeasured position distribution for confinement of a single plasmid shown in Fig.~5(a)\nof Ref.~\\onlinecite{capaldi2018probing}. Since the plasmid contour length was considerably \nshorter than that of the $\\lambda$ DNA chain and was of circular topology, its average \nsize was also much smaller. This naturally results in a distribution better approximated \nby a model system using a larger $L\/R_{{\\rm g},xy}^*$ ratio.\n\nIn the case of two confined polymers, the results for \n$L$=13 in Fig.~\\ref{fig:prob2d_N60} suggest that crowding effects cause the polymer centres \nto be pushed out from the centre to opposite halves of the box at four quasi-discrete \nlocations and leaving a probability ``hole'' at the box centre. The corresponding experimental\nresults of Fig.~4(d) of Ref.~\\onlinecite{capaldi2018probing} do indicate a crowding-induced\ndisplacement of the $\\lambda$-DNA molecules from the box centre. However, in that case\nan asymmetry was noted, likely a result of an underlying difference in the DNA contour and \npersistence lengths caused by using different stains, as required for separate observation of \neach molecule. Specifically, the YOYO-3 labeled DNA chain was slightly more concentrated in \nthe cavity centre than the YOYO-1 labeled chain. In addition, the distribution of sampled\ncentre-of-mass positions shown in Fig.~4(b) of Ref.~\\onlinecite{capaldi2018probing}\ndo not show any evidence of the quasi-discrete states, {which likely\narise from enhanced packing effects due to the small value of $l_{\\rm K}\/w$ ratio in\nthe model.} Instead, those results are more\nqualitatively consistent with our simulation results for a slightly larger box width ($L$=22),\nwhere ${\\cal P}_1$ tends to be dependent on radial distance from the box centre alone,\nindependent of the polar angle, and where the probability hole at the centre is less pronounced. \nSuch a distribution is also consistent with a collective Brownian rotation of the two molecules \naround the centre of the box, a behaviour noted in Ref.~\\onlinecite{capaldi2018probing}.\n\nIn addition to measurement of the time-dependence of the MSD, Capaldi {\\it et al.} also measured \nthe position autocorrelation function, $C_{\\rm auto}(t)$, and observed exponential decay for both\n1- and 2-polymer systems. They found a time constant of $\\tau=0.25\\pm 0.01$~s for 1-chain confinement\nand $\\tau=2.0\\pm 0.1$~s for 2-chain confinement. Thus, the crowding effect caused by the presence\nof the second chain increased $\\tau$ by a factor of 8.  It is easy to show that an exponential \ndecay of the correlation function implies an exponential decay of the MSD with exactly the same \ntime constant. Figure~\\ref{fig:tau.N.Le} shows that the values of the time constant extracted \nfrom fits to the MSD diverge for the two different systems as $L_{\\rm e}$ decreases. This\ndivergence begins at an effective box length of $L_{\\rm e}\\approx 20$.\nAs noted above, the cavity width in the experiment satisfies $L\/R_{{\\rm g},xy}^*\\approx 2.2$.\nFor the model system, Fig.~\\ref{fig:tau.N.Le}(a) shows that this ratio corresponds to the regime\nwhere the presence of a second polymer increases the time constant relative to the\ncase for single-polymer confinement. Using the data in this figure we estimate an increase\nin the time constant by a factor of 21 for $N$=60. {While this is larger than \nthe experimental value, the model does correctly predict an increase in $\\tau$ by about an\norder of magnitude for the two-polymer system. Given the simplicity of the molecular model,\nthis is a reasonably good agreement.}\n\nA final point of comparison with the experiments is the relationship between the center-of-mass\nauto- and cross-correlation functions. Capaldi {\\it et al.} reported exponential decay of\nboth $C_{\\rm auto}(t)$ and $C_{\\rm cross}(t)$ with time constants of $2.0\\pm 0.1$~s\nand $2.8\\pm 0.3$~s, respectively. This stands in contrast to the simulation results\nin which exponential decay only occurs at longer time, where the two functions converge\nand thus are characterized by the same time constant. We suspect that this discrepancy\narises from the way in which the fit to the data was carried out in Ref.~\\onlinecite{capaldi2018probing}.\nThe cross-correlation function in Fig.~7(c) of Ref.~\\onlinecite{capaldi2018probing} \nappears to show a flattening of the curve at low $t$.  This trend is consistent with our \nsimulation results, suggesting that it has a physical origin and is not merely a statistical\nanomaly. Discarding the low-$t$ experimental data will likely increase the measured rate of decay \nand thus decrease the time constant. We speculate that such a modified fit could lead to a time\nconstant that better matches that for the autocorrelation function.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this study we have used MC and Brownian dynamics simulations to probe the organization,\nconformational behaviour, and equilibrium dynamics of two polymers under confinement in\na box-like cavity with very strong confinement in one of the dimensions.  {We find} \nthe behaviour {is} highly dependent on the degree of lateral confinement. For large\nbox width, $L$, where the polymers rarely interact with each other or the lateral walls,\nthe polymer conformational statistics and dynamics are comparable to those for \na single slit-confined polymer. The combined effects of confinement and inter-polymer\ncrowding are noticeable when $L\/R_{{\\rm g},xy}^*\\lesssim 5$, where $R_{{\\rm g},xy}^*$\nis the transverse radius of gyration for the case $L\\rightarrow\\infty$ (i.e., confinement\nto a slit with a spacing equal to the height of the confining cavity).\nIn this box size regime, there is a probability hole at the box centre, and the polymer \ncentre-of-mass positions tend to be inversely correlated with respect to the box centre \n(i.e. when one polymer is at position $(x,y)$ the other tends to be at $(-x,-y)$, where\nthe box centre is at $(0,0)$). \nAt lower $L$, the polymers tend to occupy four quasi-discrete states in opposite\nsides of the box, and at very small $L$ there is an increasing tendency for polymer overlap\nat the box centre. The polymer size decreases with $L$ in this regime, principally as a\nconsequence of confinement rather than interpolymer crowding. Increasing confinement tends\nto reduce the 2D asphericity, though the interpolymer crowding tends to offset this effect.\nBoth the rate of diffusion and the internal dynamics of each polymer is significantly\nimpacted by the presence of the other polymer. {Note that the transition\nlocation of $L\/R_{{\\rm g},xy}^*\\approx 5$ is likely specific to the model employed in\nthis study. For other models (e.g. one using semiflexible chains) the transition\nwill likely occur at a somewhat different location, though still with $L\/R_{{\\rm g},xy}^*$\nsomewhat greater than unity.}\n\nThe simple molecular model employed in this study facilitates the study of a number of \ngeneric effects of confinement on the organization and dynamics of two cavity-confined polymers.\nGenerally, the observed behaviour is qualitatively consistent with the results of the recent \nexperimental study by Capaldi {\\it et al.}, which examined two $\\lambda$-DNA chains confined \nto a nanofluidic cavity.\\cite{capaldi2018probing} We view the present study as a first step \ntoward a more realistic modeling of such experimental systems. In principle, scaling up the \npolymer length and incorporating bending rigidity into the model can be used to obtain correct \nlengthscale ratios for the contour length, persistence length, effective width of $\\lambda$ DNA,\nand the cavity dimensions. In practice, however, the required simulation times for such a model \nare infeasible at present, at least for the dynamics. A promising alternative approach could be \nto model each polymer as a chain of blobs with diameter equal to the box height, each interacting \nwith other blobs via a soft repulsive potential arising from entropic repulsion. This potential \ncould be determined using a technique to measure free energy functions employed in other recent\nstudies.\\cite{polson2014polymer,polson2018segregation,polson2021free}\nSuch effective potentials were recently employed to model the entropic repulsion of side-loops\nin a model for a bacterial chromosome.\\cite{wu2019cell,swain2019confinement}\nWe anticipate that such future studies in addition to the present one will be helpful \nin elucidating recent experimental results as well motivating new nanofluidics experiments for \ncavity-confined DNA systems.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Evolution during the Asymptotic Giant Branch (AGB) phase}\n\nStars in the mass range 1-8 M$_\\odot$ eventually evolve up the asymptotic giant\nbranch (AGB) where they begin to lose mass and form circumstellar shells of \ndust and gas. During their ascent of the AGB, mass-loss rates are expected \nto increase with the luminosity of the star, \nresulting in progressive optical thickening of circumstellar shells. \nFurthermore,  the chemistry of these stars' atmospheres, and thus of their \ncircumstellar shells, changes as a result of dredge-up of newly \nformed carbon from the He-burning shell. Due to the extreme ease of formation \nand stability of CO molecules, the chemistry in circumstellar shells is \ncontrolled by the C\/O ratio. If C\/O $<$ 1 all the carbon is trapped in CO, \nleaving oxygen to dominate the chemistry. Conversely, if C\/O $>$ 1 all the \noxygen is trapped in CO, and carbon dominates the chemistry. Stars start their \nlives with cosmic C\/O ratios ($\\approx$0.4), and are thus oxygen-rich. In some \nAGB stars, the dredge-up of newly formed carbon is efficient enough to raise \nthe  C\/O above unity, and these stars are known as carbon stars. They are \nexpected to have circumstellar shells dominated by amorphous or graphitic \ncarbon grains, although other dust grains are also important (e.g.\\ \nsilicon carbide; SiC). \n\nAs the mass-loss rate increases with the evolution of the AGB stars, the dust \nshells get thicker (both optically and geometrically) \nand these stars eventually become invisible at optical \nwavelengths and very bright in the infrared (IR). Such stars are known as \n``extreme carbon stars''  \\citep{volk92,volk00}. \nAt this stage, intense mass loss depletes the remaining hydrogen in the \nstar's outer envelope, and terminates the AGB. Up to this point the star has \nbeen making and dredging-up carbon, thus these shells have even more carbon \navailable for dust production than optically bright (early) carbon stars \n(i.e.\\ the Si\/C ratio decreases with the evolution of these stars).\nThe rapid depletion of material from the outer envelope of the star means that \nthis extremely high mass-loss phase must have a relatively short duration \n\\citep[a few $\\times 10^4$ years;][]{volk00}.\n\n\n \n\\subsection{Post-AGB evolution}\n\nOnce the AGB star has exhausted its outer envelope, the AGB phase ends. \nDuring this phase the mass loss virtually stops, and the circumstellar shell \nbegins \nto drift away from the star. At the same time, the central star begins to \nshrink and heat up from 3000 K until it is hot enough to ionize the \nsurrounding gas, at which point the object becomes a planetary nebula (PN). \nThe short-lived post-AGB phase, as the star evolves toward to the PN phase, is \nalso known as the proto-planetary nebula (PPN) phase. \nThe detached dust shell drifts away from the central star \ncausing a PPN to have cool infrared colors from its cooling dust shell.\nMeanwhile, the optical depth of the dust shell decreases as it expands \nallowing the central star to be seen and making such objects optically bright.\nThe effect of decreasing optical depth and cooling dust temperatures changes \nthe spectrum of the circumstellar envelope, revealing features that were \nhidden during the AGB phase. \n\n\n\\subsection{Observing dust around carbon stars}\n\nCircumstellar shells of carbon stars are expected to be dominated by \namorphous or graphitic carbon grains. These dust species do not have \ndiagnostic infrared features and merely contribute to the dust continuum \nemission. However, silicon carbide (SiC) does exhibit a strong infrared \nfeature, which can be exploited when studying carbon stars.\n\nIn the context of stardust, SiC has been of great interest since \nits formation was predicted by condensation models for carbon-rich \ncircumstellar regions \\citep[e.g.][]{fried69,gilman69}. \\citet{gilra71,gilra72}\npredicted that SiC should re-emit absorbed radiation as a feature in the \n10--13$\\mu$m region. Following these predictions, a broad infrared (IR) \nemission feature at $\\sim$11.4$\\mu$m that is observed in the spectra of many \ncarbon stars has been attributed to solid SiC particles\n\\citep[e.g.][]{hack72,treff74}. Indeed, SiC is now \nbelieved to be a significant \nconstituent of the dust around carbon stars. Although \\citet{goebel95} and \n\\citet{clem03} adopted the name 11+$\\mu$m feature to reflect the \nvariation in observed peak positions in astronomical spectra, this implies \nthat the the features always occur longward of 11$\\mu$m. We have adopted \n$\\sim$11$\\mu$m to reflect the variations in peak positions of this feature.\n\n\nThe effect of the evolution of the density of the dust shell on observed \nfeatures, and particularly on the $\\sim$11$\\mu$m feature, have been discussed \nextensively \n\\citep[e.g.][see \n\\S~\\ref{LRS}]{cohen84,baron87,willem88,ck90,goebel95,speck97,sloan98}. \nNot all these previous works agree.\nAll studies concur that the increasing optical depth leads to decreasing color \ntemperatures from the stars as the stellar photosphere becomes hidden from \nview, and the dust from which we receive light becomes progressively cooler. \nAt the same time, the $\\sim$11$\\mu$m feature tends to become weaker (relative \nto the continuum) and flatter topped (less sharp peaked) and possibly broader. \nVarious explanations of this behavior have been proposed, but none are \nentirely satisfactory. Initially the diminution of the \n$\\sim$11$\\mu$m feature was attributed to coating of the grains by amorphous \ncarbon \n\\citep[e.g.][]{baron87,ck90}. However, meteoritic data does not support\nthis hypothesis (see \\S~\\ref{meteor}). Moreover, more recent studies of the \n$\\sim$11$\\mu$m \nfeature have shown that it is consistent with SiC self-absorption, i.e. \nabsorption by cooler SiC particles, located in the outer part of the dust \nshell, where they can absorb the SiC emission feature produced by warmer SiC \ncloser to the central star (Speck et al. 1997). Indeed, Speck et al. (1997) \nshowed that every star in their sample whose underlying continuum temperature \n(T$_{col}$) $<$ 1200~K  was best fitted by self-absorbed SiC, whether the \n$\\sim$11$\\mu$m feature was in net emission or net absorption. \nAs the dust shell reaches extreme optical depths, the $\\sim$11$\\mu$m feature \nwill \neventually be seen in net absorption. However, these absorption features are \nrare and have mostly been ignored in discussions of \nevolutionary sequences in carbon star spectra.\n\n\n\\subsection{Meteoritic Evidence \\label{meteor}}\nSilicon carbide is an important presolar grain found in meteorites. \nThe isotopic compositions of presolar grains indicate that they originated \noutside the solar system. Presolar SiC was first discovered by \\citet{bern87} \nand since then has been the focus of a great deal of laboratory work \n\\citep[see][and references therein]{bern05}. \nThe most important findings of this work are:\n(1) most of the SiC presolar grains were formed around carbon stars; \n(2) nearly all ($\\sim$90\\%) are of the cubic \n$\\beta$-polytype\\footnote{Silicon carbide can form into numerous different \ncrystal structures, known as polytypes. $\\beta$-SiC is the cubic polytype, and \nis now believed to be the dominant form of SiC forming around carbon stars; \nsee \\S~\\ref{meteor}, and \\citet{speck97,speck99,daulton03}}; \n(3) with one exception, SiC grains are never found in the cores of carbon \npresolar grains (unlike other carbides - TiC, ZrC, MoC); \n(4) the grain-size distribution is large (1.5nm to 26$\\mu$m), \nwith most grains in the 0.1 -- 1$\\mu$m range, but with single-crystal grains \ncan exceed 20$\\mu$m in size.\n\nObservations of the $\\sim$11$\\mu$m feature have been compared with \nlaboratory spectra of various forms of SiC and after some false starts, has \nnow been attributed to $\\beta$-SiC, matching the information retrieved from \nmeteoritic samples \\citep{speck99,clem03}. However, there are still some \ndiscrepancies between observational and meteoritic evidence\n(most notably related to grain size). \nStudies of meteoritic SiC grains can aid our understanding of their evolution.\n\n\n\\subsection{Purpose}\n\nThe goal of this paper is to extract the evolution of the dust from the\ninfrared spectra of carbon-rich AGB stars (both optically \nbright and ``extreme'') and post-AGB stars. In \\S~\\ref{lab} \nwe present new \ninfrared spectra of amorphous silicon carbide which may explain some of the \nfeatures seen in the carbon star spectral sequences.\nIn \\S~\\ref{spec} we review studies of the infrared spectra of \ncarbon stars which demonstrate the changes in spectral features associated \nwith stellar evolution. We also present new detections of $\\sim$11$\\mu$m \nabsorption features in extreme carbon stars, which are essential to our \nunderstanding of the dust grain evolution. \nIn \\S~\\ref{selfabs} \nwe present a\ndust evolution sequence to explain the spectral changes in carbon stars.\nThe summary and conclusions are in \\S~\\ref{conc}.\n\n\n\n\\section{New spectra of amorphous silicon carbide \\label{lab}}\n\\subsection{Experimental samples and techniques}\n\n\n\nSuperior Graphite donated the bulk $\\beta$-SiC. The purity is 99.8\\%. Bulk \n$\\alpha$-SiC (purity = 99.8\\%) in the 6H polytype was purchased from Alfa\/Aesar\n(Lot \\# c19h06).  Grain size varied from ~1 to 25 $\\mu$m. Nanocrystals of 2-5 \nnm size produced by gas-phase combustion \\citep{axel96} best matches \nthe X-ray diffraction pattern of $\\beta$-SiC but contains some 6H as well \n\\citep[for details see][]{hofm00}. \nThis sample may also contain SiC in the diamond structure, as \nsynthesized by \\citet{kimura03} and discussed further below. We \ndesignate this sample as ``nano $\\beta+\\alpha$''. These three samples were \nstudied by \\citet{speck99} and \\citet{speck04}. Nanocrystals were \nalso purchased from Nanostructured and Amorphous Materials, Inc.  \nAmorphous SiC of 97.5\\% purity consists of 10 $\\times$ 100 nm$^2$ laths \n(stock \\# 4630js).  Nano-$\\beta$ of 97\\% purity consists of 20 nm particles \n(stock \\# 4640ke). Diamond of 95\\% purity consists of 3 nm particles \n(stock \\# 1302jgy). Most SiC samples were dark grey, indicative of excess C or \nSi, e.g., inclusions of these elements. Nano-diamond is light grey, presumably \ndue to graphite impurities. The $\\alpha$-SiC is pale grey to amber and has \nlittle in the way of impurities.\n\t\nMid-IR spectra were obtained from $\\sim$450 to 4000 cm$^{-1}$ \n(2.2 to 25$\\mu$m) at 2 cm$^{-1}$ \nresolution using a liquid-nitrogen cooled HgCdTe detector, a KBr beamsplitter \nand an evacuated Bomem DA 3.02 Fourier transform interferometer. Thin films \nwere created through compression in a diamond anvil cell (DAC) which was used \nas a sample holder and for the reference spectrum. Efforts were made to cover \nthe entire diamond tip (0.6 mm diameter) with an even layer of sample, but \nslight irregularities in the thickness were inevitable.  Far-IR data were \nobtained from powder dispersed in petroleum jelly on a polyethelene card from  \n$\\sim$150 to 650 cm$^{-1}$ ($\\sim$15 to 67 $\\mu$m) using a \nDTGS\\footnote{deuterated triglycine sulfate} detector and \nmylar beamsplitter. A \nspectrum of nano-diamond (which should be featureless) was subtracted to \nremove the effect of scattering. For the bulk and nano $\\beta + \\alpha$ \nsample, a thin film \nwas made in a DAC and a helium bolometer served as the detector.  Far- and \nmid-IR spectra were scaled to match in the region of overlap and merged.  \nFor procedural details see \\citet{speck99} or \\citet{hofm03}.\n\n\\subsection{Laboratory Results}\n\nThe two nanocrystalline samples have spectra that closely resemble each other,\nwhereas the spectrum of the amorphous sample is most similar to that of bulk \n$\\beta$-SiC (Fig.~\\ref{labdata}).  \nAll samples have three peak complexes centered near \n9, 11, and 21$\\mu$m. The number and position of the peaks constituting each \ncomplex vary, and the relative intensities of the complexes vary among the 4 \nsamples (Table 1).   Bulk $\\alpha$-SiC shows only the 11 $\\mu$m complex.  \nThe shoulders in the spectrum shown are due to interference fringes as the \novertone-combination \npeaks in this area are much weaker \\citep[e.g.][]{hofm00}.  \nAs it is clear from Fig.~\\ref{labdata} that the nano \nsamples have spectra unlike that of $\\alpha$-SiC (and $\\alpha$-SiC is absent \nfrom presolar grains; see \\S~\\ref{meteor}), the remainder of the \ndiscussion concerns the amorphous and $\\beta$-SiC samples.\n\nSome of the spectral differences are due to grain size. \nFrom Fig.~\\ref{labthick}\n\\citep[and Fig.\\ 2 in ][]{hofm00}, as the sample thickens, the LO \n(longitudinal optic) shoulder increases in intensity relative to the main peak \n(TO, transverse optic).  This occurs because of light leakage \n\\citep[see][for a detailed explanation]{hofm03}.  The nanocrystalline samples \nproduce \nspectra close to the expected intrinsic profile \\citep[see][]{spitz} \nbecause their very fine grain sizes aid production of a thin, uniform film \nthat covers the diamond tip. The rounded profiles for $\\beta$- and the \namorphous SiC are attributed to production of less perfect films from these \nmuch larger particle sizes. The difference in relative intensity of the 11 and \n9 $\\mu$m peaks between the two spectra acquired for amorphous SiC may be due \nto different film thicknesses and amounts of light leakage as well.  However, \nthe relative intensity of this pair varied among the spectrum of the \n$\\beta$-SiC sample, but was consistent for nano $\\beta+\\alpha$ \n\\citep[Fig. 2 from ][]{hofm00}, \nsuggesting an impurity as the origin.\n\nAs observed by \\citet{speck04} the relative intensity of the 21 and\n11 $\\mu$m peaks varied among the samples studied.  This observation is \nsupported by the two additional samples studied here, and, intensities\nof the 21 and 9$\\mu$m peaks are roughly correlated (Table 1). \nA feature similar to the 9$\\mu$m  peak has been observed in nanocrystals\nwith varying proportions of Si and C by Kimura and colleagues.  Carbon\nfilms that contain about 30\\% Si made by evaporation have a peak at \n9.5$\\mu$m \\citep{kimura03}.  From electron diffraction, the structure\nof the films is nano-diamond and small amounts of $\\beta$-SiC are also \npresent \\citep{kimura03}. Pure diamond structure should not \nhave an IR peak, but impurity bands are well known, such as those due to \nnitrogen. Apparently, the Si-C stretch in the diamond structure differs from \nthat in the derivative structure of $\\beta$-SiC.  \nHigher frequency is consistent \nwith the lattice constant being smaller for diamond than for SiC.  \n\t\n\nIon-sputtered carbon films with proportions of 10, 30, and 50\\% Si\nsimilarly contain particles of nano-diamond and  IR absorption bands at 9.5\nand 21$\\mu$m.  The 30\\% film has the most intense 9.5$\\mu$m peak, whereas the \n50\\% Si film contains nano-crystals of $\\beta$-SiC in addition to the\nsolid-solution nano-diamonds and weak peaks at 11.3 and 12.3$\\mu$m\n\\citep{kimura05a}.\n\nNano-particles produced by radio-frequency plasma \\citep{kimura05b}\nabsorb at 8.2 (shoulder), 9.2, a doublet at 11.2 and 11.7, and a\nmoderately intense band at 21$\\mu$m.  The structure is nano-diamond.\n$\\beta$-SiC peaks were rarely seen in the electron diffraction \nresults, suggesting that the IR\nbands near 11$\\mu$m result from short-range order \\citep{kimura05b}.\nThese authors attribute the 21$\\mu$m band to excess carbon which may be\npresent in interstitial sites in the solid-solution nano-diamond particles. \nThe present data corroborate and augment their observations.  \nBulk $\\beta$-SiC, nanocrystalline material, and \namorphous nanosamples all behave similarly. \nIn the amorphous sample the structural control is lost so that the sample\nhas places where the $\\beta$-SiC structure occurs over nearest neighbors, and\nplaces where the nano-diamond structure occurs. These are about in equal\nproportions.  Then there are local sites of excess C giving the 21$\\mu$m\nband.\nThat the 21 and 9$\\mu$m peaks are strongest in the amorphous material \ncorroborates assignment of these peaks to excess C in SiC, one of the \npossibilities proposed by \\citet{speck04}.\n\t\nTherefore, the 9$\\mu$m peak is indicative of Si-C locally in a diamond \nstructure and the 11$\\mu$m of Si--C locally in an SiC polytype. \nThese peaks are found in amorphous SiC (observed here) and\nnano-diamond crystals \\citep{kimura03,kimura05a,kimura05b}.\nThe ratios of \nthese peaks are not correlated, but depend on the Si\/C ratio of the material. \nEither amorphous SiC or nanocrystalline diamond with a high\nproportion of Si substituting for C are good candidates for the carrier of the \n9$\\mu$m (see \\S~\\ref{LRS}) and 21$\\mu$m (\\S~\\ref{21um}) features\n\n\\subsection{Comparison with astronomical environments}\n\nFig.~\\ref{ycvn} compares the amorphous \nSiC spectrum with the observed,\ndouble-peaked emission feature seen in the spectrum of Y~CVn \n\\citep[original published in][]{speck97}.\nAmorphous SiC provides peaks at the same positions as those observed in\nthe  double peaked feature, but not in the same intensity ratio.\n\nThe shorter wavelength peak is due to excess carbon forming diamond-like \nstructures in the amorphous or nanocrystalline SiC grains. \nThe relative strengths of the two peaks in the \nlaboratory spectra are controlled by the concentration of excess carbon in the \ngrains. Therefore we can manipulate the composition of the grains to obtain \nbigger or smaller short wavelength peaks, relative to the longer wavelength \nSi--C peak. Furthermore, there may be a contribution to the Si--C part of the \nspectrum from $\\beta$-SiC also present in the circumstellar shells, further \nenhancing the longer wavelength peak relative to the shorter one.\n Fig.~\\ref{mix} shows the effect of mixing amorphous and crystalline SiC to \nvarying degrees.\n\nComparing Figures~\\ref{mix} and \\ref{newabsobs} shows that the \ndouble-peaked amorphous SiC spectrum, in combination with $\\beta$-SiC can \nexplain the variety of absorption features observed in the spectra of extreme \ncarbon stars (see also \\S~\\ref{sicabs}). \nThe differences in peak positions between the laboratory spectra and \nthe observed astronomical absorption features are due to the effect of \nself-absorption and grain size, which is discussed in \\S~\\ref{selfabs}.\n\n\n\\section{Spectroscopic studies of carbon stars \\label{spec}}\n\n\\subsection{{\\it IRAS LRS} studies of carbon stars \\label{LRS}}\n\n\nThe spectra of carbon stars change with the evolution of the star.\nSeveral studies exist of the evolution of the $\\sim$11$\\mu$m feature in carbon \nstar spectra based on {\\it IRAS LRS} data, but are somewhat contradictory.\nThe majority of carbon stars exhibit the $\\sim$11$\\mu$m feature in emission.\nWith the exception of \n\\citet{speck97}, all attempts to understand the sequence of spectral \nfeatures fail to include the SiC absorption feature. \nHere is the summary of the accepted evolutionary trends:\nEarly in the carbon star phase, when the mass-loss rate is low and the shell \nis optically thin, the $\\sim$11$\\mu$m SiC emission feature is strong, narrow \nand sharp. As \nthe mass loss increases and the shell becomes optically thicker, the SiC \nemission feature broadens, flattens and weakens. \nFinally, once the mass-loss rate is \nextremely high and the shell is extremely optically thick, the SiC feature \nappears in absorption.\nFig.~\\ref{cgs3} shows spectra from \\citet{speck97} which demonstrate these \ntrends. \nOnce the AGB phase ends, the dust thins and cools and we begin to see new \nfeatures that are indicative of the dust in the extreme carbon star phase.\n\n\n\\citet{lml86} found that the majority of $\\sim$11$\\mu$m emission \nfeatures peak at \n11.15$\\mu$m, with only 4\\% peaking longward of 11.6$\\mu$m.\n\\citet{baron87} found that, as the continuum temperature decreases and the \npeak-continuum strength of the $\\sim$11$\\mu$m emission feature diminishes, \nthe peak position \ntends to move to longer wavelengths (from $\\sim$11.3 to $\\sim$11.7$\\mu$m). \nThis trend is demonstrated in Fig.~\\ref{cgs3}.\n\\citet{willem88} and \\citet{goebel95} \nfound that the stars with high continuum temperatures tended \nto have the $\\sim$11$\\mu$m emission feature at 11.7$\\mu$m, with the 11.3$\\mu$m \nfeature arising in the spectra of stars with cool continuum temperatures, \nan obvious \ncontradiction to the work of \\citet{baron87}. \nSpeck et al. (1997) used a much smaller sample of ground-based UKIRT\nCGS3 spectra, and found no correlation between the peak position of the \n$\\sim$11$\\mu$m feature and the continuum temperature.\n\n\n\\citet{baron87} and \\citet{goebel95} noticed that the decreasing continuum \ntemperature is accompanied by emergence of a second spectral feature at \n$\\sim9\\mu$m.\n\\citet{sloan98} \nfound an anti-correlation between the $\\sim9\\mu$m feature and the \ndecreasing continuum temperature, which is opposite to what was observed by \nboth \\citet{baron87} and \\citet{goebel95}.\nWith such a diverse set of analyses and interpretations\nof what is essentially the same dataset \\citep[except for ][]{speck97},\nno clear correlation can be discerned between changes in the spectral\nfeatures and evolution of the dust.\nThe differences in interpretation of the {\\it IRAS LRS} data may stem \nfrom differing underestimations of the depth of the molecular \n(HCN and C$_2$H$_2$) absorptions, \nand thus the continuum level \\citep{aoki99}.\n\n\n\n\\subsection{The SiC absorption feature \\label{sicabs}}\n\nThe prototype for extreme carbon stars is AFGL~3068 which has \nan \nabsorption feature at $\\sim 11 \\mu$m, tentatively attributed to absorption by \nSiC \\citep[][]{jones78}. \\citet{speck97} reinvestigated \nAFGL~3068 and confirmed the absorption features. In addition, \\citet{speck97} \ndiscovered three more extreme carbon stars (IRAS 02408+5458, AFGL~2477 and \nAFGL~5625) with $\\sim11\\mu$m absorption features, attributable to SiC. \nTwo of these objects (AFGL~2477 and AFGL~5625) exhibited a double peaked \nabsorption feature, with the \n$\\sim11\\mu$m feature accompanied by a shorter wavelength absorption peak at \n$\\sim9\\mu$m. \\citet{speck97} attributed this shorter wavelength peak to \ninterstellar silicate absorption along the line of sight, although it is \npossible that this feature is related\nto the  $\\sim9\\mu$m emission feature that accompanies the $\\sim$11$\\mu$m \nemission \nfeature in the spectra of some carbon stars.\nIn all four cases, the $\\sim11\\mu$m absorption feature actually ``peaks'' at \n10.8$\\mu$m. \n\nThe absorption features of AFGL~3068 and IRAS 02408+5458 were revisited by \n\\citet{clem03} who showed that these feature, as seen in ISO SWS spectra, \nwere consistent with isolated $\\beta$-SiC nanoparticles. \nHowever they could not fit the \nshort wavelength side of the $\\sim$11$\\mu$m absorption\nfeature using SiC alone, which may indicate that the $\\sim9\\mu$m feature \nabsorption is intrinsic to these stars and its strength varies. This will be \ndiscussed further is \\S~6.\n\nThe double-peaked absorption features of AFGL~2477 and AFGL~5625 were \nrevisited by \\citet{clem05}. This work presented new infrared spectra of \nsilicon nitride (Si$_3$N$_4$) and found a correlation between the observed \ndouble-peaked feature and the laboratory absorption spectrum. Furthermore, \nthey were able to correlate various weaker longer wavelength absorptions in \nthe astronomical spectra, with those of Si$_3$N$_4$ observed in the \nlaboratory.  However, while the \nmatch to the longer wavelength features is good, in their laboratory spectra \nthe \nrelative strength of the $\\sim11$ and $\\sim9\\mu$m features compared to the \nlonger wavelength features indicates that Si$_3$N$_4$ is almost \ncertainly present but cannot be solely responsible for these $\\sim11$ and \n$\\sim9\\mu$m absorption features.\n\n\n\\citet{volk00} presented the ISO spectra of five extreme carbon stars. They \nproduced radiative transfer models, assuming amorphous carbon dust and \nincluding a way to fit the broad 26--30$\\mu$m feature, but without trying to \nfit the absorption features in the 8$-$13$\\mu$m range. \nWe have divided the ISO spectra \nby their model fits, and the resulting spectrum of the 7$-$13$\\mu$m region \nis shown in Fig~\\ref{newabsobs}, together with the continuum-divided \nspectra of extreme \ncarbon stars from \\citet{speck97}. It is clear that IRAS 06582+1507 shows \nthe $\\sim$11$\\mu$m absorption feature seen in AFGL~3068 and IRAS 02408+5458. \nThe spectrum of  IRAS 00210+6221 shows a double\/broad \nfeature similar to that of \nAFGL~5625. IRAS 17534-3030 seems to be intermediate between the two.\nGiven that the extreme carbon star phase is not expected to last more than a \nfew $\\times 10^4$ years, it is not surprising that stars which exhibit these \nfeatures are rare. We now have four stars which exhibit the single \n$\\sim$11$\\mu$m absorption feature, and three which exhibit the broader double \nfeature. It is no longer possible to ignore these absorption features \nwhen trying to understand the evolution of dust around carbon-stars.\n\nThe $\\sim$9$\\mu$m feature\/wing appears to correlate with optical \ndepth, appearing strongest when the $\\sim$11$\\mu$m feature is in absorption, \nbut also exists when the $\\sim$11$\\mu$m feature region weakly emits. \nThis feature may be due to amorphous SiC \nwith excess carbon (see\\S~\\ref{lab}), in which case, the formation of \nsuch grains occurs when the dust shell is denser, further up the AGB. \nThis makes \nsense, in that the high density shells may form dust grains so fast that the \natoms do not have time to migrate to the most energetically favored position \nbefore another atom sticks on. In this way we would expect to form amorphous, \nrather than crystalline grains. If it is due to nanocrystalline grains, this \nfurther supports the hypothesis that grain sizes derease as mass-loss rates \nincrease. Furthermore, early in the life of a carbon star, the \ndust forming regions will have more Si than C for dust formation (the \nmajority of the C atoms will be locked into CO molecules). As the star \nevolves, and more carbon is dredged up from deep within the star, there will \neventually be more C atoms than Si atoms and excess C will get trapped in \nthe grains. \n\n\nThe lack of amorphous SiC in meteoritic samples may be due to the relative \nscarcity of this form of SiC compared to the crystalline polytypes. A typical \nAGB star sends dust out into the interstellar medium \nfor a few hundred thousand years, but \nthe extreme carbon star phase is much shorter lived ($<10^4$ years) and may \nnot occur for all carbon stars (absorption features are rare). Therefore we \nwould expect there to be much more crystalline (mostly $\\beta$-SiC) than \namorphous SiC grains. Furthermore, there may well be a mixture of amorphous \nand $\\beta$-SiC forming in the circumstellar shells of very evolved (extreme) \nAGB stars. Alternatively, the meteoritic data may support the attribution of \nthe 9 and 21$\\mu$m features to nano-crystalline SiC grains with diamond \ninclusions. Nanometer-sized SiC grains have been found in the presolar SiC \nsamples \\citep{bern05}.\n\n\n\\subsection{Carbon-rich post-AGB spectra: the 21$\\mu$m feature \\label{21um}}\n\n\nAmong the C-rich PPNs, approximately half exhibit a feature in their infrared \nspectra at 21$\\mu$m \\citep{omont95}. Subsequent higher resolution data revised \nthe so-called 21$\\mu$m position to 20.1$\\mu$m \\citep{volk99}. PPNs that \ndisplay this feature are all C-rich and all show evidence of {\\it s-process} \nenhancements in \ntheir photospheres, indicative of efficient dredge-up during the ascent of \nAGB \\citep{vwr00}. The 21$\\mu$m feature is rarely seen in the spectra of \neither the PPN precursors, AGB stars, or in their successors, PNs \n\\citep[however, see][]{volk00,hony}. \nThe observed peak \npositions and profile shapes of the 21$\\mu$m feature are remarkably constant \n\\citep{volk99}. This enigmatic feature has been widely discussed since its \ndiscovery \\citep{kwok89} and has been attributed to a variety of both \ntransient molecular and long-lived solid-state species, \nbut most of these species have since been discarded as carriers, \nexcept for HACs\/PAHs \n\\citep{just96,volk99,buss90,grish01} and \nSiC \\citep{speck04}. \nIn the case of \nSiC, it is necessary for the dust grains to be small and contaminated with \ncarbon impurities in order for this feature to appear \n\\citep{kimura05a,kimura05b}. \nFurthermore, the \ncooling and thinning of the dust shell is essential to the emergence of this \nfeature. During the AGB phase, the dust is too warm for this feature to \nappear, but as the dust cools the $\\sim$11$\\mu$m feature of SiC is diminished \nand \nthe 21$\\mu$m feature is promoted by the underlying dust-continuum emission. \nIn this case the changing spectrum reflects the change in temperature and \noptical depth of the dust shell, whereas the spectral features are indicative \nof the last stage of carbon-rich evolution (the extreme carbon star \nphase).\n\n\n\\section{Self Absorption: the effect of changing grain sizes \\label{selfabs}}\n\n\\citet{cohen84} interpreted the change in the appearance of $\\sim$11$\\mu$m \nfeature \nfrom sharp and narrow  to broad and flat-topped as possibly an effect of \nself-absorption. This was supported by \n\\citet{speck97}, who showed that all carbon star spectra in their sample with \ndust continuum temperatures less than 1200~K needed to have self-absorbed SiC \nin order to be fitted well. Self-absorption exhibits some interesting \ncharacteristics that can be used to diagnose the nature of the dust grains \nthat produce observed spectra.\n\nFigure~\\ref{modelthick} shows how the absorption profile changes as particle \nthickness (i.e.\\ grain size) increases.  This appears to be a violation of \nBeer's Law, in that the absorbance is not simply increasing \n(for a single wavelength) with thickness of the particles.\nThis departure occurs because the\nmeasured intensity depends not only on the amount of light that the sample\nabsorbs, but also on the amount reflected at the surface facing the\nsource.  The main peak (the TO mode) becomes saturated in measured spectra\nwhen the actual amount of light transmitted at the TO frequency equals the\nreflectivity. This point is reached at lower thicknesses (grain sizes) for the \nstrong TO mode than for the more weakly\nabsorbing shoulder.  Violations of Beer's law will alter self-absorption in\nastronomical environments but not emission spectra because the latter are not\naffected by surface reflections. Therefore, emission spectra of the grains \nare not so sensitive to the grain size. As long as the grains are still small \ncompared to the wavelength at which they are observed \n(i.e. grains smaller than $\\approx 1\\mu$m for 10$\\mu$m spectra),\nthe pure emission feature will appear at the same wavelength ($\\sim11.3\\mu$m).\nHowever, once the grains are cool enough and the optical depth high enough, \nself-absorption of the 11.3$\\mu$m feature will begin and the profile of the \nfeature will depend on the relative sizes of the grains.\n\n             \nIn computing the effect of Beer's law violations on\nself-absorption, we assume that the following holds: \n(1) the absorbing particles are thicker (larger) than the emitting particles, \n(2) the light received by the absorbing particles is that of the star = \nI$_0$, i.e., the inner cloud of emitting particles is\nrarified enough that it contributes negligible intensity compared to the\nstar, and \n(3) that outer dust transmits more light that the inner dust emits \n(i.e. starlight is also transmitted).\n\n\nBeer's law states that the wavelength-dependent absorbance $a(\\lambda)$ of \na given substance is proportional \nto the mass absorption coefficient $\\kappa_{abs}(\\lambda)$, \nand the thickness through which the light has to pass $d$:\n\n\\[ a(\\lambda) = \\kappa_{abs}(\\lambda) d \\]\n\nDeviations from Beer's law may occur for many reasons. Strictly, Beer's law \napplies to transparent particles \n(i.e.\\ particles which transmit some light at all wavelengths)  \nand therefore, at high concentrations\/opacities may no longer be applicable. \nOther problems include stray light \\citep{machof}.\n\nWhen measuring absorption, we usually measure the transmittivity $T(\\lambda)$ \nof a material and thus obtain the absorptivity $A(\\lambda)$:\n\n\\[ T = \\frac{I_{trans}}{I_{0}} = 1 - \\frac{I_{abs}}{I_{0}} = 1- e^{-a}\\]\n\\[ A = \\frac{I_{abs}}{I_{0}} = e^{-a} \\]\n\nThe light received from a circumstellar shell $I_{dust}$ is given by:\n\n\\[ I_{dust} = I_{emit} + I_{trans} \\]\n\nwhere $I_{emit}$ is the emission from the inner dust and $I_{trans}$ is the \ntransmission from the outer dust.\n\nFrom Kirchhoff's law, the emissivity is the same as the \nabsorptivity, so that $I_{emit} = I_{abs}$\n\n\\noindent\nTherefore,\n\n\\[ \\frac{I_{dust}}{I_0} = f \\frac{I_{abs}}{I_0} + \\frac{I_{trans}}{I_0} \\]\n\\[ \\frac{I_{dust}}{I_0} = f e^{-a} +  1- e^{-a} \\]\n\n\\noindent\nwhere the factor $f$ is varied  from 0 to 1 as the fraction of light\noriginating from emitting dust increases.\n\nThe absorption coefficient \nwill change with grain size $d$, because of the effect of reflections. \n\n\\begin{equation}\n\\frac{I_{dust}}{I_0} = f e^{-\\kappa_{abs,inner}d_{inner}} +  \n1 - e^{-\\kappa_{abs,outer}d_{outer}} \n\\label{eq1}\n\\end{equation} \n\nEssentially, Eq.~\\ref{eq1} provides the effective transmission of the dust.  \nIf Beer's law is followed (i.e. reflectivity is low and the particles are thin\nenough to transmit light at all frequencies), \nthen $\\kappa_{abs,inner} = \\kappa_{abs,outer}$ and\nthe spectrum received is no different than the intrinsic absorptions.  \nHowever, if the absorbing particles are larger, and opaque at some frequencies \n(i.e.\\ the TO mode), the spectrum will be altered. We have computed the effect \nof particle thickness on self-absorption in circumstellar shells, and the \nresulting spectra are shown in\n Fig.~\\ref{modelthick}. \nFor $\\kappa_{abs,outer}d_{outer}$, \nwe used baseline-corrected\nabsorption spectra from the thickest sample of nano $\\beta$-SiC shown in \nFig.~\\ref{labthick}, and for $\\kappa_{abs,inner}d_{inner}$, \nwe similarly used the intermediate sample. The same\nresults would be obtained for the thinnest sample. As the contribution of\nthe emitting particles increases, the contribution of the LO component\nincreases relative to the TO component, and the TO component appears to\nshift to longer wavelengths.\n\nIf the size and absorption coefficients are identical for both the inner and \nouter regions of the dust shell, then there will be no shift in the spectral \nfeatures between emission and absorption. However, if the outer grains are \nlarger, there will be a shift in the absorption to shorter wavelengths, \nshown in Fig.~\\ref{modelthick}.\n\n\n\nIn terms of what we would expect to see in the sequence of carbon star \nspectra, the discussion above means that for optically thin dust shells, \nwhere we are seeing pure emission, the SiC feature should peak at 11.3$\\mu$m \nand be sharp. As the optical depth goes up, the SiC feature will become \nself-absorbed, \nbut if the grains in the outer part of the shell are larger than \nthose in the inner zone, the absorption will occur preferentially on the LO \nside of the feature, diminishing the blue side, which would appear as a shift \nin the feature to longer wavelengths ($\\sim$11.7$\\mu$m). \nAs the optical depth gets \nhigh enough for the emission feature to be completely absorbed, we no longer \nsee the 11.7$\\mu$m feature and the absorption will\npeak at a shorter wavelength (10.8$\\mu$m) than the regular SiC feature at \n11.3$\\mu$m. This is shown schematically in Fig.~\\ref{cartoon} \n(c.f. Fig.~\\ref{cgs3}).\nThis mechanism explains the evolution \nin the observed SiC absorption feature in extreme carbon stars. \nFurthemore, it correlates with the evolution of the spectral features as seen \nby \\citet{baron87}, who showed that as the continuum temperature decreases and \nthe peak-continuum strength of the SiC feature diminishes, the peak position \ntends to move from $\\sim$11.3$\\mu$m to $\\sim$11.7$\\mu$m. This suggests that \nthere is an evolution in the dust grains towards smaller dust \ngrains with higher mass loss.\n\nTwo separate studies of presolar SiC suggest that there is an evolution in\ngrain size that corresponds to the evolution of carbon stars. \\citet{prombo}\nfound a correlation between grain size and the concentration of \n{\\it s-process} elements in SiC grains taken from the Murchison meteorites. \nThe Indarch meteorite presolar SiC grains yielded similar results \n\\citep{jennings}. In both cases, the smaller grains have higher relative \nabundances of {\\it s-process} elements. Since newly-formed\n {\\it s-process} elements are dredged-up along with carbon from the He-burning \nshell, they are more abundant in more evolved circumstellar shells than \nearly ones. \nTherefore, these results suggest that\nthe grains formed around carbon stars decrease in size as the stars evolve and \nsupport the observations of the self-absorption effects described above. \n\nUp until now, the generally accepted wisdom has been that \nlow mass-loss rates early in the AGB phase lead to small grains, and \nincreasing mass-loss rates permit the growth of larger grains. In the new \nscenario, smaller grain sizes with increasing mass loss can be understood in \nterms of potential nucleation sites. The hardest step in grain formation is \nthe production of seeds onto which minerals can grow. In a low-density gas \nvery few seeds can nucleate, leading to the formation of few grains. However, \nthese few grains can grow large because there are not so many grains competing \nfor the same atoms. Conversely, in a high-density gas, it is \neasier to form seeds, but so many more nucleate, that there are not enough \natoms around for any individual grain to grow large. In this way, high \ndensities lead to very dense dust shells of small grains. \nThis effect may be exacerbated by the earlier outflows moving \nslower than the later outflows.\nIf the stellar winds that drive the mass loss \nand send the newly formed grains away from star accelerate with time, then the \nearlier grains will have more time closer to the star in regions dense enough \nfor grain growth to occur. Since radiation-pressure coupling should be related \nto grain composition, and this will vary with the Si\/C ratio, we may expect \nsuch an effect.\n\n\n\\section{Conclusions \\label{conc}}\n\nThe dust shells around carbon stars and their successors, carbon-rich post-AGB \nstars, evolve along with the star.\nIn early phases,  the mass-loss rate is low, and hence the dust \nshell is optically thin. As these stars evolve, the mass-loss rate increases, \nand thus the dust shell gets more optically thick, until the star is \ncompletely obscured in visible light and very bright in the infrared. \nDuring this evolution the relative abundance of silicon to carbon (Si\/C) \navailable for dust formation decreases, so that in the early carbon stars \nthere is more Si and C, whereas in the extreme carbon stars there is much\nmore C than Si. Once the AGB phase ends, the existing dust shell spreads out, \nbecoming optically thin and cooler.\nThe combination of increasing density and increasing carbon on the AGB \nmanifests itself \nin the nature of the dust grains as seen in the spectral sequence for carbon \nstars.\n\nEarly in the carbon star phase, when the mass-loss rate is low and the shell \nis optically thin, the $\\sim$11$\\mu$m SiC emission \nfeature is strong, narrow and sharp. As \nthe mass loss increases and the shell becomes optically thicker, the SiC \nfeature broadens, flattens and weakens. Finally, once the mass-loss rate is \nextremely high and the shell is extremely optically thick, the SiC feature \nappears in absorption.\nThe shifts in the peak of the $\\sim$11$\\mu$m SiC feature are attributable to a \ncombination of optical depth and grain sizes.\nThe only way to see a shift in the absorption feature to 10.8$\\mu$m is if \nthe absorbing grains are larger than the emitting grains.\nTherefore, we have observational evidence to suggest that the grains formed \nin circumstellar shell get smaller as the star evolves. Further evidence for \nthis scenario is seen both in post-AGB spectra and in meteoritic studies \nof presolar grains.\n\nWe have presented new mid-IR laboratory spectra for various forms of SiC \nincluding amorphous SiC with excess carbon structure. \nThese data corroborate and augment the laboratory work of \n\\citet{kimura03} and \\citet{kimura05a,kimura05b}, \nindicating that solid solutions of C \nreplacing Si in SiC have the diamond \nstructure on a local scale.\nThis is a good candidate\nfor the carrier of the $\\sim9\\mu$m feature seen in both emission and \nabsorption, and correlated with trends in the $\\sim$11$\\mu$m feature.\n\n\n\\acknowledgments\nWe are extremely grateful to the reviewer whose comments significantly \nimproved the paper. We would also like to thank Kevin Volk for providing us \nwith his ISO SWS spectra and model fits for extreme carbon stars.\nSupport for AMH was provided by NASA grant APRA04-0000-0041.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\t\t\\label{sec:1}\n\nNanotechnology has been highlighted for various applications such as medical systems, healthcare systems, nano-material, nano-machinary, nanoscale communication networks, and molecular communication systems.\nIn particular, molecular communication is an emerging technology for communication between nanomachines where information is conveyed by means of molecules \\cite{SEA:12:IT,NEH:13:Book, FMGCEG:19:MBSC}. In \\emph{passive transport} molecular communication, the random propagation of molecules in a fluid medium such as air, water, or blood vessels in human tissue can be solely determined by the law of diffusion or can be subjected to unpredictable turbulence caused by the environment.\\footnote{According to the type of molecule propagation, the mechanisms of molecular communication can be classified into \\emph{active transport} and \\emph{passive transport}. In active transport molecular communication, molecules propagate through predefined pathways such as a molecular motor, while in passive transport molecular communication, molecules are released through spontaneous diffusion.} Therefore, it is crucial to consider for the spontaneous diffusion law as well as non-predictable turbulence in the fluid medium in characterizing molecular communication.\n\nBrownian motion describes an ideal diffusion environment where the movement of molecules in the fluid medium is induced by their collisions. This diffusion can be modeled by Fick's laws where the homogeneous diffusion coefficient can be applied both in space and time, assuming that each molecule propagates independently. Due to the mathematical convenience---i.e., the time evolution of the probability density function (PDF) associated with the position of the molecule is normally distributed with zero mean---rather than the accuracy of diffusion models, Brownian motion has been widely used to model a molecular communication channel (see, e.g., \\cite{SEA:12:IT,KA:13:JSAC,JAJSS:16:COM,LZMY:17:CL,LHLCJ:17:MNL,CLY:18:NB} and references therein). However, the extraordinary diffusion phenomenon, called \\emph{anomalous diffusion} or non-Fickian diffusion, was discovered in crowded, heterogeneous, or complex structure systems, (e.g., the particles in heterogeneous porous media \\cite{KB:88:PF, Plo:14:JAM}, in cytoplasm \\cite{RVD:13:BJ}, and in the rotating flow \\cite{SWS:93:PRL}, magnetic resonance in excised human tissue \\cite{OBS:06:JMR}, telomeres in nucleus of mammalian cells \\cite{BIK:09:PRL}) where the anomalous diffusion process does not obey the linear relationship between mean square displacement and time, in contrast to the ideal diffusion process \\cite{MK:00:PR, MJCB:14:PCCP, ZDK:15:RMP,MMM:16:NB}. This highly motivates the use of anomalous diffusion in molecular communication for a wide range of applications \\cite{CTJS:15:CL, TJS:19:ACCESS, TJSW:18:COM}.\\footnote{Experimental data obtained using molecular communication platforms showed that the end-to-end molecular communication channel has a nonlinearity, unlike many previously developed molecular communication channel models \\cite{FKEC:14:JSAC}. } In \\cite{CTJS:15:CL}, the one-dimensional anomalous diffusion propagation was considered and the error rate was analyzed for timing and amplitude modulations. This work further extended to a connectivity problem with a random time constraint in a one-dimensional nanonetwork, where the random locations of molecules at the initial time are modeled by poisson point process \\cite{TJS:19:ACCESS}. The Cox process has been considered in \\cite{TJSW:18:COM} to capture the dynamic variation of the molecule concentration arising from the mobility of anomalously diffusive molecules, and the spatial ordering of the molecular communication performance has been characterized in terms of the error rate in the presence of interfering molecules. However, there is no comprehensive study on the modeling of anomalous diffusion channels in the context of molecular communication.\n\n\nVarious anomalous diffusion processes typically can be modeled numerous ways including continuous random walk (CTRW), generalized diffusion equation, generalized master equation, fractional Brownian motion, and fractional kinetic equation (fractional diffusion equation) \\cite{MK:00:PR,WR:83:ACP,Sch:93:PRE,Tun:74:SP}.\\footnote{The master equation for Brownian motion is the standard linear diffusion equation. Its fundamental solution is coined with the Gaussian density function, where the spatial variance increases linearly in time.}\nIn particular, the CTRW simply describes diffusion of molecules in the medium with arbitrary distributions of jump lengths and waiting times.\\footnote{The CTRW can be considered as diffusion governed by a space-time fractional diffusion equation, where probabilities for jump length and waiting time behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives, which is called \\emph{parametric subordination} \\cite{GMV:07:CSF,GVM:06:MS}.} In addition, the combination of a stochastic operational time---a \\emph{directing process}---and the \\emph{self-similar parent process} is equivalent to the subordination integral mechanism for the product of two random variables in the context of subordinated processes \\cite{MPG:03:FCAA,PP:16:FCAA,MP:08:JPA,PMM:13:PTRSA}.\\footnote{A natural way to derive subordination formulas for the fractional diffusion processes is called the \\emph{stochastic method}  \\cite{BM:01:FCAA,MBS:02:PRE,CKN:17:SPL}, in contrast to the subordination integral mechanism method.} This subordination law generates the solution of the fractional diffusion equation in purely analytical ways using the machinery offered by convolution properties of the Mellin transform.\\footnote{The Mellin and convolution operators are essential to provide a systematic language in dealing with communication performance by averaging of nonnegative random variables in wireless communication systems \\cite{JCSW:13:JSAC,JQKS:14:WCOM,JSW:15:IT,TJS:17:ACCESS} using a general integral transform---called an $H$-transform.}\n\nIn this paper, we embody anomalous diffusion according to the self-similar processes. To this end, we introduce the $H$-process by capturing the concept of $H$-variate that is the versatile family of statistical distributions \\cite{JSW:15:IT}. Therefore, the $H$-process is well diversified including various typical stochastic processes as special cases. Furthermore, the modeling and analysis framework based on $H$-theory falls into the $H$-transform framework \\cite{JSW:15:IT, TJSW:18:COM}; hence, the framework serves as a systematic method in a unified fashion to model the anomalous diffusion channel and analyze the molecular communication system by the virtue of Mellin and convolution operators of two $H$-functions, covering various typical diffusion models. The main contributions of this paper can be summarized as follows:\n\n\\begin{itemize}\n\n\\item\n\n\nBy introducing a new class of stochastic self-similar processes, namely an $H$-process and a symmetric $H$-process (see Definition~\\ref{def:Hprocess} and Definition~\\ref{def:SHprocess}), we show that the parent-directing subordination process---which consists of the parent $H$-process and directing $H$-process---is again an $H$-process (see Theorem~\\ref{thm:subordination}). Using these two self-similar $H$-processes, we introduce \\emph{$H$-diffusion} (see Definition~\\ref{def:Hdiffusion}), which can encompass most typical anomalous diffusion types such as space-time fractional diffusion (ST-FD), space fractional diffusion (S-FD), time fractional diffusion (T-FD), Erd\\'elyi-Kober fractional diffusion (EK-FD), and grey Brownian motion (GBM) as its special cases. Specifically, we define standard $H$-diffusion (SHD) which allows to model fractional Brownian motion (FBM) and Brownian motion (BM) as well as typical anomalous types of $H$-diffusion as special cases (see Definition~\\ref{def:StdHdiffuion}).\n\n\\item \nWe present a new class of molecular noise---namely, \\emph{$H$-noise}---to develop a unifying framework for characterizing statistical properties of uncertainty of the random propagation time of a molecule (see Definition~\\ref{def:Hnoise}). We derive the first passage time (FPT) of an anomalous diffusive molecule in terms of $H$-variate (see Theorem~\\ref{thm:fpt:Hdiffusion}), which acts as the unique source of uncertainty in diffusive molecular communication. The $H$-noise is well-diversified to various types of molecular noise models such as L{\\'e}vy distribution noise for various diffusion scenarios. Then, we put forth an $H$-noise tail (see Theorem~\\ref{thm:Hnoise:tails}) and finite logarithm moments of $H$-noise (see Theorem~\\ref{thm:LM:Hnoise}) to describe algebraic-tailed and heavy-tailed distribution properties of the $H$-noise. We quantify the geometric power of $H$-noise---namely, \\emph{$H$-noise power} coined with the fractional lower-order statistics and zero-order statistics (see Corollary~\\ref{cor:gmp:Hnoise} and Remark~\\ref{rem:GPF}). \n\n\\item\nWe characterize the effect of anomalous diffusion on the error probability in timing-based molecular communication. We first develop the unifying error probability analysis for multiple number of transmitted molecules with the $M$-ary modulation technique (see Theorem~\\ref{thm:SEP:Hdiffusion}). The first arrival detection method is adopted to carry out the simplified decoding process at a receive nanomachine (RN) using the statistic of the first arrival molecule among multiple released molecules at a transmit nanomachine (TN) (see Proposition~\\ref{pro:pdf:fa}). For SHD, we define a signal-to-noise ratio (SNR) based on the geometric power of $H$-noise (see Definition~\\ref{def:snr}) and characterize the high-SNR behavior of the error probability in terms of the high-SNR slope and high-SNR power offset for anomalous diffusion-based molecular communication (see Corollary~\\ref{cor:highsnr}). \n\n\\end{itemize}\n\nThroughout the paper, we shall adopt the notation that random variables are displayed in sans serif, upright font; their realizations in serif, italic font. For example, a random variable and its realization are denoted by $\\rv{x}$ and $x$, respectively. The basic operations on the $H$-function can be found in Table~\\ref{table:operation} (see also \\cite{JSW:15:IT}).  We relegate the glossary of notations and symbols used in the paper to Appendix~\\ref{sec:appendix:NS}.  In Appendix~\\ref{sec:appendix:SF}, we briefly introduce the special functions which are frequently used in the context of diffusion theory, fractional diffusion theory, and molecular communication. $H$-representation for stable distributions is provided in Appendix~\\ref{sec:appendix:Stable}, which is required to develop a framework for modeling and analysis in  molecular communication. \n\n\n\\begin{table\n\\caption{Operations on the order and parameter sequences of Fox's $H$-function}\n\\centering\n\\label{table:operation}\n\n\\begin{threeparttable}\n\n\\begin{tabular}{lllll}\n\\addlinespace\n\\midrule\n\\midrule\n\nOperation &\nSymbol &\nOrder or parameter sequence\n\\\\\n\\midrule\n\\addlinespace\n\n\nScaling &\n$\\BB{\\pDefine{P}}\\ket{\\alpha}$ &\n$\\left(\n\t\\frac{\\pDefine{k}}{\\alpha},\n\t\\frac{\\pDefine{c}}{\\alpha},\n\t\\BB{\\pDefine{a}},\n\t\\BB{\\pDefine{b}},\n\t\\BB{\\pDefine{A}},\n\t\\BB{\\pDefine{B}}\n\\right)$\n\\\\[-0.2cm]\n\\addlinespace\n\nConjugate &\n$\\Bra{\\gamma}\\BB{\\pDefine{P}}$ &\n$\\left(\n\t\\frac{\\pDefine{k}}{\\pDefine{c}^{\\gamma}},\n\t\\pDefine{c},\n\t\\BB{\\pDefine{a}}+\\gamma\\BB{\\pDefine{A}},\n\t\\BB{\\pDefine{b}}+\\gamma\\BB{\\pDefine{B}},\n\t\\BB{\\pDefine{A}},\n\t\\BB{\\pDefine{B}}\n\\right)$\n\\\\[-0.2cm]\n\\addlinespace\n\n\nElementary &\n$\\eOP{\\alpha}{\\beta}{\\gamma}\\BB{\\pDefine{P}}$ &\n$\\left(\n\t\\frac{\\pDefine{k}}{\\left(\\alpha\\pDefine{c}\\right)^{\\beta\\gamma}},\n\t\\left(\\alpha\\pDefine{c}\\right)^{\\beta},\n\t\\BB{\\pDefine{a}}+\\beta\\gamma\\BB{\\pDefine{A}},\n\t\\BB{\\pDefine{b}}+\\beta\\gamma\\BB{\\pDefine{B}},\n\t\\beta\\BB{\\pDefine{A}},\n\t\\beta\\BB{\\pDefine{B}}\n\\right)$\n\\\\[-0.2cm]\n\\addlinespace\n\nInverse &\n$\\BB{\\pDefine{O}}^{-1}$ &\n$\\left(\n\t\\pDefine{n},\n\t\\pDefine{m},\n\t\\pDefine{q},\n\t\\pDefine{p}\n\\right)$\n\\\\[-0.2cm]\n\\addlinespace\n\n&\n$\\BB{\\pDefine{P}}^{-1}$ &\n$\\left(\n\t\\pDefine{k},\n\t\\frac{1}{\\pDefine{c}},\n\t\\B{1}_\\pDefine{q} - \\BB{\\pDefine{b}},\n\t\\B{1}_\\pDefine{p} - \\BB{\\pDefine{a}},\n\t\\BB{\\pDefine{B}},\n\t\\BB{\\pDefine{A}}\n\\right)$\n\\\\[-0.2cm]\n\\addlinespace\n\nMellin &\n$\\stdOP{\\BB{\\pDefine{O}}_1}{\\BB{\\pDefine{O}}_2}$ &\n$\\left(\n\t\\pDefine{m}_1+\\pDefine{n}_2,\n\t\\pDefine{n}_1+\\pDefine{m}_2,\n\t\\pDefine{p}_1+\\pDefine{q}_2,\n\t\\pDefine{q}_1+\\pDefine{p}_2\n\\right)$\n\\\\[-0.2cm]\n\\addlinespace\n\n&\n$\\stdOP{\\BB{\\pDefine{P}}_1}{\\BB{\\pDefine{P}}_2}$ &\n$\\left(\n \t\\frac{\\pDefine{k}_1\\pDefine{k}_2}{\\pDefine{c}_2},\n\t\\frac{\\pDefine{c}_1}{\\pDefine{c}_2},\n\t\\BB{\\pDefine{a}},\n\t\\BB{\\pDefine{b}},\n\t\\BB{\\pDefine{A}},\n\t\\BB{\\pDefine{B}}\n\\right)$\n\\\\[-0.2cm]\n\\addlinespace\n\n&\n&\n\n$\n\\begin{cases}\n\\BB{\\pDefine{a}}=\n\t\t\\left(\n\t\t\t\\dot{\\BB{\\pDefine{a}}_{1}},\n\t\t\t\\B{1}_{\\pDefine{q}_2}-\\BB{\\pDefine{b}}_{2}-\\BB{\\pDefine{B}}_{2},\n\t\t\t\\ddot{\\BB{\\pDefine{a}}_{1}}\n\t\t\\right)\n\\\\\n\\BB{\\pDefine{b}}=\n\t\t\\bigl(\n\t\t\t\\dot{\\BB{\\pDefine{b}}_{1}},\n\t\t\t\\B{1}_{\\pDefine{p}_2}-\\BB{\\pDefine{a}}_{2}-\\BB{\\pDefine{A}}_{2},\n\t\t\t\\ddot{\\BB{\\pDefine{b}}_{1}}\n\t\t\\bigr)\n\\\\\n\\BB{\\pDefine{A}}=\n\t\t\\left(\n\t\t\t\\dot{\\BB{\\pDefine{A}}_{1}},\n\t\t\t\\BB{\\pDefine{B}}_{2},\n\t\t\t\\ddot{\\BB{\\pDefine{A}}_{1}}\n\t\t\\right)\n\\\\\n\\BB{\\pDefine{B}}=\n\t\t\\left(\n\t\t\t\\dot{\\BB{\\pDefine{B}}_{1}},\n\t\t\t\\BB{\\pDefine{A}}_{2},\n\t\t\t\\ddot{\\BB{\\pDefine{B}}_{1}}\n\t\t\\right)\n\\end{cases}\n$\n\n\\qquad\n(or \\cite[Eqs.~(30), (31)]{JSW:15:IT})\n\\\\[-0.2cm]\n\\addlinespace\n\nConvolution &\n$\\canOP{\\BB{\\pDefine{O}}_1}{\\BB{\\pDefine{O}}_2}$ &\n$\\left(\n\t\\pDefine{m}_1+\\pDefine{m}_2,\n\t\\pDefine{n}_1+\\pDefine{n}_2,\n\t\\pDefine{p}_1+\\pDefine{p}_2,\n\t\\pDefine{q}_1+\\pDefine{q}_2\n\\right)$\n\\\\[-0.2cm]\n\\addlinespace\n\n&\n$\\canOP{\\BB{\\pDefine{P}}_1}{\\BB{\\pDefine{P}}_2}$ &\n$\\left(\n        \t\t\\pDefine{k}_1 \\pDefine{k}_2,\n        \t\t\\pDefine{c}_1 \\pDefine{c}_2,\n\t\t\\BB{\\pDefine{a}},\n\t\t\\BB{\\pDefine{b}},\n\t\t\\BB{\\pDefine{A}},\n\t\t\\BB{\\pDefine{B}}\n\\right)$\n\\\\[-0.2cm]\n\\addlinespace\n\n&\n&\n$\n\\begin{cases}\n\\BB{\\pDefine{a}}=\n\t\t\\left(\n\t\t\t\\dot{\\BB{\\pDefine{a}}_{1}},\n\t\t\t\\BB{\\pDefine{a}}_{2},\n\t\t\t\\ddot{\\BB{\\pDefine{a}}_{1}}\n\t\t\\right)\n\\\\\n\\BB{\\pDefine{b}}=\n\t\t\\bigl(\n\t\t\t\\dot{\\BB{\\pDefine{b}}_{1}},\n\t\t\t\\BB{\\pDefine{b}}_{2},\n\t\t\t\\ddot{\\BB{\\pDefine{b}}_{1}}\n\t\t\\bigr)\n\\\\\n\\BB{\\pDefine{A}}=\n\t\t\\left(\n\t\t\t\\dot{\\BB{\\pDefine{A}}_{1}},\n\t\t\t\\BB{\\pDefine{A}}_{2},\n\t\t\t\\ddot{\\BB{\\pDefine{A}}_{1}}\n\t\t\\right)\n\\\\\n\\BB{\\pDefine{B}}=\n\t\t\\left(\n\t\t\t\\dot{\\BB{\\pDefine{B}}_{1}},\n\t\t\t\\BB{\\pDefine{B}}_{2},\n\t\t\t\\ddot{\\BB{\\pDefine{B}}_{1}}\n\t\t\\right)\n\\end{cases}\n$\n\\hspace{1.8cm} (or \\cite[Eqs.~(42), (43)]{JSW:15:IT})\n\\\\\n\\addlinespace\n\\midrule\n\\midrule\n\\end{tabular}\n\n\\begin{tablenotes}\n\\item[\\hspace{0.4cm}Note)]\n$\\alpha, \\beta \\in \\mathbbmss{R}_{++}$, $\\gamma \\in \\mathbbmss{C}$,\n$\\ell \\in \\mathbbmss{N}$\n\n\\end{tablenotes}\n\n\\end{threeparttable}\n\n\n\\end{table}\n\n\n\n\\section{$H$-Diffusion Modeling}\t\t\\label{sec:2}\n\nIn this section, we begin by reviewing the subordination law with self-similar processes for modeling anomalous diffusion \\cite{MP:08:JPA,PMM:13:PTRSA}. Then, we introduce an $H$-diffusion model, which can span a wide range of well-established anomalous diffusion types.\n\n\\subsection{$H$-Variables}\n\nWe use the representation of Fox's $H$-function defined in \\cite{JSW:15:IT} (see also Appendix~\\ref{sec:appendix:NS}) for notational simplicity as \\cite[Eq.~(245)]{JSW:15:IT}:\n\\begin{align}\n\\Fox{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{x;\\BB{\\pDefine{P}}}\n&=\n\t\t\\pDefine{k}\\FoxH{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\pDefine{c} x}{\\left(\\BB{\\pDefine{a}},\\BB{\\pDefine{A}}\\right)}{\\left(\\BB{\\pDefine{b}},\\BB{\\pDefine{B}}\\right)} \n\t\\qquad\n\t\\left(x>0\\right),\n\\end{align}\nwhere\n$\n\\BB{\\pDefine{P}}\n=\n\t\\left(\\pDefine{k},\\pDefine{c},\\BB{\\pDefine{a}},\\BB{\\pDefine{b}},\\BB{\\pDefine{A}},\\BB{\\pDefine{B}}\\right)\n$\nis the parameter sequence satisfying the necessary conditions \\cite[Remark~7]{JSW:15:IT} to be a density function. By convention, letting the \\emph{null} sequences be $\\BB{\\pDefine{P}}_\\emptyset =\\left(1,1,\\text{--},\\text{--},\\text{--},\\text{--}\\right)$ and $\\BB{\\pDefine{O}}_\\emptyset=\\left(0, 0, 0, 0\\right)$, we define\n\\begin{align}\n\\Fox{0}{0}{0}{0}{x;\\BB{\\pDefine{P}}_\\emptyset}\n=\n\t\\delta\\left(x-1\\right).\n\\end{align}\n\n\\begin{definition}[$H$-Variable \\cite{JSW:15:IT}] \\label{def:H}\nA nonnegative random variable $\\rv{x}$ is said to have an $H$-distribution with the order sequence $\\BB{\\pDefine{O}}=\\left(\\pDefine{m},\\pDefine{n},\\pDefine{p},\\pDefine{q}\\right)$ and the parameter sequence $\\BB{\\pDefine{P}} = \\left(\\pDefine{k}, \\pDefine{c}, \\BB{\\pDefine{a}}, \\BB{\\pDefine{b}}, \\BB{\\pDefine{A}}, \\BB{\\pDefine{B}} \\right)$, denoted by $\\rv{x} \\sim \\FoxRV{\\BB{\\pDefine{O}}}{\\BB{\\pDefine{P}}}$ or simply $\\rv{x} \\sim \\FoxV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}}$, if its PDF is given by\n\\begin{align} \\label{eq:Def:FV}\n\\PDF{\\rv{x}}{x}\n&=\n        \\Fox{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{x;\\BB{\\pDefine{P}}}\n        \\qquad\\quad\n        \\left(x >0 \\right),\n\\end{align}\nwith the set of parameters satisfying a distributional structure such that $\\PDF{\\rv{x}}{x} \\geq 0$ for all $x >0$ and $\\int_0^\\infty \\PDF{\\rv{x}}{x} dx=1$.\n\\end{definition}\n\n\\begin{definition}[Symmetric $H$-Variable] \\label{def:H}\nA symmetric random variable $\\rv{y}$ is said to have a symmetric $H$-distribution, denoted by $\\rv{y} \\sim \\FoxSV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}}$, \nif $\\sNorm{\\rv{y}} \\sim \\FoxV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}}$, that is, its PDF is\n$\n\\PDF{\\rv{y}}{y}=\\frac{1}{2}\\Fox{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\sNorm{y};\\BB{\\pDefine{P}}}\n$ for $y \\in \\mathbbmss{R}$.\n\\end{definition}\nThe cumulative distribution function (CDF), moment generating function (MGF), and moments of the symmetric $H$-variable are given in Table~\\ref{table:statistics:FV}. Note that the CDF, MGF, and moments of the $H$-variable can be found in  \\cite[Table~IV]{JSW:15:IT}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$H$-Diffusion}\n\nA concept of \\emph{diffusion} is originated from modeling the spread of molecular concentration in a medium caused by the random motion of molecules \\cite{KS:14:Book}. Some anomalous diffusion processes are well-defined by continuous time random walk \\cite{MPG:03:FCAA,GMV:07:CSF,PMM:13:PTRSA} or Gaussian processes with time subordination \\cite{MP:08:JPA,PMM:12:IJSA,MM:09:ITSP},  which are also well-defined by a subordinated process with a self-similar parent process and a self-similar directing process. This model enables us to generate the fundamental solution of a given diffusion equation using the Mellin convolution of two independent random variables whose density functions are governed by the subordination law \\cite{MP:08:JPA,PMM:13:PTRSA}. In the following theorem, we introduce a versatile family of diffusion established by the subordination law.\n\n\\begin{remark}[Self-Similarity]\nA stochastic process $\\left\\{\\rp{s}{t}; t \\geq 0\\right\\}$ is self-similar if, for any $a \\in \\mathbbmss{R}_{++}$, there exists $b \\in \\mathbbmss{R}_{++}$ such that  \\cite{EM:00:JMPB,EM:02:Book}\n\\begin{align}\n\\rp{s}{at}\n\\mathop{=}\\limits^{\\text{d}}\n\tb\\rp{s}{t}.\n\\end{align}\nIf the self-similar process $\\left\\{\\rp{s}{t}\\right\\}$ is (i) nontrivial,\\footnote{A process $\\left\\{\\rp{x}{t}; t\\geq 0\\right\\}$ is a trivial process or a deterministic process if $\\rp{x}{t}$ is constant or its distribution $\\PDF{\\rv{x}\\left(t\\right)}{x}$ is a $\\delta$-distribution for every $t \\geq 0$ \\cite[Definition~13.6]{Sat:99:Book}.} (ii) stochastically continuous, and (iii) $\\rp{s}{0} =0$ almost surely,  \nthen there exists the self-similarity exponent $\\omega \\in \\mathbbmss{R}_{++}$ for any $a$ such that $b=a^\\omega$, \nand for any $t>0$:\n\\begin{align}\n\\rp{s}{t}\n\\mathop{=}\\limits^{\\text{d}}\n\tt^\\omega\\rp{s}{1}.\n\\end{align}\n\n\n\\end{remark}\n\n\\begin{definition}[$H$-Process] \t\\label{def:Hprocess}\nA nonnegative self-similar process $\\left\\{\\rp{x}{t}; t \\geq 0\\right\\}$ is said to be an \\emph{$H$-process} with the exponent $\\omega \\in \\mathbbmss{R}_{++}$, denoted by  $\\left\\{\\rp{x}{t}; t \\geq 0\\right\\} \\sim \\left\\{\\FoxV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}},\\omega\\right\\}$, if $\\rp{x}{t} \\sim \\FoxV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}\\ket{t^{\\omega}}}$ for $t >0$.\n\\end{definition}\n\n\\begin{definition}[Symmetric $H$-Process] \\label{def:SHprocess}\nA symmetric $\\left\\{\\rp{y}{t}; t \\geq 0\\right\\}$ is said to be a \\emph{symmetric $H$-process} with the exponent $\\omega \\in \\mathbbmss{R}_{++}$, denoted by  $\\left\\{\\rp{y}{t}; t \\geq 0\\right\\} \\sim \\bigl\\{\\FoxSV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}},\\omega\\bigr\\}$, if $\\rp{y}{t}$ is a self-similar process and $\\rp{y}{t} \\sim \\FoxSV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}\\ket{t^{\\omega}}}$ for $t >0$. \n\\end{definition}\n\n\n\\begin{table}\n\\caption{CDF, MGF, and moments of symmetric $H$-variable $\\rv{y} \\sim \\FoxSV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}}$}\\centering%\n\\label{table:statistics:FV}\n\n\\begin{threeparttable}\n\n\\begin{tabular}{lllll}\n\\midrule\n\\midrule\nStatistics &\nSymbol &\nFunction representation &\nParameter sequence\n\\\\\n\\midrule\n\\addlinespace\n\nCDF &\n\n$\\CDF{\\rv{y}}{y}$ &\n\n$\n\t\\tfrac{1}{2}+\\sgn\\left(x\\right)\\Fox{\\pDefine{m}}{\\pDefine{n}+1}{\\pDefine{p}+1}{\\pDefine{q}+1}{\\sNorm{x};\n\t\\canOP{\\pSeq_\\FontDef{cdf}}{\\Bra{1}\\BB{\\pDefine{P}}}}\n$ &\n$\n\\pSeq_\\FontDef{cdf}\n=\n        \\left(\n\t        \t1,\n\t        \t1,\n\t        \t\\left(1,\\text{--}\\right),\n\t        \t\\left(\\text{--},0\\right),\n\t        \t\\left(1,\\text{--}\\right),\n\t        \t\\left(\\text{--},1\\right)\n\t\\right)\n$\n\\\\\n\\addlinespace\n\n\nMGF &\n\n$\\MGF{\\rv{y}}{s}$  &\n$\n\t\\tfrac{1}{2}\\Fox{\\pDefine{n}+1}{\\pDefine{m}}{\\pDefine{q}}{\\pDefine{p}+1}{s;\n\t\\stdOP{\\pSeq_\\FontDef{exp}}{\\BB{\\pDefine{P}}}}+\\tfrac{1}{2}\\Fox{\\pDefine{n}+1}{\\pDefine{m}}{\\pDefine{q}}{\\pDefine{p}+1}{-s;\n\t\\stdOP{\\pSeq_\\FontDef{exp}}{\\BB{\\pDefine{P}}}}\n$ &\n$\n\\pSeq_\\FontDef{exp}\n=\n        \\left(\n\t        \t1,\n\t        \t1,\n\t        \t\\text{--},\n\t        \t\\left(0,\\text{--}\\right),\n \t       \t\\text{--},\n\t        \t\\left(1,\\text{--}\\right)\n\t\\right)\n$\n\\\\\n\\addlinespace\n\nMoment &\n$\\mean{\\rv{y}^\\ell}$ &\n$\n\\begin{cases}\n\t\\frac{\\pDefine{k}}{\\pDefine{c}^{\\ell+1}}\n\t\\frac{\n\t\t\\prod_{j=1}^{\\pDefine{m}}\\GF{\\pB{j}+\\left(\\ell+1\\right)\\pT{j}}\n\t\t\\prod_{j=1}^{\\pDefine{n}}\\GF{1-\\pA{j}-\\left(\\ell+1\\right)\\pS{j}}\n\t}{\n\t\t\\prod_{j=\\pDefine{n}+1}^{\\pDefine{p}}\\GF{\\pA{j}+\\left(\\ell+1\\right)\\pS{j}}\n\t\t\\prod_{j=\\pDefine{m}+1}^{\\pDefine{q}}\\GF{1-\\pB{j}-\\left(\\ell+1\\right)\\pT{j}}\n\t},\\\\\n0, \\text{if $\\ell$ is odd number}\n\\end{cases}\n$ &\n$\n-\n$\n\\\\\n\\addlinespace\n\\midrule\n\\midrule\n\\end{tabular}\n\\end{threeparttable}\n\n\n\\end{table}\n\n\\begin{theorem}[$H$-Subordination]\t\\label{thm:subordination}\nLet $\\left\\{\\rp{p}{t}; t \\geq 0\\right\\} \\sim \\left\\{\\FoxV{\\pDefine{m}_1}{\\pDefine{n}_1}{\\pDefine{p}_1}{\\pDefine{q}_1}{\\BB{\\pDefine{P}}_1},\\omega_1\\right\\}$ be a parent $H$-process\nindependent of a directing $H$-process $\\left\\{\\rp{d}{t}; t \\geq 0\\right\\} \\sim \\left\\{\\FoxV{\\pDefine{m}_2}{\\pDefine{n}_2}{\\pDefine{p}_2}{\\pDefine{q}_2}{\\BB{\\pDefine{P}}_2},\\omega_2\\right\\}$.\nThen, a parent-directing \nsubordinated process $\\left\\{\\rp{x}{t}=\\rp{p}{\\rp{d}{t}}; t \\geq 0\\right\\}$ is again an $H$-process with the self-similar exponent $\\omega_1\\omega_2$, that is,\n\\begin{align} \\label{eq:gf:hv}\n\\left\\{\\rp{x}{t};t \\geq 0\\right\\}\n\t\\sim\n\t\\bigl\\{\\FoxV{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{n}_1+\\pDefine{n}_2}{\\pDefine{p}_1+\\pDefine{p}_2}{\\pDefine{q}_1+\\pDefine{q}_2}\n\t{\n\t\\BB{\\pDefine{P}}_{\\left(\\omega_1\\right)}},  \n\t\\omega_1 \\omega_2 \\bigr\\}\n\\end{align}\nwith  \n\\begin{align} \\label{eq:thm:1}\n\\BB{\\pDefine{P}}_{\\left(\\omega_1\\right)}\n\t&=\n\t\t\\canOP{\\BB{\\pDefine{P}}_1}{\\eOP{1}{\\omega_1}{1\/\\omega_1-1}\\BB{\\pDefine{P}}_2}\n\\end{align}\nwhere $\\eOP{\\alpha}{\\beta}{\\gamma}$, and $\\canOP{}{}$ are the elementary and convolution operations on the parameter sequence \\cite[Table~III]{JSW:15:IT}.\n\\begin{proof}\nSince the parent process $\\left\\{\\rp{p}{t}\\right\\}$ and the directing process $\\left\\{\\rp{d}{t}\\right\\}$ are self-similar processes, the subordinated process $\\left\\{\\rp{x}{t}\\right\\}$ is again a self-similar process as follows:\n\\begin{align} \\label{eq:thm:pf:1}\n\t\\rp{x}{t}\n\t&\\mathop{=}\\limits^{\\text{d}}\n\t\\rp{p}{t^{\\omega_2}\\rp{d}{1}}\n\t\\nonumber \\\\\n\t&\n\t\\mathop{=}\\limits^{\\text{d}}\n\tt^{\\omega_1\\omega_2}\n\t\\rp{p}{\\rp{d}{1}}\n\t\\nonumber \\\\\n\t&\n\t=\n\tt^{\\omega_1\\omega_2}\n\t\\rp{p}{1}\n\t\\rp{d}{1}^{\\omega_1},\n\\end{align}\nwhere the last equality follows from the subordination formula\n\\begin{align}\t\\label{eq:thm:pf:2}\n\t\\PDF{\\rp{x}{1}}{x}\n\t&=\n\t\t\\int_0^\\infty\n\t\t\\PDF{\\rp{p}{\\tau}}{x}\n\t\t\\PDF{\\rp{d}{1}}{\\tau}d\\tau\n\t\\nonumber \\\\\n\t&\n\t=\n\t\t\\int_0^\\infty\n\t\t\\PDF{\\rp{p}{1}}{\\frac{x}{\\tau^{\\omega_1}}}\n\t\t\\PDF{\\rp{d}{1}}{\\tau}\n\t\t\\frac{d\\tau}{\\tau^{\\omega_1}}.\n\\end{align}\nIn addition, $\\rp{p}{1}\\rp{d}{1}^{\\omega_1}$ has an $H$-distribution by \\cite[Theorem~1]{JSW:15:IT}. Using \\eqref{eq:thm:pf:1}, and the fact that $\\alpha \\rv{w} \\sim \\FoxV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}\\Ket{\\alpha}}$ if $\\rv{w} \\sim \\FoxV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}}$ for $\\alpha >0$, we have \n\\begin{align}  \n\t\\rp{x}{t}\n\t\\sim\n\t\\FoxV{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{n}_1+\\pDefine{n}_2}{\\pDefine{p}_1+\\pDefine{p}_2}{\\pDefine{q}_1+\\pDefine{q}_2}\n\t{\n\t\\BB{\\pDefine{P}}_{\\left(\\omega_1\\right)} \n\t\\Ket{t^{\\omega_1\\omega_2}}\n\t}\n\\end{align}\nwhich completes the proof.\n\\end{proof}\n\n\\end{theorem}\n\n\n\n\\begin{definition}[$H$-Diffusion] \t\t\\label{def:Hdiffusion} \nLet $\\left\\{\\rp{p}{t}; t \\geq 0\\right\\} \\sim \\bigl\\{\\FoxSV{\\pDefine{m}_1}{\\pDefine{n}_1}{\\pDefine{p}_1}{\\pDefine{q}_1}{\\BB{\\pDefine{P}}_1},\\omega_1\\bigr\\}$ \nbe a symmetric $H$-process independent of a directing $H$-process $\\left\\{\\rp{d}{t}; t \\geq 0\\right\\} \\sim \\bigl\\{\\FoxV{\\pDefine{m}_2}{\\pDefine{n}_2}{\\pDefine{p}_2}{\\pDefine{q}_2}{\\BB{\\pDefine{P}}_2},\\omega_2\\bigr\\}$. Then, a subordinated process $\\left\\{\\rp{x}{t}=\\rp{p}{\\rp{d}{t}}; t \\geq 0\\right\\}$ is said to be \\emph{$H$-diffusion}, denoted by \n$$\\left\\{\\rp{x}{t}; t \\geq 0 \\right\\} \\sim \\left\\{\\FoxSV{\\pDefine{m}_1:\\pDefine{m}_2}{\\pDefine{n}_1:\\pDefine{n}_2}{\\pDefine{p}_1:\\pDefine{p}_2}{\\pDefine{q}_1:\\pDefine{q}_2}{\\BB{\\pDefine{P}}_1,\\BB{\\pDefine{P}}_2;\\omega_1,\\omega_2}\\right\\}.$$\n\\end{definition}\n\n\\begin{corollary} \\label{crl:Hdiffusion}\nUsing the same argument in Theorem~\\ref{thm:subordination}, we can see that $H$-diffusion is again a symmetric $H$-process with the self-similar exponent $\\omega_1\\omega_2$:\n\\begin{align} \n\\left\\{\\rp{x}{t};t \\geq 0\\right\\}\n\t\\sim\n\t\\bigl\\{\\FoxSV{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{n}_1+\\pDefine{n}_2}{\\pDefine{p}_1+\\pDefine{p}_2}{\\pDefine{q}_1+\\pDefine{q}_2}\n\t{\n\t\\BB{\\pDefine{P}}_{\\left(\\omega_1\\right)}},  \n\t\\omega_1 \\omega_2 \\bigr\\}\n\\end{align}\nwhere $\\BB{\\pDefine{P}}_{\\left(\\omega_1\\right)}$ is given in \\eqref{eq:thm:1}. If a particle is released into a fluid medium governed by this $H$-diffusion process, at time $t=0$ at position $x=0$, then the particle's position at time $t>0$ is the symmetric $H$-variable\n\\begin{align} \\label{eq:gf:hv}\n\\rp{x}{t}\n\t&\\mathop{=}\\limits^{\\text{d}}\n\tt^{\\omega_1\\omega_2}\n\t\\rp{p}{1}\n\t\\rp{d}{1}^{\\omega_1}\n\\\\\n\t&\n\t\\sim\n\t\\FoxSV{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{n}_1+\\pDefine{n}_2}{\\pDefine{p}_1+\\pDefine{p}_2}{\\pDefine{q}_1+\\pDefine{q}_2}\n\t{\n\t\\BB{\\pDefine{P}}_{\\left(\\omega_1\\right)}  \n\t\\Ket{t^{\\omega_1 \\omega_2}}}\n\\end{align}\nwhere $\\rp{p}{1}  \\sim \\FoxSV{\\pDefine{m}_1}{\\pDefine{n}_1}{\\pDefine{p}_1}{\\pDefine{q}_1}{\\BB{\\pDefine{P}}_1}$, and $\\rp{d}{1}  \\sim \\FoxV{\\pDefine{m}_2}{\\pDefine{n}_2}{\\pDefine{p}_2}{\\pDefine{q}_2}{\\BB{\\pDefine{P}}_2}$. \n\\end{corollary}\n\n\n\\begin{remark}[Mean-Square Displacement]\nA particle is released into a fluid medium, governed by this $H$-diffusion process, at time $t=0$ at position $x=0$. Thus, the particle's position at time $t>0$ is the symmetric $H$-variable (see Corollary~\\ref{crl:Hdiffusion})\n\\begin{align} \\label{eq:gf:hv}\n\\rp{x}{t}\n\t&\\sim\n\t\\FoxSV{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{n}_1+\\pDefine{n}_2}{\\pDefine{p}_1+\\pDefine{p}_2}{\\pDefine{q}_1+\\pDefine{q}_2}\n\t{\n\t\\BB{\\pDefine{P}}_{\\left(\\omega_1\\right)} \n\t\\Ket{t^{\\omega_1 \\omega_2}}}.\n\\end{align}\nHence, the mean-square displacement (MSD) of the particle is\n\\begin{align}  \n\\bigl<\\sNorm{\\rp{x}{t}}^2\\bigr>\n\t\\propto\n\tt^{2\\omega_1 \\omega_2}\n\\end{align}\nwhere $2\\omega_1 \\omega_2$ represents the diffusion exponent.\nUsing this exponent, we \ncan classify diffusion into three types: i) subdiffusion for $0 <\\omega_1 \\omega_2 < 1\/2$, ii)\nnormal diffusion for $\\omega_1 \\omega_2=1\/2$, and iii) superdiffusion for $\\omega_1 \\omega_2 >1\/2$.\n\n\\end{remark}\n\n\n\n\n\\begin{remark} \\label{rmk:limitation}\nSince the $H$-distribution can span a wide range of statistical distributions, $H$-diffusion encompasses various types of anomalous diffusion processes as special cases, including L\\'evy flight, space-time fractional diffusion, and Erd\\'elyi-Kober fractional diffusion (generalized grey Brownian motion). Moreover, the probability density of molecule location $x$ at given time $t$ can be found from Mellin and convolution operations of two $H$-functions \\cite{JSW:15:IT}.\nHowever, since the $H$-process needs to be a nontrivial process (it is a necessary condition for the existence of self-similarity exponent), $H$-diffusion cannot cover the family of Brownian motions, whose \\emph{directing process} is a trivial process.\\footnote{By the definition, the family of Brownian motions is $H$-process but cannot be called $H$-diffusion because its directing process is the $\\delta$-distribution.  See Corollary~\\ref{corl:fbm} and Table~\\ref{table:TDM:HD}.} For example, the directing process of FBM and BW has the form of dirac delta function.\n\\end{remark}\n\n\n\\begin{corollary} [Fractional Brownian Motion] \\label{corl:fbm}\nLet $\\PDF{\\rp{d}{t}}{\\tau}=\\Fox{0}{0}{0}{0}{\\frac{\\tau}{t};\\BB{\\pDefine{P}}_\\emptyset}$. Then, the subordinated process $\\left\\{\\rp{x}{t}; t \\geq 0\\right\\}$ becomes the parent $H$-process $\\left\\{\\rp{p}{t}; t \\geq 0\\right\\} $ as\n\\begin{align} \\label{eq:crl:Hps}\n\\PDF{\\rp{x}{t}}{x}\n\t&=\n\t\t\\int_0^\\infty\n\t\t\\PDF{\\rp{p}{1}}{\\frac{x}{\\tau^{\\omega_1}}}\n\t\t\\Fox{0}{0}{0}{0}{\\frac{\\tau}{t};\\BB{\\pDefine{P}}_\\emptyset}\n\t\t\\frac{d\\tau}{\\tau^{\\omega_1}}\n\t\t\\nonumber \\\\\t\t\n\t&\n\t=\n\t\t\\int_0^\\infty\n\t\t\\PDF{\\rp{p}{1}}{\\frac{x}{\\tau^{\\omega_1}}}\n\t\t\\delta\\left(\\frac{\\tau}{t}-1\\right)\n\t\t\\frac{d\\tau}{\\tau^{\\omega_1}}\n\\nonumber \\\\\n\t&\n\t=\n\t\t\\PDF{\\rp{p}{t}}{x}.\n\\end{align}\nWith the symmetric Gaussian process of the parent $H$-process, that is,\n\\begin{align}\n\\left\\{\\rp{x}{t}; t \\geq 0\\right\\} \n=\n\\left\\{\\rp{p}{t}; t \\geq 0\\right\\} \n\\sim \\bigl\\{\\FoxSV{1}{0}{0}{1}{\\BB{\\pDefine{P}}_1},\\omega_1\\bigr\\} \n\\end{align}\nwhere $\\BB{\\pDefine{P}}_1=\\Bigl(\\frac{1}{2\\sqrt{\\pi}},\\frac{1}{2},\\text{--},0,\\text{--},\\frac{1}{2}\\Bigr)$, $\\rp{x}{t}$ is FBM. Specifically, $\\rp{x}{t}$ is called BM when $\\omega_1=1\/2$.\n\\end{corollary}\n\nTable~\\ref{table:TDM:HD} shows the typical anomalous diffusion models as special cases of $H$-diffusion and fractional Brownian motion.\n\n\n\n\n\n\\begin{table}[t]\n\\caption\nTypical Anomalous Diffusion Models as Special Cases of $H$-diffusion and Normal Diffusion\n} \\centering\n\\label{table:TDM:HD}\n\n\\begin{threeparttable}\n\n\\begin{adjustbox}{max width=1\\textwidth}\n\\begin{tabular}{lllllll}\n\\midrule\n\\midrule\nDiffusion &\n\\multicolumn{2}{l}{Parent $\\rp{p}{1}\\sim\\FoxSD{\\BB{\\pDefine{O}}_1,\\BB{\\pDefine{P}}_1}$} &\n\\multicolumn{2}{l}{Directing $\\rp{d}{1}\\sim\\FoxD{\\BB{\\pDefine{O}}_2,\\BB{\\pDefine{P}}_2}$} &\n&\n\n\\\\[-0.1cm]\n\n$\\rp{h}{t}$ &\n$\\BB{\\pDefine{O}}_1$ \n$\\BB{\\pDefine{P}}_1$ \n$\\BB{\\pDefine{O}}_2$ \n$\\BB{\\pDefine{P}}_2$ \n$\\omega_1$ &\n$\\omega_2$ \n\n\n\\\\\n\n\\midrule\n\nST-FD &\n$\\left(1,1,2,2\\right)$ &\n$\\Bigl(\\frac{2}{\\alpha},1,\\left(1-\\frac{1}{\\alpha},\\frac{1}{2}\\right),\\left(0,\\frac{1}{2}\\right),\\left(\\frac{1}{\\alpha},\\frac{1}{2}\\right),\\left(1,\\frac{1}{2}\\right)\\Bigr)$ &\n$\\left(1,0,1,1\\right)$ &\n$\\left(1,1,1-\\beta,0,\\beta,1\\right)$ &\n$1\/\\alpha$ &\n$\\beta$ \n\n\\\\ \\midrule\n\nS-FD &\n$\\left(1,1,2,2\\right)$ &\n$\\Bigl(\\frac{2}{\\alpha},1,\\left(1-\\frac{1}{\\alpha},\\frac{1}{2}\\right),\\left(0,\\frac{1}{2}\\right),\\left(\\frac{1}{\\alpha},\\frac{1}{2}\\right),\\left(1,\\frac{1}{2}\\right)\\Bigr)$ &\n$\\left(0,1,1,1\\right)$ &\n$\\left(\\frac{\\cos\\left(\\frac{\\pi\\beta}{2}\\right)^{\\alpha_d}}{\\beta},\\cos\\left(\\frac{\\pi\\beta}{2}\\right)^{\\beta},1-\\frac{1}{\\beta},0,\\frac{1}{\\beta},1\\right)$ &\n$1\/\\alpha$ &\n$1\/\\beta$\n\n\\\\ \\midrule\n\nT-FD &\n$\\left(1,0,1,1\\right)$ &\n$\\Bigl(1,1,\\frac{1}{2},0,\\frac{1}{2},1\\Bigr)$ &\n$\\left(1,0,1,1\\right)$ &\n$\\left(1,1,1-\\beta,0,\\beta,1\\right)$ &\n$1\/2$ &\n$\\beta$\n\n\\\\ \\midrule\n\nEK-FD &\n$\\left(1,0,0,1\\right)$ &\n$\\Bigl(\\frac{1}{2\\sqrt{\\pi}},\\frac{1}{2},\\text{--},0,\\text{--},\\frac{1}{2}\\Bigr)$ &\n$\\left(1,0,1,1\\right)$ &\n$\\left(1,1,1-\\beta,0,\\beta,1\\right)$ &\n$1\/2$ &\n$\\alpha$\n\n\\\\ \\midrule\n\nGBM &\n$\\left(1,0,0,1\\right)$ &\n$\\Bigl(\\frac{1}{2\\sqrt{\\pi}},\\frac{1}{2},\\text{--},0,\\text{--},\\frac{1}{2}\\Bigr)$ &\n$\\left(1,0,1,1\\right)$ &\n$\\left(1,1,1-\\beta,0,\\beta,1\\right)$ &\n$1\/2$ &\n$\\beta$\n\n\\\\ \n\\midrule\n\nFBM &\n$\\left(1,0,0,1\\right)$ &\n$\\Bigl(\\frac{1}{2\\sqrt{\\pi}},\\frac{1}{2},\\text{--},0,\\text{--},\\frac{1}{2}\\Bigr)$ &\n$\\left(0,0,0,0\\right)$ &\n$\\left(1,1,\\text{--},\\text{--},\\text{--},\\text{--}\\right)$ &\n$\\alpha\/2$ &\n$-$\n\n\n\\\\ \\midrule\n\nBM &\n$\\left(1,0,0,1\\right)$ &\n$\\Bigl(\\frac{1}{2\\sqrt{\\pi}},\\frac{1}{2},\\text{--},0,\\text{--},\\frac{1}{2}\\Bigr)$ &\n$\\left(0,0,0,0\\right)$ &\n$\\left(1,1,\\text{--},\\text{--},\\text{--},\\text{--}\\right)$ &\n$1\/2$ &\n$-$\n\n\n\\\\\n\\midrule\n\\midrule\n\\end{tabular}\n\n\\end{adjustbox}\n\n\\begin{tablenotes}\n\n\\item[\\hspace{0.5cm}(Note)]\n\n$\\alpha \\in \\left(0,2\\right]$, $\\beta \\in \\left(0,1\\right)$,\n$\\omega_1,\\omega_2\\in \\mathbbmss{R}_{++}$\n\n\\end{tablenotes}\n\n\\end{threeparttable}\n\n\\end{table}\n\n\n\\subsection{Standard $H$-diffusion}\nIn this subsection, we define SHD coined with special cases of the $H$-process, namely, symmetric L\\'evy stable, $M$-Wright, and one-sided L\\'evy  stable processes \\cite{PMM:13:PTRSA,PP:16:FCAA, MTM:08:PA}. Furthermore, SHD allows to model the FBM and BM as special cases, which have a shared boundary with $H$-diffusion. \n\n\\begin{definition}[Standard $H$-Diffusion]\t\t\\label{def:StdHdiffuion}\nLet\n\\begin{align}\n\\left\\{\\rp{x}{t}; t \\geq 0 \\right\\} \\sim \\left\\{\\FoxV{1:1}{1:0}{2:1}{2:1}{\\BB{\\pDefine{P}}_{\\left(\\alpha_1\\right)}\\Ket{\\beta_1},\\BB{\\pDefine{P}}_{\\left(\\alpha_2\\right)}\\Ket{\\beta_2};\n\\omega_1,\\omega_2}\\right\\}\n\\end{align}\nwith the parameter sequences\n\\begin{align}\n\\BB{\\pDefine{P}}_{\\left(\\alpha_1\\right)}\n&=\n\t\\Bigl(\n\t\t\\tfrac{2}{\\alpha_1},\n\t\t1,\n\t\t\\bigl(1-\\tfrac{1}{\\alpha_1},\\tfrac{1}{2}\\bigr),\n\t\t\\bigl(0,\\tfrac{1}{2}\\bigr),\n\t\t\\bigl(\\tfrac{1}{\\alpha_1},\\tfrac{1}{2}\\bigr),\n\t\t\\bigl(1,\\tfrac{1}{2}\\bigr)\n\t\\Bigr)\n\\nonumber \\\\\n\\BB{\\pDefine{P}}_{\\left(\\alpha_2\\right)}\n&=\n\t\\Bigl(\n\t\t1,\n\t\t1,\n\t\t1-\\alpha_2,\n\t\t0,\n\t\t\\alpha_2,\n\t\t1\n\t\\Bigr)\n\\end{align}\nfor $\\alpha_1 \\in \\left(0,2\\right]$, $\\alpha_2 \\in \\left(0,1\\right]$, and $\\beta_1,\\beta_2\\in \\mathbbmss{R}_{++}$ be SHD. Then, the molecule's position $\\rp{x}{t}$ governed by SHD is the symmetric $H$-variate\n\\begin{align}\n\t\\rp{x}{t}\n\t&\\sim\n\t\t\\FoxSV{2}{1}{3}{3}{\\hat{\\BB{\\pDefine{P}}}_{\\left(\\omega_1,\\alpha_1,\\alpha_2\\right)}\\Ket{\\beta_1\\beta_2^{\\omega_1}t^{\\omega_1\\omega_2}}}\n\\end{align}\nwhere the parameter sequence $\\hat{\\BB{\\pDefine{P}}}_{\\left(\\omega_1,\\alpha_1,\\alpha_2\\right)}$ is given by\n\\begin{align}\n\\hat{\\BB{\\pDefine{P}}}_{\\left(\\omega_1,\\alpha_1,\\alpha_2\\right)}=\n\\left(\n\t\\tfrac{2}{\\alpha_1},\n\t1,\n\t\\Bigl(1-\\tfrac{1}{\\alpha_1},1-\\omega_1 \\alpha_2,\\tfrac{1}{2}\\Bigr),\n\t\\Bigl(0,1-\\omega_1,\\tfrac{1}{2}\\Bigr),\n\t\\Bigl(\\tfrac{1}{\\alpha_1},\\omega_1\\alpha_2,\\tfrac{1}{2}\\Bigr),\n\t\\Bigl(1,\\omega_1,\\tfrac{1}{2}\\Bigr)\n\\right).\n\\end{align}\n\\end{definition}\n\n\n\\begin{remark}\t\\label{rem:sHD}\nThe symmetric $H$-variate $\\rp{p}{1}$ in SHD follows a stable distribution with a characteristic exponent $\\alpha_1$ and scaling parameter $\\beta_1^{\\alpha_1}$, i.e., $\\rp{p}{1} \\sim \\Stable{\\alpha_1}{0}{\\beta_1^{\\alpha_1}}{0}$ (see Appendix~\\ref{sec:appendix:Stable}), while the $H$-variate $\\rp{d}{1}$\nis distributed as an $M$-Wright function with the parameter $\\alpha_2$ and scaling parameter $\\beta_2$ \\cite{MMP:10:JDE}. Note that $\\rp{d}{1}$ can be obtained from the extremal nonnegative strictly stable random variable $\\rv{s}\\sim\\Stable{\\alpha_2}{1}{\\cos\\left(\\pi\\alpha_2\/2\\right)\/\\beta_2}{0}$ such that\n\\begin{align}\n\\rp{d}{1} \\sim \\rv{s}^{-\\alpha_2}.\n\\end{align}\nThe stable distribution can explain the heavy-tailed distribution of the stochastic jump process. The $M$-Wright function can be used as a generalization of the Gaussian density for fractional diffusion processes, which plays the key role to describe both slow and fast types of diffusion phenomena in anomalous diffusion \\cite{Pag:13:FCAA, MMP:10:JDE, PS:14:CAIM}.\\footnote{The nonnegative stable random variable is also known as a possible solution for the waiting time distribution in CTRW. Specifically, the Mittag-Leffler distribution, which has a stretched (natural generalization of) exponential distribution, is used for the waiting time distribution in CTRW as a special case \\cite{MRGS:00:PA, Cah:13:CSTM, GKMR:14:Book,Lin:98:SPI}.} Therefore, SHD can well describe the various typical anomalous diffusion models with unit scaling parameters $\\beta_1=\\beta_2=1$: i) ST-FD with a set of parameters $\\left(\\alpha_1,\\alpha_2,\\omega_1,\\omega_2\\right)=\\left(\\alpha,\\beta,1\/\\alpha,\\beta\\right)$; ii) EK-FD with $\\left(\\alpha_1,\\alpha_2,\\omega_1,\\omega_2\\right)=\\left(2,\\beta,1\/2,\\alpha\\right)$; iii) grey Brownian motion when $\\left(\\alpha_1,\\alpha_2,\\omega_1,\\omega_2\\right)=\\left(2,\\beta,1\/2,\\beta\\right)$; and iv) standard normal diffusion (Brownian motion) with $\\left(\\alpha_1,\\alpha_2,\\omega_1,\\omega_2\\right)=\\left(2,1,1\/2,1\\right)$. A Venn diagram for anomalous diffusion sets with their relationship is shown in Fig.~\\ref{fig:shd}. SHD covers the shaded area.   \n\\end{remark}\n\n\\begin{figure}\n    \\centerline{\\includegraphics[width=0.55\\textwidth]{shd.eps}}\n   \n    \\caption{\n        A Venn diagram displays anomalous diffusion sets with their relationship. SHD covers the shaded area.  \n    }\n    \\label{fig:shd}\n\\end{figure}\n\n\n\\begin{remark}[Role of Scaling Parameters and Diffusion Coefficient]\nTwo positive scaling parameters $\\beta_1$ and $\\beta_2$ in SHD are determined by the diffusion medium. For example, the diffusion equation for ST-FD is well established with the diffusion coefficient $K$, which acts as a scaling factor on the spatial density function. Under our framework, SHD can connect to an equivalent fractional differential diffusion equation form as\n\\begin{align}   \\label{eq:gdiff}\n    \\frac{\\partial ^{\\alpha_2}}{\\partial t^{\\alpha_2}} g\\left(x,t\\right)\n    &=\n       \\beta_1^{\\alpha_1} \\beta_2\\, \\frac{\\partial ^{\\alpha_1}}{\\partial \\sNorm{x}^{\\alpha_1}} g\\left(x,t\\right)\n\\end{align}\nwhere the fundamental solution $g\\left(x,t\\right)$ can be obtained from the Fourier transform of $\\rp{p}{1}$ and the Laplace transform of $\\rv{s}$ in Remark~\\ref{rem:sHD}. In this case, the diffusion coefficient $K=\\beta_1^{\\alpha_1} \\beta_2$ \\cite{VIKH:08:OC}.\n\\end{remark}\n\n\n\n\n\n\\iffalse\n\\begin{table}[t]\n\\caption{Typical Anomalous Diffusion Models as Special Cases of $H$-Diffusion\n} \\centering\n\\label{table:TDM:HD}\n\n\\begin{threeparttable}\n\n\\begin{adjustbox}{max width=0.9\\textwidth}\n\\begin{tabular}{lllllll}\n\\toprule\nDiffusion &\n\\multicolumn{2}{l}{Parent $\\sNorm{\\rp{p}{1}}\\sim\\FoxD{\\BB{\\pDefine{O}}_1,\\BB{\\pDefine{P}}_1}$} &\n\\multicolumn{2}{l}{Directing $\\rp{d}{1}\\sim\\FoxD{\\BB{\\pDefine{O}}_2,\\BB{\\pDefine{P}}_2}$} &\n&\n\n\\\\[-0.1cm]\n\n$\\rp{h}{t}$ &\n$\\BB{\\pDefine{O}}_1$ \n$\\BB{\\pDefine{P}}_1$ \n$\\BB{\\pDefine{O}}_2$ \n$\\BB{\\pDefine{P}}_2$ \n$\\omega_1$ &\n$\\omega_2$ \n\n\n\\\\\n\n\\midrule\n\\midrule\n\nST-FD &\n$\\left(1,1,2,2\\right)$ &\n$\\Bigl(\\frac{2}{{\\alpha_\\mathrm{st}}},1,\\left(1-\\frac{1}{{\\alpha_\\mathrm{st}}},\\frac{1}{2}\\right),\\left(0,\\frac{1}{2}\\right),\\left(\\frac{1}{{\\alpha_\\mathrm{st}}},\\frac{1}{2}\\right),\\left(1,\\frac{1}{2}\\right)\\Bigr)$ &\n$\\left(1,0,1,1\\right)$ &\n$\\left(1,1,1-{\\beta_\\mathrm{st}},0,{\\beta_\\mathrm{st}},1\\right)$ &\n$1\/{\\alpha_\\mathrm{st}}$ &\n${\\beta_\\mathrm{st}}$ \n\n\\\\ \\midrule\n\nS-FD &\n$\\left(1,1,2,2\\right)$ &\n$\\Bigl(\\frac{2}{{\\alpha_\\mathrm{st}}},1,\\left(1-\\frac{1}{{\\alpha_\\mathrm{st}}},\\frac{1}{2}\\right),\\left(0,\\frac{1}{2}\\right),\\left(\\frac{1}{{\\alpha_\\mathrm{st}}},\\frac{1}{2}\\right),\\left(1,\\frac{1}{2}\\right)\\Bigr)$ &\n$\\left(0,0,0,0\\right)$ &\n$\\left(1,1,\\text{--},\\text{--},\\text{--},\\text{--}\\right)$ &\n$1\/{\\alpha_\\mathrm{st}}$ &\n$1$\n\n\\\\ \\midrule\n\nT-FD &\n$\\left(1,0,1,1\\right)$ &\n$\\Bigl(1,1,\\frac{1}{2},0,\\frac{1}{2},1\\Bigr)$ &\n$\\left(1,0,1,1\\right)$ &\n$\\left(1,1,1-{\\beta_\\mathrm{st}},0,{\\beta_\\mathrm{st}},1\\right)$ &\n$1\/2$ &\n${\\beta_\\mathrm{st}}$\n\n\\\\ \\midrule\n\nEK-FD &\n$\\left(1,0,0,1\\right)$ &\n$\\Bigl(\\frac{1}{2\\sqrt{\\pi}},\\frac{1}{2},\\text{--},0,\\text{--},\\frac{1}{2}\\Bigr)$ &\n$\\left(1,0,1,1\\right)$ &\n$\\left(1,1,1-{\\beta_\\mathrm{ek}},0,{\\beta_\\mathrm{ek}},1\\right)$ &\n$1\/2$ &\n${\\alpha_\\mathrm{ek}}$\n\n\\\\ \\midrule\n\nGBM &\n$\\left(1,0,0,1\\right)$ &\n$\\Bigl(\\frac{1}{2\\sqrt{\\pi}},\\frac{1}{2},\\text{--},0,\\text{--},\\frac{1}{2}\\Bigr)$ &\n$\\left(1,0,1,1\\right)$ &\n$\\left(1,1,1-{\\alpha_\\mathrm{ek}},0,{\\alpha_\\mathrm{ek}},1\\right)$ &\n$1\/2$ &\n${\\alpha_\\mathrm{ek}}$\n\n\\\\ \\midrule\n\nFBM &\n$\\left(1,0,0,1\\right)$ &\n$\\Bigl(\\frac{1}{2\\sqrt{\\pi}},\\frac{1}{2},\\text{--},0,\\text{--},\\frac{1}{2}\\Bigr)$ &\n$\\left(0,0,0,0\\right)$ &\n$\\left(1,1,\\text{--},\\text{--},\\text{--},\\text{--}\\right)$ &\n$1\/2$ &\n${\\alpha_\\mathrm{ek}}$\n\n\\\\ \\midrule\n\nBM &\n$\\left(1,0,0,1\\right)$ &\n$\\Bigl(\\frac{1}{2\\sqrt{\\pi}},\\frac{1}{2},\\text{--},0,\\text{--},\\frac{1}{2}\\Bigr)$ &\n$\\left(0,0,0,0\\right)$ &\n$\\left(1,1,\\text{--},\\text{--},\\text{--},\\text{--}\\right)$ &\n$1\/2$ &\n$1$\n\n\\\\\n\n\\bottomrule\n\\end{tabular}\n\n\\end{adjustbox}\n\n\\begin{tablenotes}\n\n\\item[(Note)]\n\n${\\alpha_\\mathrm{st}}, {\\alpha_\\mathrm{ek}} \\in \\left(0,2\\right]$, ${\\beta_\\mathrm{st}}, {\\beta_\\mathrm{ek}} \\in \\left(0,1\\right]$,\n$\\omega_1,\\omega_2\\in \\mathbbmss{R}_{++}$,\n$\\Fox{0}{0}{0}{0}{z;\\left(k,1,\\text{--},\\text{--},\\text{--},\\text{--}\\right)}=k \\delta\\left(z-1\\right)$, $k \\in \\mathbbmss{R}_{++}$\n\n\\end{tablenotes}\n\n\\end{threeparttable}\n\n\\end{table}\n\\fi\n\n\n\n\\section{$H$-Noise Modeling}\nIn this section, we characterize the effect of \\emph{molecular noise}---influences on the conversation between two nanomachines.  Since the random motion of the molecule emitted from the TN directly effects the uncertainty of absorbing time at the RN, the modeling of molecular noise to unveil the intrinsic characteristic of molecules' movements governed by anomalous diffusion laws is important in designing a molecular communication system.\n\n\\subsection{$H$-Noise Model}\n\nLet $\\rv{s}$ be the random released time of molecules at the TN. Then, the arrival time $\\rv{y}$ of the molecule at the RN located away from the TN is \n\\begin{align}\t\\label{eq:at}\n\\rv{y}=\\rv{s}+\\rv{t}\n\\end{align}\nwhere $\\rv{t}$ is an additional random time of the molecule to arrive at the RN. This additional random time $\\rv{t}$ evidently plays the role of an \\emph{additive} random noise in molecular communication. Since the RN acts as a boundary with a perfect absorbing process, the noise $\\rv{t}$ can be thought of as a \\emph{FPT} such that the molecule emitted from the TN reaches the boundary for the first time. This random noise has been unveiled as a L\\'evy distribution  \\cite{NOL:12:CL, KLYFEC:16:JSAC,MMM:16:NB,HAASG:17:CL}, an inverse Gaussian distribution \\cite{SEA:12:IT,LZMY:17:CL}, a stable distribution \\cite{FMGCEG:19:MBSC}, and (or more generally) $H$-variate \\cite{CTJS:15:CL,TJSW:18:COM} for various diffusion scenarios. \n\nWe begin by deriving the FPT of the molecule governed by $H$-diffusion. Then we introduce a general class of molecular noise---namely, \\emph{$H$-noise}---to develop a unifying framework for characterizing  statistical properties of uncertainty or distribution of random propagation time.\n\n\\begin{theorem}[First Passage Time]\t\t\\label{thm:fpt:Hdiffusion}\nLet $\\left\\{\\rp{x}{t}; t \\geq 0 \\right\\} \\sim \\left\\{\\FoxSV{\\pDefine{m}_1:\\pDefine{m}_2}{\\pDefine{n}_1:\\pDefine{n}_2}{\\pDefine{p}_1:\\pDefine{p}_2}{\\pDefine{q}_1:\\pDefine{q}_2}{\\BB{\\pDefine{P}}_1,\\BB{\\pDefine{P}}_2;\\omega_1,\\omega_2}\\right\\}$ and $\\rv{t}$ be an FPT, which is defined such that the molecule starting at $x=0$ reaches distance $x=a$, $a\\in \\mathbbmss{R}_{+}$ for the first time: \n\\begin{align}\n\t\\rv{t}=\n\t\\inf\\left\\{t: x\\left(t\\right)=a\\right\\}.\n\\end{align}\nGiven an initial condition $\\PDF{\\rp{x}{0}}{x}=\\delta\\left(x\\right)$ and a boundary condition $\\PDF{\\rp{x}{t}}{a}=0$ for the absorbing process, the FPT of the molecule in $H$-diffusion is the $H$-variate:\\footnote{The FPT of molecules highly depends on the boundary condition. It is nontrivial to find an accurate statistic of FPT with an arbitrary boundary condition in multi-dimensional space. See \\cite{Red:01:Book,TJS:19:ACCESS}.}\n\\begin{align}\t\\label{eq:foxv:fpt}\n\\rv{t}\n\\sim\n\t\\FoxV{\\pDefine{n}_1+\\pDefine{n}_2}{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{q}_1+\\pDefine{q}_2}{\\pDefine{p}_1+\\pDefine{p}_2}\n\t{\n\t\t\\BB{\\pDefine{P}}_\\rv{t}\n\t\t\\Ket{\\left(a \\pDefine{c}_{\\left(\\omega_1\\right)} \\right)^{1\/\\left(\\omega_1\\omega_2\\right)}}\n\t}\n\\end{align}\nwhere the parameter sequence $\\BB{\\pDefine{P}}_\\rv{t}$ is given b\n\\begin{align}\n\\BB{\\pDefine{P}}_\\rv{t}\n&=\n\t\\Bigl(\n\t\t\\tfrac{\\pDefine{k}_{\\left(\\omega_1\\right)} }{{\\pDefine{c}_{\\left(\\omega_1\\right)} }},\n\t\t1,\n\t\n\t\t\\left(\\B{1}_{\\pDefine{q}_1+\\pDefine{q}_2}-\\BB{\\pDefine{b}}_{\\left(\\omega_1\\right)}-\\BB{\\pDefine{B}}_{\\left(\\omega_1\\right)}-\\tfrac{1}{\\omega_1\\omega_2}\\BB{\\pDefine{B}}_{\\left(\\omega_1\\right)}\\right),\n\t\t\t\\nonumber \\\\\n\t&\\hspace{3cm}\n\t\t\\left(\\B{1}_{\\pDefine{p}_1+\\pDefine{p}_2}-\\BB{\\pDefine{a}}_{\\left(\\omega_1\\right)}-\\BB{\\pDefine{A}}_{\\left(\\omega_1\\right)}-\\tfrac{1}{\\omega_1\\omega_2}\\BB{\\pDefine{A}}_{\\left(\\omega_1\\right)}\\right), \n\t\t\\tfrac{1}{\\omega_1\\omega_2}\\BB{\\pDefine{B}}_{\\left(\\omega_1\\right)},\n\t\t\\tfrac{1}{\\omega_1\\omega_2}\\BB{\\pDefine{A}}_{\\left(\\omega_1\\right)}\n\t\\Bigr).\n\\end{align} \n\n\n\\begin{proof}\nWith the absorbing boundary condition $\\PDF{\\rp{x}{t}}{a}=0$, the density function of the position of molecule $x$ at time $t$, denoted by $\\PDF{\\rp{\\tilde{x}}{t}}{x}$ for $x<a$, can be found using the image method \\cite[Section~3.2]{Red:01:Book} as\n\\begin{align}\t\\label{eq:barw}\n\t\\PDF{\\rp{\\tilde{x}}{t}}{x}\n\t&=\n\t\t\\PDF{\\rp{x}{t}}{x}-\\PDF{\\rp{x}{t}}{-x+2a}.\n\\end{align}\nLet \n$\nS_{\\rv{t}}\\left(t\\right)\\triangleq\n\t\t\\int_{-\\infty}^{a}\n\t\t\\PDF{\\rp{\\tilde{x}}{t}}{x}\n\t\tdx\n$\nbe the \\emph{survival probability} that the molecule is located at $x<a$ for all times up to $t$. Then, we have\n\\begin{align}\n\t\\PDF{\\rv{t}}{t}\n\t&\\overset{\\text{(a)}}{=}\n\t\t-\\frac{\\partial S_{\\rv{t}}\\left(t\\right)}{\\partial t}\n\t\\nonumber \\\\\n\t&\n\t\\overset{\\text{(b)}}{=}\n\t\t-\n\t\t\\frac{\\partial}{\\partial t}\n\t\t\\int_{-\\infty}^{a\/t^{\\omega_1\\omega_2}}\n\t\t\\PDF{\\sNorm{\\rp{x}{1}}}{y}\n\t\tdy\n\t\t+\n\t\t\\frac{\\partial}{\\partial t}\n\t\t\\int_{a\/t^{\\omega_1\\omega_2}}^{\\infty}\n\t\t\\PDF{\\sNorm{\\rp{x}{1}}}{y}\n\t\tdy\n\t\\nonumber \\\\\n\t&\\overset{\\text{(c)}}{=}\n\t\t-2\n\t\t\\frac{\\partial}{\\partial t}\n\t\t\\int_{0}^{a\/t^{\\omega_1\\omega_2}}\n\t\t\\PDF{\\sNorm{\\rp{x}{1}}}{y}\n\t\tdy\n\t\\nonumber \\\\\n\t&\n\t=\n\t\t\\frac{2a\\omega_1\\omega_2}{t^{\\omega_1\\omega_2+1}}\n\t\t\\PDF{\\sNorm{\\rp{x}{1}}}{\\frac{a}{t^{\\omega_1\\omega_2}}}\n\\end{align}\nwhere (a) follows from the relationship between the FPT and survival probability as $\\PDF{\\rv{t}}{t}dt=S_\\rv{t}\\left(t\\right)-S_\\rv{t}\\left(t+dt\\right)$ with the first-order Taylor approximation \\cite{Red:01:Book}; (b) is obtained from the self-similar property of $\\rp{x}{t}$ in \\eqref{eq:barw} and by making changing the  variable $x\/t^{\\omega_1\\omega_2}=y$; and (c) follows from the fact that $\\PDF{\\sNorm{\\rp{x}{1}}}{x}$ has a symmetric relation. Finally, using the modulation property of $H$-function \\cite[Table~II]{JSW:15:IT}, the desired result \\eqref{eq:foxv:fpt} is obtained.\n\\end{proof}\n\\end{theorem}\n\n\n\\begin{definition}[$H$-Noise]\t\\label{def:Hnoise}\nThe FPT $\\rv{t}$ acts as the \\emph{unique} source of uncertainty or distortion in molecular communication. We refer to the FPT of $H$-diffusion molecules in Theorem~\\ref{thm:fpt:Hdiffusion} as the \\emph{$H$-noise}. Specifically, we define the FPT of SHD molecules in Definition~\\ref{def:StdHdiffuion} as the \\emph{standard $H$-noise}. \n\\end{definition}\n\n\\begin{remark}[Standard $H$-noise] \nThe  standard $H$-noise, denoted by $\\rv{t}_\\mathrm{sHn}$, is the $H$-variate\n\\begin{align}\n\\rv{t}_\\mathrm{sHn}\n\\sim\n\t\\FoxV{1}{2}{3}{3}\n\t{\\BB{\\pDefine{P}}_\\mathrm{sHn}\\Ket{\\left(\\frac{a}{\\beta_1\\beta_2^{\\omega_1}}\\right)^{1\/\\left(\\omega_1\\omega_2\\right)}}}\n\\end{align}\nwhere the parameter sequence $\\BB{\\pDefine{P}}_\\mathrm{sHn}$ is\n\\begin{align}\n\\BB{\\pDefine{P}}_\\mathrm{sHn}\n&=\n\\Bigl(\n\t\\tfrac{2}{\\alpha_1},\n\t1,\n\t\\left(-\\tfrac{1}{\\omega_2},-\\tfrac{1}{\\omega_1\\omega_2},-\\tfrac{1}{2\\omega_1\\omega_2}\\right),\n\t\\left(-\\tfrac{1}{\\alpha_1\\omega_1\\omega_2},-\\tfrac{1}{2\\omega_1\\omega_2},-\\tfrac{\\alpha_2}{\\omega_2}\\right),\n\t\\nonumber \\\\\n&\\hspace{5.5cm}\n\t\\left(\\tfrac{1}{\\omega_2},\\tfrac{1}{\\omega_1\\omega_2},\\tfrac{1}{2\\omega_1\\omega_2}\\right),\n\t\\left(\\tfrac{1}{\\alpha_1\\omega_1\\omega_2},\\tfrac{1}{2\\omega_1\\omega_2},\\tfrac{\\alpha_2}{\\omega_2}\\right)\n\\Bigr).\n\\end{align}\n\\end{remark}\n\n\n\\subsection{$H$-Noise Power}\n\nSince the \\emph{algebraic-tailed} random variable $\\rv{z}$ exhibits finite absolute moments only for the order $\\ell$ less than $\\eta$, i.e., $\\mathbb{E}\\bigl\\{\\left|\\rv{z}\\right|^\\ell\\bigr\\}< \\infty$ for $\\ell<\\eta$, the second moment that is widely used as a standard signal power measure does not exist for the class of algebraic distributions when $\\eta<2$. The fractional lower-order statistics (FLOS) theory is a useful tool to measure and characterize the behavior of impulsive signals with an algebraic distribution \\cite{MN:96:SP}, but it cannot provide a universal framework for characterization of algebraic-tailed processes. \nIn this subsection, we provide a concept of \\emph{$H$-noise power} using the finite logarithm moments based on zero-order statistics \\cite[Theorem~1]{GPA:06:SP}.\n\n\n\\begin{theorem}[$H$-Noise Tails]\t\\label{thm:Hnoise:tails}\nLet \n\\begin{align}\t\\label{def:Hnoise:tails}\n\t\\kappa\n\t\\triangleq\n\t\\lim_{t\\rightarrow \\infty}\n\t\\frac{\n\t\t-\\log S_{\\rv{t}}\\left(t\\right)\n\t\t}\n\t\t{\n\t\t\\log t}\n\\end{align}\nbe the tail constant of $\\rv{t}$. Then, we have\n\\begin{align}\n\\kappa\n=\n\t\\omega_1\\omega_2\\cdot\n\t\\min_{j=1,\\ldots,\\pDefine{m}_1+\\pDefine{m}_2}\n\t\\left\\{\n\t\t\t1+\\frac{\\Re\\left(\\BB{\\pDefine{b}}_{\\left(\\omega_1\\right),j}\\right)}{\n\t\t\t\\BB{\\pDefine{B}}_{{\\left(\\omega_1\\right)},j}\n\t\t\t}\n\t\\right\\}\n\\end{align}\nwhere $\\BB{\\pDefine{b}}_{\\left(\\omega_1\\right),j}$ and $\\BB{\\pDefine{B}}_{{\\left(\\omega_1\\right)},j}$ are the $j$th element of the sequences $\\BB{\\pDefine{b}}_{\\left(\\omega_1\\right)}$ and $\\BB{\\pDefine{B}}_{{\\left(\\omega_1\\right)}}$, respectively.\n\\begin{proof}\nUsing the formulation for the CDF of $H$-variate \\cite[Table~VI]{JSW:15:IT}, we have\n\\begin{align}\t\\label{eq:cdf:Hn}\n\tS_{\\rv{t}}\\left(t\\right)\n\t&=\n\t\\Prob{\\rv{t}>t}\n\t\\nonumber \\\\\n\n&=\n\t\t1-\\FoxHT{\\pDefine{n}_1+\\pDefine{n}_2,\\pDefine{m}_1,\\pDefine{m}_2}{\\pDefine{q}_1+\\pDefine{q}_2,\\pDefine{p}_1+\\pDefine{p}_2}\n\t\t{\n\t\t\t\\Bra{1}\n\t\t\t\\left(\\BB{\\pDefine{P}}_\\rv{t}\\Ket{\\left(a\\pDefine{c}_{\\left(\\omega_1\\right)}\\right)^{1\/\\left(\\omega_1\\omega_2\\right)}}\\right)\n\t\t}\n\t\t{\\frac{1}{t}\\IndF{\\left[0,1\\right]}{t}}\n\t\t{t}\n\t\\nonumber \\\\\n\t&=\n\t\t\\Fox{\\pDefine{n}_1+\\pDefine{n}_2+1}{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{q}_1+\\pDefine{q}_2+1}{\\pDefine{p}_1+\\pDefine{p}_2+1}\n\t\t{\\frac{t}{\\left(a\\pDefine{c}_{\\left(\\omega_1\\right)}\\right)^{1\/\\left(\\omega_1\\omega_2\\right)}};\\canOP{\\pSeq_\\FontDef{cdf}^{-1}}\n\t\t\t{\n\t\t\t\t\\Bra{1}\n\t\t\t\t\\BB{\\pDefine{P}}_\\rv{t}\n\t\t\t}\n\t\t}.\n\\end{align}\nThen, using the algebraic asymptotic expansion of the $H$-function \\cite[Proposition~3]{JSW:15:IT}, we get\n\\begin{align}\nS_{\\rv{t}}\\left(t\\right)\n&\\doteq\n\t\t\\Fox{\\pDefine{n}_1+\\pDefine{n}_2+1}{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{q}_1+\\pDefine{q}_2+1}{\\pDefine{p}_1+\\pDefine{p}_2+1}\n\t\t{t;\\canOP{\\pSeq_\\FontDef{cdf}^{-1}}\n\t\t\t{\n\t\t\t\t\\Bra{1}\n\t\t\t\t\\BB{\\pDefine{P}}_\\rv{t}\t\t\t\n\t\t\t}}\n\\nonumber\\\\\n&\n\\doteq\n\t\tt^{-\\omega_1\\omega_2\\cdot\n\t\t\\min_{j=1,\\ldots,\\pDefine{m}_1+\\pDefine{m}_2}\n\t\t\\left\\{\n\t\t\t1+\n\t\t\t\\frac{\\Re\\left(\\BB{\\pDefine{b}}_{{\\left(\\omega_1\\right)},j}\\right)}{\n\t\t\t\\BB{\\pDefine{B}}_{{\\left(\\omega_1\\right)},j}\n\t\t\t}\n\t\t\\right\\}\n\t\t}\n\\end{align}\nwhich completes the proof.\n\\end{proof}\n\\end{theorem}\n\n\\begin{remark}[Algebraic-Tailed or Heavy-Tailed Distribution]\nSince $S_{\\rv{t}}\\left(t\\right)$ (also called a tail function $\\Prob{\\rv{t}>t}$) in Theorem~\\ref{thm:Hnoise:tails} has a polynomial decay rate $\\kappa \\in \\mathbbmss{R}_{++}$, the $H$-noise can be said to be an algebraic-tailed random variable \\cite{Arc:04:Book}. In addition, since all algebraic-tailed random variables possess heavier tails than a family of exponential distributions, we can also call it the heavy-tailed distribution. Hence, the $H$-noise $\\rv{t}$ has a finite moment $\\EX{\\rv{t}^\\ell}$ for $\\ell < \\kappa$.\n\\end{remark}\n\n\\begin{remark}[Standard $H$-Noise Tails] \nThe tail constant for the standard $H$-noise $\\kappa_\\mathrm{sHn}$ is \n\\begin{align}\n\\kappa_\\mathrm{sHn}\n&=\n\t\\begin{cases}\n\t\\omega_1\\omega_2, & \\omega_1 < 1 \\\\\n\t\\omega_2, & \\omega_1 \\geq 1.\n\t\\end{cases}\n\\end{align}\n\\end{remark}\n\n\\begin{theorem}[Logarithm Moments of $H$-Noise]\t\\label{thm:LM:Hnoise}\nNote that any algebraic-tailed distribution $\\rv{x}$ has a finite logarithm moment \\cite[Theorem~2.5]{Arc:04:Book}, i.e., $\\EX{\\ln\\left(\\rv{x}\\right)}<\\infty$. Hence, the logarithm moment of $H$-noise $\\rv{t}$ exists for all ranges of parameters and is given by\n\\begin{align}\t\\label{eq:LM:Hnoise:HT}\n\t\\EX{\\ln\\left(\\rv{t}\\right)}\n\t&=\n\t\t\\FoxHT{2,2}{2,2}\n\t\t{\\BB{\\pDefine{P}}_\\mathrm{ln}}\n\t\t{\\left(t-1\\right)\\PDF{\\rv{t}}{t}}\n\t\t{1}\n\t\\\\ \\label{eq:LM:Hnoise:HF}\n\t&=\n\t\t\\Fox{\\pDefine{m}_1+\\pDefine{m}_2+2}{\\pDefine{n}_1+\\pDefine{n}_2+2}{\\pDefine{p}_1+\\pDefine{p}_2+2}{\\pDefine{q}_1+\\pDefine{q}_2+2}\n\t\t{1;\\stdOP{\\BB{\\pDefine{P}}_\\mathrm{ln}}{\\Bra{1\\frac{}{}}\\left(\\BB{\\pDefine{P}}_\\rv{t}}\\Ket{\\left(a\\pDefine{c}_{\\left(\\omega_1\\right)}\\right)^{\\frac{1}{\\omega_1\\omega_2}}}\\right)}\n\t\\nonumber \\\\\n\t&\\hspace{2.5cm}\n\t\t-\n\t\t\\Fox{\\pDefine{m}_1+\\pDefine{m}_2+2}{\\pDefine{n}_1+\\pDefine{n}_2+2}{\\pDefine{p}_1+\\pDefine{p}_2+2}{\\pDefine{q}_1+\\pDefine{q}_2+2}\n\t\t{\\left(a\\pDefine{c}_{\\left(\\omega_1\\right)}\\right)^{\\frac{1}{\\omega_1\\omega_2}};\\stdOP{\\BB{\\pDefine{P}}_\\mathrm{ln}}{\\BB{\\pDefine{P}}_\\rv{t}}}\n\\end{align} \nwhere $\\BB{\\pDefine{P}}_\\mathrm{\\ln}=\\left(1,1,\\left(\\B{0}_2,\\text{--}\\right),\\left(\\B{0}_2,\\text{--}\\right),\\left(\\B{1}_2,\\text{--}\\right),\\left(\\B{1}_2,\\text{--}\\right)\\right)$. For standard $H$-noise, the logarithm moment of $H$-noise in \\eqref{eq:LM:Hnoise:HF} reduces to\n\\begin{align}\t\\label{eq:LM:stdHnoise}\n\t\\EX{\\ln\\left(\\rv{t}_\\mathrm{sHn}\\right)}\n\t=\n\t\t\\left(\n\t\t\t\\frac{\n\t\t\t1-1\/\\alpha_1+\\left(1-\\alpha_2\\right)\\omega_1}{\n\t\t\t\\omega_1\\omega_2}\n\t\t\\right)\n\t\t\\gamma_\\mathrm{e}\n\t\t+\n\t\t\\frac{1}{\\omega_1\\omega_2}\n\t\t\\ln\\left(\\frac{a}{\\beta_1\\beta_2^{\\omega_1}}\\right)\n\\end{align}\nwhere $\\gamma_\\mathrm{e}\\approx 0.57721$ is the Euler-Mascheroni constant.\n\\begin{proof}\nUsing \\eqref{eq:foxv:fpt} and \\cite[Example~4]{JSW:15:IT}, the $H$-transform expression is obtained for the logarithm moment of $H$-noise \\eqref{eq:LM:Hnoise:HT}, from which along with the Mellin operation \\cite[Proposition~4]{JSW:15:IT}, we arrive at the desired result \\eqref{eq:LM:Hnoise:HF}. For standard $H$-noise, using the relation between logarithm moment and derivative of moment such that\n\\begin{align}\n\t\\EX{\\ln\\left(\\rv{t}_\\mathrm{sHn}\\right)}\n\t=\n\t\t\\left.\\frac{\\partial \\EX{\\rv{t}_\\mathrm{sHn}^\\ell}}{\\partial\\ell}\\right|_{\\ell=0}\n\\end{align}\nwe obtained \\eqref{eq:LM:stdHnoise}, which completes the proof.\n\\end{proof}\n\\end{theorem}\n\n\n\\begin{corollary}[Geometric Power of Standard $H$-Noise]\t\\label{cor:gmp:Hnoise}\nLet \n\\begin{align}\n\t\\GMP{\\rv{t}}\n\t\\triangleq\n\t\\exp\\left\\{\\EX{\\ln\\left(\\rv{t}\\right)}\\right\\}\n\\end{align}\nbe the \\emph{geometric power} of random variable $\\rv{t}$. Then, the geometric power of $H$-noise $\\GMP{\\rv{t}}$ can be obtained generally using the logarithm moments of $H$-noise given in \\eqref{eq:LM:Hnoise:HT}. Specifically, for the standard $H$-noise, $\\GMP{\\rv{t}_\\mathrm{sHn}}$ has a compact form of\n\\begin{align}\n\t\\GMP{\\rv{t}_\\mathrm{sHn}}\n\t=\n\t\t\\left(\\frac{a\\mathcal{G}^{1-1\/\\alpha_1+\\left(1-\\alpha_2\\right)\\omega_1}}{\\beta_1\\beta_2^{\\omega_1}}\\right)^{\\frac{1}{\\omega_1\\omega_2}}\n\\end{align}\nwhere $\\mathcal{G}\\triangleq e^{\\gamma_{\\mathrm{e}}}\\approx 1.78107$ denotes the exponential Euler-Mascheroni constant.\\footnote{The measure of geometric power was introduced in \\cite{GPA:06:SP} for the processing and characterization of very impulsive signals with the concept of zero-order statistics. Specifically, the symmetric $\\alpha$-stable distribution $\\rv{s} \\sim \\Stable{\\alpha}%{{\\alpha_\\mathrm{s}}}{0}{\\gamma}%{{\\gamma_\\mathrm{s}}}{0}$ was considered where its geometric power was shown as $\\GMP{\\rv{s}} = \\left(\\mathcal{G}\\gamma}%{{\\gamma_\\mathrm{s}}\\right)^{1\/\\alpha}%{{\\alpha_\\mathrm{s}}}\/\\mathcal{G}$.} \n\\begin{proof}\nIt follows readily from Theorem~\\ref{thm:LM:Hnoise}.\n\\end{proof}\n\n\\end{corollary}\n\n\\begin{remark}[Geometric Mean, Power, and FLOS]\t\\label{rem:GPF}\nThe geometric power $\\GMP{\\rv{t}}$ has a relation to the geometric mean of nonnegative random variable $\\rv{t}$ by\n\\begin{align}\n\\GMP{\\rv{t}}\n&=\\exp\\left\\{ \\lim_{N \\rightarrow \\infty}\\frac{1}{N}\\sum_{i=1}^{N}\\ln\\left(t_i\\right)\n\\right\\}\n\\nonumber \\\\\n&\n=\\lim_{N \\rightarrow \\infty}\\left\\{\\prod_{i=1}^{N} t_i\\right\\}^{1\/N}\n\\end{align}\nwhere $\\left(t_1, \\ldots, t_N\\right)$ is a sequence of independent samples initiated by random variable $\\rv{t}$. Compared to the arithmetic mean, the geometric mean is said to be not overly influenced by the very large values in a skewed distribution. This advantage is appropriate for $H$-noise. Since the geometric power is linked to the geometric mean, we use the square of $\\GMP{\\rv{t}}$ for the \\emph{$H$-noise power}, denoted by $\\mathcal{N}\\left(\\rv{t}\\right)=\\left\\{\\GMP{\\rv{t}}\\right\\}^2$.\\footnote{\nThe geometric power also can be linked to the FLOS method if there exists a sufficiently small value $\\ell$ satisfying \\cite{GPA:06:SP}\n$\n\\GMP{\\rv{t}}\n\t=\\lim_{\\ell \\rightarrow 0} \\left\\{\\EX{\\rv{t}^\\ell}\\right\\}^{1\/\\ell}\n$. \nThis reveals that the geometric power can be used mathematically and conceptually in a rich set of heavy-tailed distributions.\n}\n\\end{remark}\n\n\n\\begin{remark}[Normal Diffusion]\nThe $H$-noise $\\rv{t}$ in Brownian motion without drift has a nonnegative stable distribution with the characteristic exponent $1\/2$ (L{\\'e}vy distribution) where the PDF $\\PDF{\\rv{t}}{t}$ is given by\\footnote{The $H$-noise in Brownian motion with nonzero drift follows an inverse Gaussian distribution \\cite{SEA:12:IT}.}\n\\begin{align}\n\t\\PDF{\\rv{t}}{t}\n\t&=\n\t\\frac{4}{a^2\\sqrt{\\pi}}\\FoxH{0}{1}{1}{0}{\\frac{4t}{a^2}}{\\left(-\\frac{1}{2},1\\right)}{\\text{---}}\n\t\\nonumber \\\\\n\t&\n\t=\n\t\\frac{a}{\\sqrt{4\\pi t^3}}\\exp\\left(-\\frac{a^2}{4t}\\right)\n\\end{align}\nand its corresponding geometric power $\\GMP{\\rv{t}}$ is given by $\\GMP{\\rv{t}} = a^2\\mathcal{G}$ \\cite{FMGCEG:19:MBSC}. \nTable~\\ref{table:Hnoise} shows the $H$-noise $\\rv{t}$ and its geometric power $\\GMP{\\rv{t}}$ for the typical anomalous diffusion models in Table~\\ref{table:TDM:HD}.\n\\end{remark}\n\n\\begin{table}[t]\n\\caption{$H$-Noise $\\rv{t}$ and Its Geometric Power $\\GMP{\\rv{t}}$ for Typical Anomalous Diffusion \nin Table~\\ref{table:TDM:HD}:\n$$\n\t\\rv{t}\\sim\\FoxD{\\BB{\\pDefine{O}},\\BB{\\pDefine{P}}\\Ket{a^{1\/\\omega}}},\\quad \n\t\\GMP{\\rv{t}}=a^{1\/\\omega}\\mathcal{G}^{1\/\\omega-c}\n$$\n} \\centering\n\\label{table:Hnoise}\n\n\n\\begin{threeparttable}\n\\begin{adjustbox}{max width=0.9\\textwidth}\n\\begin{tabular}{lllll}\n\\midrule\n\\midrule\nDiffusion &\n\\multicolumn{2}{l}{$H$-noise $\\rv{t} \\sim \\FoxD{\\BB{\\pDefine{O}},\\BB{\\pDefine{P}}}$} &\n\\multicolumn{2}{l}{Geometric Power $\\GMP{\\rv{t}}$}\n\\\\[-0.1cm]\n\n$\\rp{h}{t}$ &\n$\\BB{\\pDefine{O}}$ \n$\\BB{\\pDefine{P}}$ \n$\\omega$ &\n$c$\n\\\\\n\n\\midrule\n\nST-FD &\n$\\left(1,2,3,3\\right)$ &\n$\\left(\n\t\\frac{2}{\\alpha}, 1,\\left(-\\frac{1}{\\beta},-\\frac{\\alpha}{\\beta},-\\frac{\\alpha}{2\\beta}\\right),\\left(-\\frac{1}{\\beta},-\\frac{\\alpha}{2\\beta},-1\\right),\\left(\\frac{1}{\\beta},\\frac{\\alpha}{\\beta},\\frac{\\alpha}{2\\beta}\\right),\\left(\\frac{1}{\\beta},\\frac{\\alpha}{2\\beta},1\\right)\n\\right)$ &\n$\\beta\/\\alpha$ &\n$1$\n\\\\ \\midrule\n\nS-FD &\n$\\left(1,1,2,2\\right)$ &\n$\\Bigl(\n\t\\frac{2}{\\alpha}, 1,\\left(-\\alpha,-\\frac{\\alpha}{2}\\right),\\left(-1,-\\frac{\\alpha}{2}\\right),\\left(\\alpha,\\frac{\\alpha}{2}\\right),\\left(1,\\frac{\\alpha}{2}\\right)\t\n\\Bigr)$ &\n$1\/\\alpha$ &\n$1$\n\\\\ \\midrule\n\nT-FD &\n$\\left(0,1,1,1\\right)$ &\n$\\Bigl(\n\t1, 1,-\\frac{2}{\\beta},-1,\\frac{2}{\\beta},1\n\\Bigr)$ &\n$\\beta\/2$ &\n$1$\n\\\\ \\midrule\n\nEK-FD &\n$\\left(0,2,2,1\\right)$ &\n$\\Bigl(\\frac{4^{1\/\\alpha}}{\\sqrt{\\pi}},4^{1\/\\alpha},\\left(-\\frac{1}{\\alpha},\\frac{1}{2}-\\frac{1}{\\alpha}\\right),-\\frac{\\beta}{\\alpha},\\frac{1}{\\alpha}\\B{1}_2,\\frac{\\beta}{\\alpha}\n\\Bigr)$ &\n$\\alpha\/2$ &\n$\\beta\/\\alpha$\n\\\\ \\midrule\n\nGBM &\n$\\left(0,2,2,1\\right)$ &\n$\\Bigl(\\frac{4^{1\/\\beta}}{\\sqrt{\\pi}},4^{1\/\\beta},\\left(-\\frac{1}{\\beta},\\frac{1}{2}-\\frac{1}{\\beta}\\right),-1,\\frac{1}{\\beta}\\B{1}_2,1\n\\Bigr)$ &\n$\\beta\/2$ &\n$1$\n\\\\ \\midrule\n\nFBM &\n$\\left(0,1,1,0\\right)$ &\n$\\Bigl(\\frac{4^{1\/\\alpha}}{\\sqrt{\\pi}},4^{1\/\\alpha},\\frac{1}{2}-\\frac{1}{\\alpha},\\text{--},\\frac{1}{\\alpha},\\text{--}\n\\Bigr)$ &\n$\\alpha\/2$ &\n$1\/\\alpha$\n\\\\ \\midrule\n\nBM &\n$\\left(0,1,1,0\\right)$ &\n$\\Bigl(\\frac{4}{\\sqrt{\\pi}},4,-\\frac{1}{2},\\text{--},1,\\text{--}\\Bigr)$ &\n$1\/2$ &\n$1$\n\\\\\n\\midrule\n\\midrule\n\\end{tabular}\n\\end{adjustbox}\n\\end{threeparttable}\n\n\\end{table}\n\n\n\\subsection{Numerical Examples}\n\nIn what follows, we use $\\left(\\alpha_1,\\alpha_2\\right)$-SHD to denote the particularized SHD where the diffusion parameters are $\\omega_1=1\/\\alpha_1$, $\\omega_2=\\alpha_2$, and $\\beta_1\\beta_2^{1\/\\alpha_1}=K^{1\/{\\alpha_1}}$. To exemplify the different types of diffusion scenarios, we consider:\n       i) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,1\\right)$ for normal diffusion; \n       ii) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,0.5\\right)$ for subdiffusion; and  \n       iii) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(1.8,1\\right)$ for superdiffusion.\n \n \\begin{figure}[t!]\n    \\centerline{\\includegraphics[width=0.55\\textwidth]{fig6.eps}}\n   \n\\caption{\n       CDF $\\CDF{\\rv{t}_{\\mathrm{sHn}}}{t}$ of the standard $H$-noise $\\rv{t}_\\mathrm{sHn}$ in the $\\left(\\alpha_1,\\alpha_2\\right)$-SHD at the distance $a=10^{-5}$~[m] for:  \n       i) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,1\\right)$; \n       ii) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,0.5\\right)$; and  \n       iii) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(1.8,1\\right)$;\n       with $K=10^{-10}$~[m$^2$\/s].\n}\n\\label{fig:6}\n\\end{figure}\n\\begin{figure}[t!]\n    \\centerline{\\includegraphics[width=0.55\\textwidth]{fig7.eps}}\n   \n\\caption{\n       Survival probability $S_{\\rv{t}_{\\mathrm{sHn}}}\\left(t\\right)$ of the standard $H$-noise $\\rv{t}_\\mathrm{sHn}$ in the $\\left(\\alpha_1,\\alpha_2\\right)$-SHD for: \n       i) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,1\\right)$; \n       ii) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,0.5\\right)$; and  \n       iii) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(1.8,1\\right)$;\n       with $K=10^{-10}$~[m$^2$\/s]. The black dashed-dotted line stands for the tail constant $\\kappa$ of $H$-noise in~\\eqref{def:Hnoise:tails}.\n       }\n\\label{fig:7}\n\\end{figure}\n\nFig.~\\ref{fig:6} shows the CDF $\\CDF{\\rv{t}_{\\mathrm{sHn}}}{t}$ of the standard $H$-noise $\\rv{t}_\\mathrm{sHn}$ in the $\\left(\\alpha_1,\\alpha_2\\right)$-SHD at distance $a=10^{-5}$~[m]. We can observe that the anomalous diffusions for $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,0.5\\right)$ and $\\left(\\alpha_1,\\alpha_2\\right)=\\left(1.8,1\\right)$ have a large dispersion in propagation compared to the normal diffusion for $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,1\\right)$.\\footnote{The simulation is conducted based on the particle-based simulator \\cite{CTJS:15:CL, TJS:19:ACCESS, TJSW:18:COM, AJS:18:COM}. The simulation of FPT in superdiffusion is overestimated due to the long jump property of  molecules \\cite{TJSW:18:COM}. } To demonstrate the heavy-tailed property of $H$-noise, the survival probability $S_{\\rv{t}_{\\mathrm{sHn}}}\\left(t\\right)$ of the standard $H$-noise $\\rv{t}_\\mathrm{sHn}$ in $\\left(\\alpha_1,\\alpha_2\\right)$-SHD for the three diffusion scenarios  is depicted in Fig.~\\ref{fig:7}. It can be seen from the figure that the $H$-noise distribution follows the asymptotic tail constant scaling behaviours as discussed in Theorem~\\ref{thm:Hnoise:tails}. In this example, the tail constants (slope of black dashed-dotted line in the figure) are equal to $\\kappa=0.5$, $0.25$, and $0.56$ for $\\left(2,1\\right)$-, $\\left(2,0.5\\right)$-, and $\\left(1.8,1\\right)$-SHD, respectively.\n\n\n\n\n\n\n\\begin{figure}[t!]\n    \\subfigure[$a=10^{-5}~\\text{[m]}$]{\n        \\centerline{\\includegraphics[width=0.6\\textwidth]{fig8a.eps}}\n        \\label{fig:8:a}\n    }\\hfill\n    \\subfigure[$a=10^{-8}~\\text{[m]}$]{\n        \\centerline{\\includegraphics[width=0.6\\textwidth]{fig8b.eps}}\n        \\label{fig:8:b}\n    }\\hfill\n    \\subfigure[$a=10^{-10}~\\text{[m]}$]{\n        \\centerline{\\includegraphics[width=0.6\\textwidth]{fig8c.eps}}\n        \\label{fig:8:c}\n    }\\hfill    \n    \\caption{\n       $H$-noise power $\\mathcal{P}\\left(\\rv{t}_\\mathrm{sHn}\\right)$ [dB] in the $\\left(\\alpha_1,\\alpha_2\\right)$-SHD as a function of $\\left(\\alpha_1,\\alpha_2\\right)$ at the distance (a) $a=10^{-5}$~[m], (b) $a=10^{-8}$~[m], and (c) $a=10^{-10}$~[m] with $K=10^{-10}$~[m$^2$\/s].\n    }\n    \\label{fig:8}\n\\end{figure}\n\n\nFig.~\\ref{fig:8} shows the $H$-noise power  $\\mathcal{N}\\left(\\rv{t}_\\mathrm{sHn}\\right)$ [dB] in the $\\left(\\alpha_1,\\alpha_2\\right)$-SHD as a function of $\\left(\\alpha_1,\\alpha_2\\right)$ at distance (a) $a=10^{-5}$~[m], (b) $a=10^{-8}$~[m], and (c) $a=10^{-10}$~[m] with $K=10^{-10}$~[m$^2$\/s]. The three diffusion scenarios are indicated as the blue square at $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,1\\right)$ for normal diffusion, the red square at $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,0.5\\right)$ for subdiffusion, and the green square at $\\left(\\alpha_1,\\alpha_2\\right)=\\left(1.8,1\\right)$ for superdiffusion. Given a fixed diffusion coefficient, the $H$-noise power increases with large distance $a$ and low value of $\\alpha_1$. However, with fixed $\\alpha_1$, the noise power decreases with $\\alpha_2$ when $a^{\\alpha_1}\/K > 1$ (Fig.~\\ref{fig:8:a}), while it increases in the opposite case (Fig.~\\ref{fig:8:b} and Fig.~\\ref{fig:8:c}). Thus, the error performance in $\\left(2,0.5\\right)$-SHD outperforms that of $\\left(2,1\\right)$-SHD in the low-SNR regime (see also Fig.~\\ref{fig:10}).  Similarity, with fixed $\\alpha_2$, the noise power decreases with $\\alpha_1$ when $a<1$ (Fig.~\\ref{fig:8}), while it increases with $\\alpha_1$ when $a>1$. This leads to intersection of the BER curves in $\\left(2,1\\right)$-SHD and $\\left(1.8,1\\right)$-SHD as in Fig.~\\ref{fig:10}.\n\n\n\\section{Error Probability Analysis}\n\nIn this section, we characterize the effect of $H$-diffusion on the error performance of molecular communication. Specifically, we consider an $M$-ary transmission scheme to boost the data rate as well as a  $N$-molecule transmission scheme to increase reliability for molecular communication.\n\n\\subsection{Molecular Communication System Model}\n\nWe consider a molecular communication system, as illustrated in Fig.~\\ref{fig:9}, where a TN located at $x=0$ emits molecules (information carrier) to an RN located  $a$~[m] from the TN. The emitted information molecules are assumed to be randomly and freely propagated in the fluid medium, e.g., blood vessels or tissues, under the $H$-diffusion laws with the diffusion coefficient $K$~[m$^2$\/s]. In this paper, we consider an molecular communication system with the following assumptions: 1) the TN can perfectly control the releasing time and the number of molecules for each symbol message; 2) the clock of the TN is perfectly synchronized with that at the RN; 3) movements of each molecule in the fluid medium are independent and identically distributed; 4) the RN acts as a perfectly absorbing boundary and perfectly measures the arrival time of molecules; 5) the molecules that arrived at the RN are absorbed and removed from the system; 6) the TN uses different types of molecule for each symbol to avoid inter-symbol interference.\\footnote{The number of molecule types used for this model can be minimized by introducing the lifetime of molecules \\cite{SML:15:WCOM,TJSW:18:COM} and the chemical reactions in the medium \\cite{AMGMKF:17:MBSC,MGMK:18:COM,FMGCEG:19:MBSC}.}\n\n\n\n\\begin{figure}[t]\n    \\centerline{\\includegraphics[width=0.95\\textwidth]{fig9.eps}}\n    \\caption{\n        Molecular communication system: a TN emits an information molecule to a RN in a fluid medium where the motion of information molecule is determined by the relationship between mean square displacement and time.\n    }\n    \\label{fig:9}\n\\end{figure}\n\n\nThe information is encoded based on the release time. Let $\\mathcal{S}\\triangleq \\left\\{s_0, s_1, \\ldots, s_{M-1} \\right\\}$ be the set of molecular release times  at the TN for corresponding symbol constellation where $s_i = i T_\\mathrm{s}\/M$, $M$ is a modulation order, and $T_\\mathrm{s}$ is a symbol time. The TN is able to emit $N$ molecules for a one-symbol transmission at the release time $\\rv{s} \\in \\mathcal{S}$. Hence, for the $n$th molecule among $N$ emitted molecules at the release time $\\rv{s}$, the arrival time $\\rv{y}_n$ at the RN is\n\\begin{align}\t\\label{eq:at}\n\\rv{y}_n=\\rv{s}+\\rv{t}_{n}\n\\end{align}\nwhere $\\rv{t}_{n}$ is the $H$-noise of the $n$th molecule. Then, the transmitted symbol $\\rv{s}$ can be decoded using the set of arrival times $\\mathcal{Y}=\\left\\{\\rv{y}_1, \\rv{y}_2, \\ldots,\\rv{y}_N\\right\\}$ at the RN. Since $H$-noise can be the heavy-tailed random variable, it may have a large waiting time for molecules arriving at the RN, i.e., there exists a positive probability that the molecule will not have arrived at TN within finite time. Furthermore, with the large number of released molecules, the RN needs to wait until all molecules are absorbed. Hence, we consider a \\emph{first arrival detection} that uses the time of first arrival molecule at the RN among $N$ released molecules to decode the transmitted symbol.\\footnote{Several detection schemes are proposed in \\cite{MFCG:16:GLOBECOM} to decode the transmitted signal using multiple transmitted molecules. In this work, we use the simple detection scheme because the complex implementation for a signal detection using multiple arrival molecules is impractical at the biological circuit level \\cite{AFSFH:12:MWCOM,NSOMV:14:NB}.}. Then, the explicit signal model for a one-symbol transmission is \n\\begin{align}\t\\label{eq:at2}\n\\rv{y}=\\rv{s}+\\rv{t}_\\mathrm{min}\n\\end{align}\nwhere \n$\n\\rv{t}_\\mathrm{min}=\\min\\left\\{\\rv{t}_{1},\\rv{t}_{2},\\ldots,\\rv{t}_{N}\\right\\}\n$ \nis referred as the \\emph{first arrival $H$-noise}.\n\n\n\n\n\\subsection{Error Probability Analysis}\n\n\nThe information can be decoded using the maximum likelihood detection for $M$-ary modulation:\n\\begin{align}\n\\hat{\\rv{s}}\n&=\n\t\\mathop{\\mathrm{arg\\,max}}_{\\rv{s}=\\left\\{s_0,s_1,\\ldots,s_{M-1}\\right\\}}\n\t\\PDF{\\rv{y}|\\rv{s}}{y|s}\n\\end{align}\nwhere\n\\begin{align}\n\t\\PDF{\\rv{y}|\\rv{s}}{y|s}\n\t&=\n\t\t\\begin{cases}\n\t\t\t\\PDF{\\rv{t}_\\mathrm{min}}{y-s},\t&\ty> s, \\\\\n\t\t\t0,\t&\ty \\leq s.\n\t\t\\end{cases}\n\\end{align}\n\\begin{proposition}[Density Function of First Arrival $H$-Noise]\t\\label{pro:pdf:fa}\nLet $\\rv{t}_{1}, \\rv{t}_2, \\ldots, \\rv{t}_{N}$ be the i.i.d. $H$-noise. Then, the PDF of first arrival  $H$-noise $\\PDF{\\rv{t}_\\mathrm{min}}{t}$ is given by\n\\begin{align}\n\t\\PDF{\\rv{t}_\\mathrm{min}}{t}\n\t&=\n\t\tN\n\t\t\\Fox{\\pDefine{n}_1+\\pDefine{n}_2}{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{q}_1+\\pDefine{q}_2}{\\pDefine{p}_1+\\pDefine{p}_2}\n\t\t{\n\t\tt;\\BB{\\pDefine{P}}_\\rv{t}\n\t\t\\Ket{\\frac{^{}}{_{}}\\left(a \\pDefine{c}_{\\left(\\omega_1\\right)}\\right)^{1\/\\left(\\omega_1\\omega_2\\right)}}\n\t\t}\n\t\\nonumber \\\\\n\t&\\hspace{0.5cm}\\times\n\t\t\\left[\n\t\t\\Fox{\\pDefine{n}_1+\\pDefine{n}_2+1}{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{q}_1+\\pDefine{q}_2+1}{\\pDefine{p}_1+\\pDefine{p}_2+1}\n\t\t{\\frac{t}{\\left(a\\pDefine{c}_{\\left(\\omega_1\\right)}\\right)^{1\/\\left(\\omega_1\\omega_2\\right)}};\\canOP{\\pSeq_\\FontDef{cdf}^{-1}}\n\t\t\t{\n\t\t\t\t\\Bra{1}\n\t\t\t\t\\BB{\\pDefine{P}}_\\rv{t}\n\t\t\t}\n\t\t}\n\t\t\\right]^{N-1}.\n\\end{align}\n\\begin{proof}\nThe CDF $\\CDF{\\rv{t}_\\mathrm{min}}{t}$ can be obtained as\n\\begin{align}\t\\label{eq:cdf:minHn}\n\t\\CDF{\\rv{t}_\\mathrm{min}}{t}\n\t&=\n\t\t\\Prob{\\min\\left\\{\\rv{t}_{1}, \\rv{t}_2, \\ldots,\\rv{t}_{N}\\right\\}<t}\n\t\\nonumber \\\\\n\t&\n=\n\t\t1-\\left(1-\\CDF{\\rv{t}}{t}\\right)^N.\n\\end{align}\nThen, we have\n\\begin{align}\n\t\\PDF{\\rv{t}_\\mathrm{min}}{t}\n\t&=\n\t\td\\CDF{\\rv{t}_\\mathrm{min}}{t}\/dt\n\t\\nonumber \\\\\n&\n\t= N \\PDF{\\rv{t}}{t}\\left(1-\\CDF{\\rv{t}}{t}\\right)^{N-1}. \n\\end{align}\nfrom which and with the $H$-function representation, we complete the proof.\n\\end{proof}\n\\end{proposition}\n\nSince $\\PDF{\\rv{t}_\\mathrm{min}}{t}$ consists of the $H$-functions, the evaluation of exact error probability requires numerical calculation of an optimal detection threshold. Hence, we consider a simple upper bound expression, which can be achieved using a fixed detection threshold \\cite{CTJS:15:CL,TJSW:18:COM}.\n\n\n\\begin{theorem}[Upper bound on the SEP]\t\\label{thm:SEP:Hdiffusion}\nThe symbol error probability (SEP) $P_\\mathrm{e}$ for $M$-ary and $N$-molecule transmission with the first arrival detection in $H$-diffusion is bounded as\n\\begin{align}\t\\label{eq:sep:Hdiffuion}\nP_{\\mathrm{e}}\n&\\leq\n\t\\frac{M-1}{M}\t\n\t\\left[\n\t\t\\Fox{\\pDefine{n}_1+\\pDefine{n}_2+1}{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{q}_1+\\pDefine{q}_2+1}{\\pDefine{p}_1+\\pDefine{p}_2+1}\n\t\t{\\frac{T_{\\mathrm{s}}\/M}{\\left(a\\pDefine{c}_{{\\left(\\omega_1\\right)}}\\right)^{1\/\\left(\\omega_1\\omega_2\\right)}};\t\n\t\t\t\\BB{\\pDefine{P}}_\\mathrm{e}\n\t\t}\n\t\\right]^N\n\\end{align}\nwhere the parameter sequence\n$\\BB{\\pDefine{P}}_\\mathrm{e}=\n\\canOP{\\pSeq_\\FontDef{cdf}^{-1}}\n\t\t\t{\n\t\t\t\t\\Bra{1}\n\t\t\t\\BB{\\pDefine{P}}_\\rv{t}\n\t\t\t}$. \n\\begin{proof}\nFor equally-likely symbols $s_i$, that is $\\Prob{\\rv{s}=s_i}=1\/M$, we have\n\\begin{align}\n\tP_\\mathrm{e}\n\t&=\n\t\t\\frac{1}{M}\n\t\t\\sum_{i=0}^{M-1}\n\t\t\\Prob{\\left.\\hat{\\rv{s}} \\neq s_i \\right|\\rv{s}=s_i}\n\t\\nonumber \\\\\n\t&\n\t\\leq\n\t\t\\frac{1}{M}\n\t\t\\sum_{i=0}^{M-2}\n\t\t\\Prob{\\left.\\rv{y}>\\left(i+1\\right)\\frac{T_\\mathrm{s}}{M}\\right|\\rv{s}=i\\frac{T_\\mathrm{s}}{M}}\n\t\\nonumber \\\\\n\t&=\n\t\t\\frac{M-1}{M}\n\t\t\\left(\n\t\t1-\n\t\t\\CDF{\\rv{t}_\\mathrm{min}}{\\frac{T_\\mathrm{s}}{M}}\n\t\t\\right).\n\\end{align}\nUsing the CDF $\\CDF{\\rv{t}_\\mathrm{min}}{t}$, which can be obtained from \\eqref{eq:cdf:Hn} and \\eqref{eq:cdf:minHn} in Proposition~\\ref{pro:pdf:fa} as \n\\begin{align}\n\\CDF{\\rv{t}_\\mathrm{min}}{t}\n&=\n\t1-\\left[\n\t\\Fox{\\pDefine{n}_1+\\pDefine{n}_2+1}{\\pDefine{m}_1+\\pDefine{m}_2}{\\pDefine{q}_1+\\pDefine{q}_2+1}{\\pDefine{p}_1+\\pDefine{p}_2+1}\n\t\t{\\frac{t}{\\left(a\\pDefine{c}_{\\left(\\omega_1\\right)}\\right)^{1\/\\left(\\omega_1\\omega_2\\right)}};\\canOP{\\pSeq_\\FontDef{cdf}^{-1}}\n\t\t\t{\n\t\t\t\t\\Bra{1}\n\t\t\t\t\\BB{\\pDefine{P}}_\\rv{t}\n\t\t\t}\n\t\t}\n\t\\right]^N,\n\\end{align}\nwe arrived at the desired result.\n\\end{proof}\n\\end{theorem}\n\\iffalse\nTable~\\ref{table:SEP} shows the upper bound on the SEP $P_\\mathrm{e}$ for typical anomalous diffusion in Table~\\ref{table:TDM:HD}.\n\\fi\n\n\\iffalse\n\\begin{table}[t]\n\\caption{Upper Bound on the SEP $P_\\mathrm{e}$ for Typical Anomalous Diffusion in Table~\\ref{table:TDM:HD}:\n$$\nP_\\mathrm{e}\\leq\n\\frac{M-1}{M}\n\\left[\n\t\\Fox{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}\n\t{\\frac{T_\\mathrm{s}\/M}{a^{1\/\\omega}};\\BB{\\pDefine{P}}}\n\\right]^N\n$$\n}\\centering\n\\label{table:SEP}\n\n\\begin{threeparttable}\n\n\\begin{adjustbox}{max width=0.6\\textwidth}\n\\begin{tabular}{llll}\n\\toprule\nDiffusion &\n\\multicolumn{2}{l}{Upper bound on the SEP $P_\\mathrm{e}$} &\n\n\n\\\\[-0.1cm]\n\n$\\rp{h}{t}$ &\n$\\BB{\\pDefine{O}}$ \n$\\BB{\\pDefine{P}}$ \n$\\omega$\n\\\\\n\n\\midrule\n\\midrule\n\nST-FD &\n$\\left(2,2,4,4\\right)$ &\n$\\Bigl(\n\t\\frac{2}{{\\alpha_\\mathrm{st}}},1,\\left(\\B{0}_3,1\\right),\\B{0}_4,\\left(\\frac{1}{{\\beta_\\mathrm{st}}},\\frac{{\\alpha_\\mathrm{st}}}{{\\beta_\\mathrm{st}}},\\frac{{\\alpha_\\mathrm{st}}}{2{\\beta_\\mathrm{st}}},1\\right),\\left(1,\\frac{1}{{\\beta_\\mathrm{st}}},\\frac{{\\alpha_\\mathrm{st}}}{2{\\beta_\\mathrm{st}}},1\\right)\n\\Bigr)$ &\n${\\beta_\\mathrm{st}}\/{\\alpha_\\mathrm{st}}$\n \n\\\\ \\midrule\n\nS-FD &\n$\\left(2,1,3,3\\right)$ &\n$\\Bigl(\n\t\\frac{2}{{\\alpha_\\mathrm{st}}},1,\\left(\\B{0}_2,1\\right),\\B{0}_3,\\left({\\alpha_\\mathrm{st}},\\frac{{\\alpha_\\mathrm{st}}}{2},1\\right),\\left(\\B{1}_2,\\frac{{\\alpha_\\mathrm{st}}}{2}\\right)\n\\Bigr)$ &\n$1\/{\\alpha_\\mathrm{st}}$\n\n\\\\ \\midrule\n\nT-FD &\n$\\left(1,1,2,2\\right)$ &\n$\\Bigl(\n\t1,1,\\left(0,1\\right),\\B{0}_2,\\left(\\frac{2}{{\\beta_\\mathrm{st}}},1\\right),\\B{1}_2\n\\Bigr)$ &\n${\\beta_\\mathrm{st}}\/2$\n\n\\\\ \\midrule\n\nEK-FD &\n$\\left(1,2,3,2\\right)$ &\n$\\Bigl(\n\t\\frac{1}{\\sqrt{\\pi}},2^{2\/{\\alpha_\\mathrm{ek}}},\\left(0,\\frac{1}{2},1\\right),\\B{0}_2,\\left(\\frac{1}{{\\alpha_\\mathrm{ek}}}\\B{1}_2,1\\right),\\left(1,\\frac{{\\beta_\\mathrm{ek}}}{{\\alpha_\\mathrm{ek}}}\\right)\n\\Bigr)$ &\n${\\alpha_\\mathrm{ek}}\/2$\n\n\\\\ \\midrule\n\nGBM &\n$\\left(1,2,3,2\\right)$ &\n$\\Bigl(\n\t\\frac{1}{\\sqrt{\\pi}},2^{2\/{\\alpha_\\mathrm{ek}}},\\left(0,\\frac{1}{2},1\\right),\\B{0}_2,\\left(\\frac{1}{{\\alpha_\\mathrm{ek}}}\\B{1}_2,1\\right),\\B{1}_2\n\\Bigr)$ &\n${\\alpha_\\mathrm{ek}}\/2$\n\n\\\\ \\midrule\n\nFBM &\n$\\left(1,1,2,1\\right)$ &\n$\\Bigl(\n\\frac{1}{\\sqrt{\\pi}},2^{2\/{\\alpha_\\mathrm{ek}}},\\left(\\frac{1}{2},1\\right),0,\\left(\\frac{1}{{\\alpha_\\mathrm{ek}}},1\\right),1\n\\Bigr)$ &\n${\\alpha_\\mathrm{ek}}\/2$\n\n\\\\ \\midrule\n\nBM &\n$\\left(1,1,2,1\\right)$ &\n$\\Bigl(\\frac{1}{\\sqrt{\\pi}},4,\\left(\\frac{1}{2},1\\right),0,\\B{1}_2,1\n\\Bigr)$ &\n$1\/2$ \n\n\\\\\n\n\\bottomrule\n\\end{tabular}\n\n\\end{adjustbox}\n\n\n\\end{threeparttable}\n\n\\end{table}\n\n\n\\fi\n\n\\begin{definition}[Signal-to-Noise Power Ratio]\t\t\\label{def:snr}\nThe SNR for a molecular communication link can be defined as\n\\begin{align}\n\\mathrm{SNR}\n\\triangleq\n\\frac{1}{2\\mathcal{G}}\n\\left(\n\\frac{T_\\mathrm{s}}{\\GMP{\\rv{t}}}\n\\right)^2\n\\end{align}\nwhere the constant $2\\mathcal{G}$ is used to normalize the SNR corresponding to the standard SNR with Gaussian noise \\cite[Table 1]{GPA:06:SP}.\n\\end{definition}\n\n\n\n\n\\subsubsection{Standard $H$-Diffusion}\nFor SHD, the SEP \\eqref{eq:sep:Hdiffuion} is reduced in terms of $\\mathrm{SNR}$ defined in Definition~\\ref{def:snr} as\n\\begin{align}\t\\label{eq:sep:std}\nP_\\mathrm{e}\n&\\leq\n\t\\frac{M-1}{M}\n\t\\left[\n\t\t\\Fox{2}{2}{4}{4}\n\t\t{\n\t\t\t\\frac{M^2}{2\\mathcal{G}^\\star \\mathrm{SNR}}\n\t\t\t;\n\t\t\t\\hat{\\BB{\\pDefine{P}}}_\\mathrm{e}\n\t\t}\n\t\\right]^N\n\\end{align}\nwhere \n\\begin{align}\n\\hat{\\BB{\\pDefine{P}}}_\\mathrm{e}\n&=\n\t\\left(\n\t\t\\tfrac{4}{\\alpha_1},\n\t\t1,\n\t\t\\B{1}_4,\n\t\t\\left(\\B{1}_3,0\\right),\n\t\t\\left(2,\\tfrac{2}{\\alpha_1\\omega_1\\omega_2},\\tfrac{1}{\\omega_1\\omega_2},\\tfrac{2\\alpha_2}{\\omega_2}\\right),\n\t\t\\left(\\tfrac{2}{\\omega_2},\\tfrac{2}{\\omega_1\\omega_2},\\tfrac{1}{\\omega_1\\omega_2},2\\right)\n\t\\right)\n\\end{align}\nwith\n$\n\\mathcal{G}^{\\star}\n\t= \n\t\t\\mathcal{G}^{2\\left(1-1\/\\alpha_1+\\left(1-\\alpha_2\\right)\\omega_1\\right)\/\\left(\\omega_1\\omega_2\\right)+1}$.\n\n\n\\subsubsection{High-SNR Expansions}\nIn the high-SNR regime, the SEP behaves as\n\\begin{align}\n\tP_\\mathrm{e}\n\t&=\n\t\t\\left(p_\\infty \\cdot \\mathrm{SNR}\\right)^{-s_\\infty}+o\\left(\\mathrm{SNR}^{-s_\\infty}\\right),\n\t\t\\quad\\quad\n\t\t\\mathrm{SNR} \\rightarrow \\infty\n\\end{align}\nwhere the quantity $s_\\infty$ denotes the \\emph{high-SNR slope} of the SEP-$\\mathrm{SNR}$ curve in a log-log scale, and $p_\\infty$ represents the \\emph{high-SNR power offset} in decibels of the SEP-$\\mathrm{SNR}$ curve relative to a reference of $\\mathrm{SNR}^{-s_\\infty}$. \n\\begin{corollary}[High-SNR Expansions]\t\\label{cor:highsnr}\nAt the high-SNR regime, two quantities $p_\\infty$ and $s_\\infty$ for SHD are given by\n\\begin{align}\t\n\ts_\\infty\n\t&=\n\t\tN\\cdot \\min\\left\\{\\frac{\\omega_2}{2},\\frac{\\omega_1\\omega_2}{2}\\right\\}\n\t\\label{eq:sinfty}\n\t\\\\\n\tp_\\infty\n\t&=\n\t\t\\left(\\frac{1}{g\\left(M,N\\right)}\\right)^{1\/s_\\infty}\n\t\\label{eq:pinfty}\n\\end{align}\nwhere\n\\begin{align}\ng\\left(M,N\\right)\n&=\n\t\\begin{cases}\n\t\\left(M-1\\right)M^{N\\omega_1\\omega_2-1}\n\t\\left(\n\t\t\\frac{\\GF{1-\\omega_1}\\GF{1\/\\alpha_1}}\n\t\t{\\alpha_1\\pi\\GF{1-\\alpha_2\\omega_1}}\n\t\t\\frac{\\left(\\mathcal{G}^{\\star}\\right)^{-\\omega_1\\omega_2\/2}}{2^{\\omega_1\\omega_2\/2-1}}\n\t\\right)^N \n\t& \\omega_1 < 1, \n\t\\\\\n\t\\left(M-1\\right)M^{N\\omega_2-1}\n\t\t\\left(\n\t\t\\frac{\n\t\t\t\\sin\\left(\\pi\/\\left(2\\omega_1\\right)\\right)\\GF{1-1\/\\omega_1}\\GF{1\/\\left(\\alpha_1\\omega_1\\right)}\n\t\t\t}\n\t\t\t{\\alpha_1\\pi\\GF{1-\\alpha_2}}\n\t\t\\frac{\\left(\\mathcal{G}^{\\star}\\right)^{-\\omega_2\/2}}{2^{\\omega_2\/2-1}}\n\t\\right)^N, &   \\omega_1 > 1.\n\t\\end{cases}\n\\end{align}\n\\begin{proof}\nUsing the algebraic asymptotic expansion of the $H$-function near zero, we have\n\\begin{align}\t\\label{eq:expansion:std}\n\t\t&\n\t\t\\Fox{2}{2}{4}{4}\n\t\t{\n\t\t\t\\frac{M^2}{2\\mathcal{G}^\\star \\mathrm{SNR}}\n\t\t\t;\n\t\t\t\\hat{\\BB{\\pDefine{P}}}_\\mathrm{e}\n\t\t}\n=\n\t\t\\frac{4}{\\alpha_1}\n\t\t\\sigma^\\star\\left(\\hat{\\BB{\\pDefine{P}}}_\\mathrm{e}\\right)\n\t\t\\left(\\frac{M^2}{2\\mathcal{G}^\\star \\mathrm{SNR}}\\right)^{\\omega^\\star\\left(\\hat{\\BB{\\pDefine{P}}}_\\mathrm{e}\\right)}\n\t\t+o\\left(\\mathrm{SNR}^{-\\omega^\\star\\left(\\hat{\\BB{\\pDefine{P}}}_\\mathrm{e}\\right)}\\right),\n\t\t\\quad\\quad\n\t\t\\mathrm{SNR} \\rightarrow \\infty \n\\end{align}\nwhere\n\\begin{align}\n\t\\omega^\\star\\left(\\hat{\\BB{\\pDefine{P}}}_\\mathrm{e}\\right)\n\t&=\n\t\t\\min\n\t\t\\left\\{\n\t\t\t\\frac{\\omega_1 \\omega_2}{2},\n\t\t\t\\frac{\\omega_2}{2}\n\t\t\\right\\}\n\t\\\\\n\t\\sigma^\\star\\left(\\hat{\\BB{\\pDefine{P}}}_\\mathrm{e}\\right)\n\t&=\n\t\t\\begin{cases}\n\t\t\\frac{\\GF{1-\\omega_1}\\GF{1\/\\alpha_1}}{2\\pi\\GF{1-\\alpha_2\\omega_1}},\t&\t\\omega_1<1 \\\\\n\t\t\\frac{\\sin\\left(\\pi\/\\left(2\\omega_1\\right)\\right)\\GF{1-1\/\\omega_1}\\GF{1\/\\left(\\alpha_1\\omega_1\\right)}}{2\\pi\\GF{1-\\alpha_2}}, \t&\t\\omega_1> 1.\n\t\t\\end{cases}\n\\end{align}\nFrom \\eqref{eq:expansion:std} and some manipulations, the desired result is obtained.\n\\end{proof}\n\\end{corollary}\n\n\\iffalse\n\\begin{table}[t]\n\\caption{High-SNR Expansions of $P_\\mathrm{e}$ for Typical Anomalous Diffusion in Table~\\ref{table:TDM:HD}\n}\\centering\n\\label{table:expansions}\n\n\\begin{threeparttable}\n\n\\begin{adjustbox}{max width=0.5\\textwidth}\n\\begin{tabular}{lll}\n\\toprule\nDiffusion &\nHigh-SNR slope &\nHigh-SNR power offset \n\n\n\\\\[-0.1cm]\n\n$\\rp{h}{t}$ &\n$s_\\infty=N\\omega$ &\n$p_\\infty=\\left[M^{1-2 s_\\infty}\/\\left(\\left(M-1\\right) p_\\mathrm{c}^N\\right)\\right]^{1\/s_\\infty}$ \n\n\\\\[-0.2cm]\n\n\\addlinespace\n\\addlinespace\n\\cline{2-3}\n\n\n\\\\[-0.5cm]\n\n&\n$\\omega$ &\n$p_\\mathrm{c}$ \n\n\\\\\n\n\\midrule\n\\midrule\n\nST-FD &\n$\\min\\left\\{\\frac{{\\beta_\\mathrm{st}}}{2},\\frac{{\\beta_\\mathrm{st}}}{2{\\alpha_\\mathrm{st}}}\\right\\}$ &\n$\n\\begin{cases}\n\\frac{\n\\sin\\left(\\pi{\\alpha_\\mathrm{st}}\/2\\right)\\GF{1-{\\alpha_\\mathrm{st}}}\n}{\n\\pi{\\alpha_\\mathrm{st}}\\GF{1-{\\beta_\\mathrm{st}}}\n}\n\\frac{\n\t\\mathcal{G}^{{\\beta_\\mathrm{st}}\/2-{\\alpha_\\mathrm{st}}}\n}{\n\t2^{{\\beta_\\mathrm{st}}\/2-1}\n},\n&\n\\text{if~} {\\alpha_\\mathrm{st}} < 1\n\\\\\n\\frac{\n\\csc\\left(\\pi\/{\\alpha_\\mathrm{st}}\\right)\n}{\n{\\alpha_\\mathrm{st}}\\GF{1-{\\beta_\\mathrm{st}}\/{\\alpha_\\mathrm{st}}}\n}\n\\left(\n\\frac{\\mathcal{G}}{2}\n\\right)^{{\\beta_\\mathrm{st}}\/\\left(2{\\alpha_\\mathrm{st}}\\right)-1},\n&\n\\text{if~} {\\alpha_\\mathrm{st}} > 1\n\\end{cases}\n$\n \n\\\\ \\midrule\n\nS-FD &\n$1\/\\left(2{\\alpha_\\mathrm{st}}\\right)$ &\n$\n\\frac{\\GF{1+1\/{\\alpha_\\mathrm{st}}}}{\\pi}\n\\left(\n\\frac{\\mathcal{G}}{2}\n\\right)^{1\/\\left(2{\\alpha_\\mathrm{st}}\\right)-1},\n\\quad\n\\text{for~}{\\alpha_\\mathrm{st}} > 1\n$ \n\n\\\\ \\midrule\n\nT-FD &\n${\\beta_\\mathrm{st}}\/4$ &\n$\n\\frac{1}{2\\GF{1-{\\beta_\\mathrm{st}}\/2}}\n\\left(\n\\frac{\\mathcal{G}}{2}\n\\right)^{{\\beta_\\mathrm{st}}\/4-1}\n$\n\n\\\\ \\midrule\n\nEK-FD &\n${\\alpha_\\mathrm{ek}}\/4$ &\n$\n\\frac{1}{\\GF{1-{\\beta_\\mathrm{ek}}\/2}}\n\\frac{\\mathcal{G}^{{\\beta_\\mathrm{ek}}\/2-1}}{\\left(2\\mathcal{G}\\right)^{{\\alpha_\\mathrm{ek}}\/4}}\n$\n\n\\\\ \\midrule\n\nGBM &\n${\\alpha_\\mathrm{ek}}\/4$ &\n$\n\\frac{1}{\\GF{1-{\\alpha_\\mathrm{ek}}\/2}}\n\\frac{\\mathcal{G}^{{\\alpha_\\mathrm{ek}}\/2-1}}{\\left(2\\mathcal{G}\\right)^{{\\alpha_\\mathrm{ek}}\/4}}\n$\n\n\\\\ \\midrule\n\nFBM &\n${\\alpha_\\mathrm{ek}}\/4$ &\n$\\frac{1}{\\sqrt{\\pi\\mathcal{G}}}\\frac{1}{\\left(2\\mathcal{G}\\right)^{{\\alpha_\\mathrm{ek}}\/4}}$\n\n\\\\ \\midrule\n\nBM &\n$1\/4$ &\n$\\frac{1}{\\sqrt{\\pi\\mathcal{G}}}\\frac{1}{\\left(2\\mathcal{G}\\right)^{1\/4}}$\n\n\\\\\n\n\\bottomrule\n\\end{tabular}\n\n\\end{adjustbox}\n\n\n\\end{threeparttable}\n\n\\end{table}\n\\fi\n\n\\iffalse\nThe high-SNR expansions of the SEP $P_\\mathrm{e}$ for typical anomalous diffusion in Table~\\ref{table:TDM:HD} are tabulated in Table~\\ref{table:expansions} as special cases of \\eqref{eq:sinfty} and \\eqref{eq:pinfty}.\n\\fi\n\n\\begin{remark}[High-SNR Slope]\nAs shown in Corollary~\\ref{cor:highsnr}, the high-SNR slope $s_\\infty$  increases linearly with the number of released molecules $N$. This can be interpreted as the benefits arising from consuming molecular resources, which is synonymous with \\emph{transmit diversity} in wireless multiple-antenna systems.\n\\end{remark}\n\n\n \\begin{figure}[t!]\n    \\centerline{\\includegraphics[width=0.55\\textwidth]{fig10.eps}}\n   \n\\caption{\n       SEP $P_\\mathrm{e}$ as a function of $\\mathrm{SNR}$ [dB] in the $\\left(\\alpha_1,\\alpha_2\\right)$-SHD for:  \n       i) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,1\\right)$; \n       ii) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(2,0.5\\right)$; and  \n       iii) $\\left(\\alpha_1,\\alpha_2\\right)=\\left(1.8,1\\right)$; \n       with $N=1$ and $M=2$.\n}\n\\label{fig:10}\n\\end{figure}\n\n \\begin{figure}[t!]\n    \\centerline{\\includegraphics[width=0.55\\textwidth]{fig11.eps}}\n   \n\\caption{\n       SEP $P_\\mathrm{e}$ as a function of $\\mathrm{SNR}$ [dB] in the $\\left(1.8,1\\right)$-SHD for $M=2$, $4$, $8$, $16$, with $N=2$.\n       }\n\\label{fig:11}\n\\end{figure}\n\n\\subsection{Numerical Examples}\nFig.~\\ref{fig:10} shows the SEP as a function of $\\mathrm{SNR}$ [dB] in $\\left(\\alpha_1,\\alpha_2\\right)$-SHD for the three diffusion scenarios with single molecule transmission ($N=1$) and binary modulation ($M=2$). Obviously, the SEP decreases with SNR, which implies that we can improve the molecular communication reliability with a large symbol time $T_\\mathrm{s}$ (with a low  symbol rate). We can observe that the high-SNR slope in $\\left(\\alpha_1,\\alpha_2\\right)$-SHD can be determined by  $s_\\infty=\\alpha_2\/\\left(2\\alpha_1\\right)$ for $\\alpha_1>1$, as stated in Corollary~\\ref{cor:highsnr}. In this example, $s_\\infty=0.25$, $0.125$, and $0.278$ for $\\left(2,1\\right)$-, $\\left(2,0.5\\right)$-, and $\\left(1.8,1\\right)$-SHD, respectively. Therefore, the SEP in superdiffusion outperforms that in other scenarios in the high-SNR regime due to the large value of $s_\\infty$. It is also noteworthy that the upper bound becomes tight at the high-SNR regime since the detection thresholds approach to $i T_s\/M$, $i=1,\\ldots, M-1$ \\cite{TJSW:18:COM}.\n\n \\begin{figure}[t!]\n    \\centerline{\\includegraphics[width=0.55\\textwidth]{fig12.eps}}\n   \n\\caption{\n       SEP $P_\\mathrm{e}$ as a function of $\\mathrm{SNR}$ [dB] in the $\\left(2,0.5\\right)$-SHD for $N=1$, $2$, $3$, $4$, with $M=4$.\n}\n\\label{fig:12}\n\\end{figure}\n\n\n\n\nFig.~\\ref{fig:11} shows the SEP $P_\\mathrm{e}$ as a function of $\\mathrm{SNR}$ [dB] in the $\\left(1.8,1\\right)$-SHD with $N=2$ for $M=2$, $4$, $8$, $16$. The figure shows that we can boost the data rate with modulation order $M$ by sacrificing the reliability of SEP performance. Fig.~\\ref{fig:12} demonstrates the effect of the number of released molecules on error performance, where the SEP $P_\\mathrm{e}$ is a  function of $\\mathrm{SNR}$ [dB] in the $\\left(2,0.5\\right)$-SHD with $M=4$ for $N=1$, $2$, $3$, $4$. The  released molecule diversity can be obtained linearly with multiple released molecules. In this example, the high-SNR slopes are equal to $s_\\infty=0.125$, $0.25$, $0.375$, and $0.5$ for $N=1$, $2$, $3$, and  $4$, respectively. As can be seen from both Figs.~\\ref{fig:11} and \\ref{fig:12}, the derived high-SNR expansion expression has close agreement in slope and power offset of the SEP curves. The high-SNR slope $s_\\infty$ of SEP in the $\\left(\\alpha_1,\\alpha_2\\right)$-SHD can be further examined by referring to Fig.~\\ref{fig:13}, where $\\left(\\alpha_1,\\alpha_2\\right)$-lines are depicted for $s_\\infty=0.45N$, $0.40N$, $0.35N$, $0.30N$, and $0.25N$. The high-SNR slope $s_\\infty$ increases with $\\alpha_2$, while it decreases with $\\alpha_1$ in the region $\\alpha_1 >1$, as expected. \n\n\n\n \\begin{figure}[t!]\n    \\centerline{\\includegraphics[width=0.55\\textwidth]{fig13.eps}}\n   \n\\caption{\n       High-SNR slope $s_\\infty$ of SEP $P_\\mathrm{e}$ as a function of $\\left(\\alpha_1,\\alpha_2\\right)$ in the $\\left(\\alpha_1,\\alpha_2\\right)$-SHD. $\\left(\\alpha_1,\\alpha_2\\right)$-line is for $s_\\infty=0.45N$, $0.40N$, $0.35N$, $0.30N$, and $0.25N$.\n       }\n\\label{fig:13}\n\\end{figure}\n\n\n\n\\section{Conclusion}\nIn this paper, we developed a new mathematical framework for modeling and analysis in molecular communication with anomalous diffusion. We first systemized the method to generate the PDF---involved in the subordination law---of the molecule's position evolving in time by introducing the general class of diffusion processes, namely $H$-diffusion. The $H$-diffusion modeling subsumes most known anomalous diffusion models obtained from two $H$-variates, which play a role in explaining the anomalous evolution of molecules in space and in time, respectively. We then provided the new notion of molecular noise, namely $H$-noise, to account for the statistical properties of uncertainty of the random propagation time under $H$-diffusion laws. The error performance in the subdiffusion scenario outperforms that in the low-SNR regime, while the superdiffusion scenario is outperformed in the high-SNR regime.\nIn summary, our framework proved by $H$-theory serves a method to model a molecular communication channel with anomalous diffusion and unified analysis for emerging nanoscale communication.\n\nThis work opens several open problems. For a practice, the multi-dimensional anomalous diffusion channel model should be considered with suitable initial and boundary conditions depending on various molecular communication system setups. Since the molecules obey different diffusion rules compared to normal diffusion, new inter-symbol-interference avoidance schemes, synchronization schemes between nanomahcine, and receiver designs are required.\n\n\n\\newpage\n\n\\appendices\n\n\n\n\\section{Glossary of Notations and Symbols}\t\\label{sec:appendix:NS}\n\n\n\n\\begin{basedescript}{\\desclabelwidth{2.8cm}}\n{\n\n\\item[~$\\mathbbmss{R}$]\n\nReal numbers \\\\[-0.8cm]\n\n\\item[~$\\mathbbmss{R}_+$]\n\nNonnegative real numbers \\\\[-0.8cm]\n\n\\item[~$\\mathbbmss{R}_{++}$]\n\nPositive real numbers \\\\[-0.8cm]\n\n\n\n\\item[~$\\B{1}_n$] \n\nAll-one sequence or vector of $n$ elements \\\\[-0.8cm]\n\n\\item[~$\\B{0}_n$] \n\nAll-zero sequence or vector of $n$ elements \\\\[-0.8cm]\n\n\\item[~$o\\left(\\cdot\\right)$]\n\nBachmann--Landau notation: \\\\[-0.8cm]\n\\begin{flalign}\n&f\\left(x\\right)\n=o\\left(g\\left(x\\right)\\right)\n\t\\quad\n\t\\left(x \\rightarrow x_0\\right)\n\\quad\n    \\Leftrightarrow ~\n    \\lim_{x\\rightarrow x_0}\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0\n&\n\\end{flalign}\n~\\\\[-1.8cm]\n\\item[~$\\mathop{=}\\limits^{\\text{d}}$]\n\nDistributional equality \\\\[-0.8cm]\n\n\\item[~$\\doteq$]\n\nAsymptotically exponential equality \\\\[-0.8cm]\n\\begin{flalign}\n&f\\left(x\\right) \\doteq x^y\n    ~\\Leftrightarrow~\n    \\lim_{x \\rightarrow \\infty}\n        \\frac{\\log f\\left(x\\right)}{\\log x}\n    =y\n&\n\\end{flalign}\n~\\\\[-0.8cm]\nwhere $y$ is the exponential order of $f\\left(x\\right)$.\\\\[-0.8cm]\n\n\n\\item[~$\\EX{\\cdot}$] \n\nExpectation operator \\\\[-0.8cm]\n\n\n\n\\item[~$\\PDF{\\rv{x}}{x}$]\n\nProbability density function of $\\rv{x}$ \\\\[-0.8cm]\n\n\\item[~$\\CDF{\\rv{x}}{x}$] \n\nCumulative distribution function of $\\rv{x}$  \\\\[-0.8cm]\n\n\\item[~$\\CF{\\rv{x}}{x}$]\n\nCharacteristic function of $\\rv{x}$ \\\\[-0.8cm]\n\n\\item[~$\\delta\\left(x\\right)$]\n\nDirac delta function \\\\[-0.8cm]\n\n\n\n\\item[~$\\GF{\\cdot}$] \n\nGamma function \\cite[eq.~(8.310.1)]{GR:07:Book} \\\\[-0.8cm]\n\n\n\n\n\n\n\n\n\n\n\n\n\\item[~$H^{\\pDefine{m},\\pDefine{n}}_{\\pDefine{p},\\pDefine{q}}{\\left[\\cdot\\right]}$]\n\nFox's $H$-function \\cite{JSW:15:IT}: \\\\[-0.7cm]\n\\begin{flalign}\n\\Fox{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{x;\\BB{\\pDefine{P}}}\n&=\n\t\\pDefine{k}\n\t\\FoxH{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\pDefine{c} x}{\n        \\left(\\pA{1},\\pS{1}\\right),\n\t\t\\left(\\pA{2},\\pS{2}\\right),\n\t\t\\ldots,\n\t\t\\left(\\pA{\\pDefine{p}},\\pS{\\pDefine{p}}\\right)}{\n\t\t\\left(\\pB{1},\\pT{1}\\right),\n\t\t\\left(\\pB{2},\\pT{2}\\right),\n\t\t\\ldots,\n\t\t\\left(\\pB{\\pDefine{q}},\\pT{\\pDefine{q}}\\right)}\n\t\\nonumber\\\\\n&=\n\t\\pDefine{k}\n\t\\FoxH{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\pDefine{c} x}\n\t\t{\\left(\\BB{\\pDefine{a}},\\BB{\\pDefine{A}}\\right)}\n\t\t{\\left(\\BB{\\pDefine{b}},\\BB{\\pDefine{B}}\\right)}\n&\n\\end{flalign}\nwhere the parameter sequence is \n$\n\\BB{\\pDefine{P}}\n=\n\t\\left(\\pDefine{k},\\pDefine{c},\\BB{\\pDefine{a}},\\BB{\\pDefine{b}},\\BB{\\pDefine{A}},\\BB{\\pDefine{B}}\\right)\n$\nwith\\\\[-0.7cm]\n\\begin{flalign}\n&\n\\begin{cases}\n\\BB{\\pDefine{a}}=\\left(\\pA{1},\\pA{2},\\ldots,\\pA{\\pDefine{n}},\\pA{\\pDefine{n}+1},\\pA{\\pDefine{n}+2}\\ldots,\\pA{\\pDefine{p}}\\right) \n\\\\[-0.3cm]\n\\BB{\\pDefine{b}}=\\left(\\pB{1},\\pB{2},\\ldots,\\pB{\\pDefine{m}},\\pB{\\pDefine{m}+1},\\pB{\\pDefine{m}+2},\\ldots,\\pB{\\pDefine{q}}\\right)\n\\\\[-0.3cm]\n\\BB{\\pDefine{A}}=\\left(\\pS{1},\\pS{2},\\ldots,\\pS{\\pDefine{n}},\\pS{\\pDefine{n}+1},\\pS{\\pDefine{n}+2},\\ldots,\\pS{\\pDefine{p}}\\right)\n\\\\[-0.3cm]\n\\BB{\\pDefine{B}}=\\left(\\pT{1},\\pT{2},\\ldots,\\pT{\\pDefine{m}},\\pT{\\pDefine{m}+1},\\pT{\\pDefine{m}+2},\\ldots,\\pT{\\pDefine{q}}\\right)\n\\end{cases}\n&\n\\end{flalign}\nA Mellin-Barnes type integral form of Fox's $H$-function is\\\\[-0.8cm]\n\\begin{flalign}\n\\Fox{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{x;\\BB{\\pDefine{P}}}\n&=\n\t\\frac{1}{2\\pi \\jmath}\n    \\int_\\mathfrak{L}\n    \\theta\\left(s\\right)\n    x^s\n    ds,\\quad x\\neq 0\n&\n\\end{flalign}\nwhere $\\mathfrak{L}$ is a suitable contour, $\\jmath=\\sqrt{-1}$, \n$\nx^s\n=\n    \\exp\\left\\{s\\left(\\ln\\left|x\\right|+\\jmath \\arg x\\right)\\right\\}\n$,\nand \n\\begin{flalign}\n\\theta\\left(s\\right)\n&=\n    \\frac{\n            \\prod_{j=1}^{\\pDefine{m}}\\GF{\\pB{j}-\\pT{j}s}\n            \\prod_{j=1}^{\\pDefine{n}}\\GF{1-\\pA{j}+\\pS{j}s}\n         }\n         {\n            \\prod_{j=\\pDefine{m}+1}^{\\pDefine{q}}\\GF{1-\\pB{j}+\\pT{j}s}\n            \\prod_{j=\\pDefine{n}+1}^{\\pDefine{p}}\\GF{\\pA{j}-\\pS{j}s}\n         }\n&\n\\end{flalign}\n\n\n\\item[~$\\FoxHT{\\pDefine{m},\\pDefine{n}}{\\pDefine{p},\\pDefine{q}}{\\BB{\\pDefine{P}}}{f\\left(t\\right)}{s}$]\n\n$H$-transform of a function $f\\left(t\\right)$ with Fox's $H$-kernel of the order sequence \\\\[-0.8cm]\n\\item[]\n$\\BB{\\pDefine{O}}=\\left(\\pDefine{m}, \\pDefine{n}, \\pDefine{p}, \\pDefine{q}\\right)$ and the parameter sequence $\\BB{\\pDefine{P}}=\\left(\\pDefine{k},\\pDefine{c},\\BB{\\pDefine{a}},\\BB{\\pDefine{b}},\\BB{\\pDefine{A}},\\BB{\\pDefine{B}}\\right)$ \\cite{JSW:15:IT}: \\\\[-0.7cm]\n\\begin{flalign} \\label{eq:Def:HT}\n\\FoxHT{\\pDefine{m},\\pDefine{n}}{\\pDefine{p},\\pDefine{q}}{\\BB{\\pDefine{P}}}{f\\left(t\\right)}{s}\n&=\n\t\\pDefine{k}\n    \t\\int_0^\\infty\n        \t\\FoxH{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}\n        \t{\\pDefine{c} st}\n        \t{\\left(\\BB{\\pDefine{a}}, \\BB{\\pDefine{A}}\\right)}\n        \t{\\left(\\BB{\\pDefine{b}}, \\BB{\\pDefine{B}}\\right)}\n        \tf\\left(t\\right)\n        dt,\n        \\quad\n        s>0\n&\n\\end{flalign}\n\n\\item[~$\\FoxV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}}$]\n\n$H$-variate with the order sequence $\\BB{\\pDefine{O}}=\\left(\\pDefine{m},\\pDefine{n},\\pDefine{p},\\pDefine{q}\\right)$ and the parameter sequence \\\\[-0.8cm]\n\\item[]\n$\\BB{\\pDefine{P}} = \\left(\\pDefine{k}, \\pDefine{c}, \\BB{\\pDefine{a}}, \\BB{\\pDefine{b}}, \\BB{\\pDefine{A}}, \\BB{\\pDefine{B}} \\right)$ \\cite{JSW:15:IT}: if $\\rv{x} \\sim \\FoxV{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}{\\BB{\\pDefine{P}}}$, then \\\\[-0.7cm]\n\\begin{flalign} \\label{eq:Def:FV}\n\\PDF{\\rv{x}}{x}\n&=\n        \\pDefine{k}\n        \\FoxH{\\pDefine{m}}{\\pDefine{n}}{\\pDefine{p}}{\\pDefine{q}}\n        {\\pDefine{c} x}\n        {\\left(\\BB{\\pDefine{a}},\\BB{\\pDefine{A}}\\right)}\n        {\\left(\\BB{\\pDefine{b}},\\BB{\\pDefine{B}}\\right)},\n        \\qquad\n        x \\geq 0\n&\n\\end{flalign} \\\\[-0.7cm]\nwith the set of parameters satisfying a distributional structure such that \\\\[-0.8cm]\n\\item[]\n$\\PDF{\\rv{x}}{x} \\geq 0$ for all $x \\in \\mathbbmss{R}_+$ and $\\FoxHT{\\pDefine{m},\\pDefine{n}}{\\pDefine{p},\\pDefine{q}}{\\BB{\\pDefine{P}}}{1}{1}=1$ \\\\[-0.8cm]\n}\n\\end{basedescript}\n\n\\section{Special Functions}\t\\label{sec:appendix:SF}\n\nIn this appendix, we briefly introduce the special functions which are frequently used in the context of diffusion theory, fractional calculation theory, and molecular communication. \n\n\n\\subsection{$M$-Wright Function}\t\\label{sec:appendix:SF:A}\nThe Wright function (of the second kind) is defined as\n\\begin{align}\nW_{\\lambda,\\mu}\\left(t\\right)\n=\n\t\\sum_{n=0}^\\infty\n\t\\frac{t^n}{n! \\GF{\\lambda n + \\mu}}\n\\end{align}\nwhere $\\lambda > -1$, $\\mu \\in \\mathbbmss{C}$, and $t \\in \\mathbbmss{C}$.\nThe $M$-Wright function $M_\\nu\\left(t\\right)$ is the one of the auxiliary function of the Wright function by setting $\\lambda=-\\nu$ and $\\mu=1-\\nu$, whose series representation is given by\n\\begin{align}\nM_\\nu\\left(t\\right)\n&=\n\t\\sum_{n=0}^\\infty\n\t\\frac{\\left(-t\\right)^n}{n!\\GF{-\\nu n + \\left(1-\\nu\\right)}}\n\t\\nonumber \\\\\n&\n=\n\t\\frac{1}{\\pi}\n\t\\sum_{n=1}^\\infty\n\t\\frac{\\left(-t\\right)^{n-1}}{\\left(n-1\\right)!}\\GF{\\nu n}\\sin\\left(\\pi n \\nu\\right)\n\\end{align}\nwhere $\\nu$ is defined on the positive real axis for $0<\\nu<1$. Note that $M_{\\nu=1}\\left(t\\right)=\\delta\\left(z\\right)$. It also can be represented in terms of the $H$-function as\n\\begin{align}\nM_\\nu\\left(t\\right)\n=\n\t\\Fox{1}{0}{1}{1}{t;\\BB{\\pDefine{P}}_\\mathrm{MW}=\\left(1,1,1-\\nu,0,\\nu,1\\right)}.\n\\end{align}\nAn important particular case is for $\\nu=1\/2$ where the $M$-Wright function reduces to\n\\begin{align}\nM_{\\nu=1\/2}\\left(t\\right)\n&=\n\t\\Fox{1}{0}{1}{1}{t;\\left(1,1,1\/2,0,1\/2,1\\right)}\n\\nonumber \\\\\n&\n=\n\t\\frac{1}{\\sqrt{\\pi}}\n\t\\,\n\t\\exp\\left(-t^2\/4\\right)\n\\end{align}\nwhich can be interpreted as a natural generalization of the Gaussian density for fractional diffusion processes. Since the $M$-Wright function has a relation to Mittag-Leffler function $E_\\nu\\left(z\\right)$ by means of the Laplace transform \n\\begin{align}\n\\LT{M_{\\nu}\\left(t\\right)}{s}\n&=\n\\FoxHT{1,0}{0,1}{\\left(1,1,\\text{--},0,\\text{--},1\\right)}{M_{\\nu}\\left(t\\right)}{s}\n\\nonumber \\\\\n&\n=\n\\Fox{1}{1}{1}{2}{s;\\left(1,1,0,\\B{0}_2,1,\\left(1,\\nu\\right)\\right)}\n\\nonumber \\\\\n&\n=\n\\sum_{n=0}^{\\infty}\n\\frac{\\left(-s\\right)^n}{\\GF{\\nu n +1}}\n\\nonumber \\\\\n&\n=\nE_\\nu\\left(-s\\right),\n\\end{align}\nthe $M$-Wright function for nonnegative variable $t$ is known as a non-Markovian model to generalize the evolution in time of fractional diffusion processes. The $M$-Wright function is also related to the inverse stable subordinator to the generalized grey Brownian motion \\cite{Pag:13:FCAA, MMP:10:JDE,PS:14:CAIM}, and hence the $M$-Wright function for the symmetric random variable can be represented as all solutions of EK-FD process.\n\n\\subsection{Mittag-Leffler Function}\t\t\\label{sec:appendix:SF:B}\n\nThe Mittag-Leffler function appears as the solution of fractional differential equations and fractional order integral equations \\cite{Lef:05:AM, Cah:13:CSTM, Gor:09:Proc, GKMR:14:Book, Lin:98:SPI}. \nThe series and $H$-function representations for the generalized Mittag-Leffler function $E_{\\alpha,\\beta}\\left(t\\right)$ are given by\n\\begin{align}\nE_{\\alpha,\\beta}\\left(t\\right)\n&=\n\t\\sum_{n=0}^\\infty\n\t\\frac{t^n}{\\GF{\\alpha n + \\beta}}\n\\nonumber \\\\\n&\n=\n\t\\Fox{1}{1}{1}{2}{t;\\BB{\\pDefine{P}}_\\mathrm{GML}=\\left(1,-1,0,\\left(0,1-\\beta\\right),1,\\left(1,\\alpha\\right)\\right)}\n\\end{align}\nwhere $\\alpha,\\beta \\in \\mathbbmss{C}$ and $\\Re\\left(\\alpha\\right), \\Re\\left(\\beta\\right) > 0$. When $\\alpha=\\nu$ and $\\beta=1$, $E_{\\alpha,\\beta}\\left(t\\right)$ reduces to Mittag-Leffler function, denoted by $E_{\\nu}\\left(t\\right)$, as\n\\begin{align}\nE_{\\nu}\\left(t\\right)\n&=\n\t\\sum_{n=0}^\\infty\n\t\\frac{t^n}{\\GF{\\nu n + 1}}\n\\nonumber \\\\\n&\n=\n\t\\Fox{1}{1}{1}{2}{t;\\BB{\\pDefine{P}}_\\mathrm{ML}=\\left(1,-1,0,\\B{0}_2,1,\\left(1,\\nu\\right)\\right)}.\n\\end{align}\nSince $E_\\nu\\left(t\\right)$ is a increasing function for all $\\nu\\in \\left(0,1\\right]$ with the convergences $E_\\nu\\left(-\\infty\\right)=0$ and $E_\\nu\\left(0\\right)=1$, the cumulative distribution function of a probability measure on the nonnegative real numbers can be defined as $\\CDF{\\rv{t}}{t;\\nu}=1-E_\\nu\\left(-t^\\nu\\right)$ called the Mittag-Leffler distribution of order $\\nu$. For $\\nu \\in \\left(0,1\\right)$, the Mittag-Leffler distribution of order $\\nu$ is a heavy-tailed generalization of the exponential distribution since the Laplace transform of \n\\begin{align}\t\\label{eq:mld}\n\\rv{t}\\sim\\FoxV{1}{1}{1}{2}{\\BB{\\pDefine{P}}_\\mathrm{MLD}=\\left(\\tfrac{1}{\\nu},1,1-\\tfrac{1}{\\nu},\\left(1-\\tfrac{1}{\\nu},0\\right),\\tfrac{1}{\\nu},\\left(\\tfrac{1}{\\nu},1\\right)\\right)}\n\\end{align}\nhas a for\n\\begin{align}\n\\EX{e^{-\\rv{t}s}}=\\frac{1}{1+s^{\\nu}}.\n\\end{align}\nNote that $\\rv{t}$ is a completely skewed geometric stable distribution and also an exponential distribution with the order $\\nu=1$.\n\n\n\\section{$H$-Representation Theory of Strictly Stable Distributions}\t\\label{sec:appendix:Stable}\n\nThe stable distribution is a rich class of probability distributions that allow skewness and\nheavier (algebraic) tails.  \nThis distribution has been used in modeling and analyzing physical, statistical, engineering, and economic systems. However, due to lack of analytical expression for the density function of stable distributions for all but a few cases---for example, normal, Cauchy, and L\\'evy distributions---the use of stable distributions meets many technical challenges. In this appendix, we provide an analytical expression for the stable distributions in terms of $H$-functions.\n\nLet $\\rv{x}_1$ and $\\rv{x}_2$ be independent copies of a random variable $\\rv{x}$. Then $\\rv{x}$ is said to be \\emph{stable} if, for any constants $a>0$ and $b>0$,\n\\begin{align}\t\\label{eq:def:stable}\na\\rv{x}_1 + b\\rv{x}_2 \\mathop{=}\\limits^{\\text{d}} c\\rv{x} + d\n\\end{align}\nfor constants $c>0$ and $d$. The distribution is said to be \\emph{strictly stable} when \\eqref{eq:def:stable} holds with $d=0$.\nIn what follows, we use $\\Stable{\\alpha}%{{\\alpha_\\mathrm{s}}}{\\beta}%{{\\beta_\\mathrm{s}}}{\\gamma}%{{\\gamma_\\mathrm{s}}}{\\mu}%{{\\mu_\\mathrm{s}}}$ to denote the distribution of a stable random variable with\n\n    \\vspace{0.5cm}\n    \\hspace{3cm}\\begin{tabular}{ll}\n    $\\alpha}%{{\\alpha_\\mathrm{s}} \\in \\left(0,2\\right]$ & characteristic exponent (index of stability) \\\\\n    $\\beta}%{{\\beta_\\mathrm{s}} \\in \\left[-1,1\\right]$ & skewness parameter \\\\\n    $\\gamma}%{{\\gamma_\\mathrm{s}} \\in \\left[0,\\infty\\right)$ & scale (dispersion) parameter \\\\\n    $\\mu}%{{\\mu_\\mathrm{s}} \\in \\mathbbmss{R} $ & location parameter. \\\\\n    \\end{tabular}\n    \\vspace{0.5cm}\n\n\\noindent\nIn general, the canonical characteristic function of stable distribution $\\rv{x}\\sim\\Stable{\\alpha}%{{\\alpha_\\mathrm{s}}}{\\beta}%{{\\beta_\\mathrm{s}}}{\\gamma}%{{\\gamma_\\mathrm{s}}}{\\mu}%{{\\mu_\\mathrm{s}}}$ is\n\\begin{align}\n        \\CF{\\rv{x}}{\\omega}\n        &=\n        \\begin{cases}\n        \\exp\\left\\{-\\gamma}%{{\\gamma_\\mathrm{s}}\\left|\\omega\\right|^\\alpha}%{{\\alpha_\\mathrm{s}}\n        \\left[1-\\jmath\\beta}%{{\\beta_\\mathrm{s}}\\sign\\left(\\omega\\right)\\tan\\left(\\frac{\\pi\\alpha}%{{\\alpha_\\mathrm{s}}}{2}\\right)\n        \\right]\n        +\\jmath\\mu}%{{\\mu_\\mathrm{s}}\\omega\\right\\},    &   \\textrm{if}~\\alpha}%{{\\alpha_\\mathrm{s}} \\neq 1 \\\\\n        \\exp\\left\\{-\\gamma}%{{\\gamma_\\mathrm{s}}\\left|\\omega\\right|\n        \\left[1+\\jmath\\beta}%{{\\beta_\\mathrm{s}}\\sign\\left(\\omega\\right)\\frac{2}{\\pi}\\ln|\\omega|\\right]\n        +\\jmath\\mu}%{{\\mu_\\mathrm{s}}\\omega\\right\\},    &   \\textrm{if}~\\alpha}%{{\\alpha_\\mathrm{s}} = 1.\n        \\end{cases}\n\\end{align}\n\nWe discuss the special cases for the stable distribution as follows:\n\\begin{itemize}\n\n\\item   \n\nWhen $\\alpha}%{{\\alpha_\\mathrm{s}} \\neq 1$: The representation problem of stable densities with $\\alpha}%{{\\alpha_\\mathrm{s}}\\neq 1$ is equivalent to finding the inverse Fourier transform of the form\n    \\begin{align}\n        \\psi_{\\alpha}%{{\\alpha_\\mathrm{s}},\\theta}\\left(\\omega\\right)\n        =\n            \\exp\\left\\{-\\left|\\omega\\right|^\\alpha}%{{\\alpha_\\mathrm{s}} e^{-\\jmath \\frac{\\pi}{2}\\theta \\sign\\left(\\omega\\right)}\\right\\}\n    \\end{align}\n    where $\\left|\\theta\\right|\\leq\\min\\left\\{\\alpha}%{{\\alpha_\\mathrm{s}},2-\\alpha}%{{\\alpha_\\mathrm{s}}\\right\\}$. The PDF of $\\rv{x}\\sim\\Stable{\\alpha}%{{\\alpha_\\mathrm{s}}}{\\beta}%{{\\beta_\\mathrm{s}}}{\\gamma}%{{\\gamma_\\mathrm{s}}}{\\mu}%{{\\mu_\\mathrm{s}}}$ for $\\alpha}%{{\\alpha_\\mathrm{s}} \\neq 1$ is given by\n    \\begin{align}\n        \\PDF{\\rv{x}}{x}\n        =\n            \\Fox{1}{1}{2}{2}{\\left|x-\\mu}%{{\\mu_\\mathrm{s}}\\right|; \\BB{\\pDefine{P}}_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}},\\gamma}%{{\\gamma_\\mathrm{s}},\\mu}%{{\\mu_\\mathrm{s}}}}\n    \\end{align}\n    where the parameter sequence $\\BB{\\pDefine{P}}_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}},\\gamma}%{{\\gamma_\\mathrm{s}},\\mu}%{{\\mu_\\mathrm{s}}}$ is\n    \\begin{align}\t\t\\label{eq:pSeq:stable}\n    \\BB{\\pDefine{P}}_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}},\\gamma}%{{\\gamma_\\mathrm{s}},\\mu}%{{\\mu_\\mathrm{s}}}\n    &=\n        \\biggl(\n                \\frac{\\omega_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}},\\gamma}%{{\\gamma_\\mathrm{s}}}}{\\alpha}%{{\\alpha_\\mathrm{s}}},\n                \\omega_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}},\\gamma}%{{\\gamma_\\mathrm{s}}},\n                \\BB{\\pDefine{a}}_\\mathrm{s}, \\BB{\\pDefine{b}}_\\mathrm{s},\\BB{\\pDefine{A}}_\\mathrm{s}, \\BB{\\pDefine{B}}_\\mathrm{s}\n        \\biggr)\n    \\end{align}\n    with\n    \\begin{align}\n    \\begin{cases}\n    \\BB{\\pDefine{a}}_\\mathrm{s}\n    &\\hspace{-0.35cm}=\n    \t\t\\Bigl(1-\\dfrac{1}{\\alpha}%{{\\alpha_\\mathrm{s}}},\\dfrac{1}{2}-\\sign\\left(x-\\mu}%{{\\mu_\\mathrm{s}}\\right)\\theta_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}}}\\Bigr)\\\\\n    \\BB{\\pDefine{b}}_\\mathrm{s}\n    &\\hspace{-0.35cm}=\n    \t\t\\Bigl(0,\\dfrac{1}{2}-\\sign\\left(x-\\mu}%{{\\mu_\\mathrm{s}}\\right)\\theta_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}}}\\Bigr)\\\\\n    \\BB{\\pDefine{A}}_\\mathrm{s}\n    &\\hspace{-0.35cm}=\n                \\Bigl(\\dfrac{1}{\\alpha}%{{\\alpha_\\mathrm{s}}},\\dfrac{1}{2}+\\sign\\left(x-\\mu}%{{\\mu_\\mathrm{s}}\\right)\\theta_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}}}\\Bigr)\\\\\n    \\BB{\\pDefine{B}}_\\mathrm{s}\n    &\\hspace{-0.35cm}=\n                \\Bigl(1,\\dfrac{1}{2}+\\sign\\left(x-\\mu}%{{\\mu_\\mathrm{s}}\\right)\\theta_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}}}\\Bigr)\n    \\end{cases}\n    \\end{align}\nand\n    \\begin{align}\n    \\omega_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}},\\gamma}%{{\\gamma_\\mathrm{s}}}\n    &=\n        \\left(\\gamma}%{{\\gamma_\\mathrm{s}}\\sqrt{1+\\beta}%{{\\beta_\\mathrm{s}}^2\\tan^2\\left(\\pi\\alpha}%{{\\alpha_\\mathrm{s}}\/2\\right)}\\right)^{-1\/\\alpha}%{{\\alpha_\\mathrm{s}}}\n    \\\\\n    \\theta_{\\alpha}%{{\\alpha_\\mathrm{s}},\\beta}%{{\\beta_\\mathrm{s}}}\n    &=\n        \\frac{1}{\\pi\\alpha}%{{\\alpha_\\mathrm{s}}}\n        \\tan^{-1}\\left(\\beta}%{{\\beta_\\mathrm{s}}\\tan\\frac{\\pi\\alpha}%{{\\alpha_\\mathrm{s}}}{2}\\right).\n    \\end{align}\nThe result shows that \\emph{all} stable densities with $\\alpha}%{{\\alpha_\\mathrm{s}}\\neq 1$ can be expressed in terms of $H$-functions.\n\n\\item\t\t\n\nWhen $\\alpha}%{{\\alpha_\\mathrm{s}}=1$ (\\emph{Terra Incognito}): Due to the presence of the logarithm $\\ln\\left|\\omega\\right|$ in the characteristic function, little information is available about the distributional structure for this case. If $\\rv{x}\\sim \\Stable{1}{\\beta}%{{\\beta_\\mathrm{s}}}{\\gamma}%{{\\gamma_\\mathrm{s}}}{\\mu}%{{\\mu_\\mathrm{s}}}$ is \\emph{strictly} stable (i.e., $\\beta}%{{\\beta_\\mathrm{s}}=0$), we have\n    \\begin{align}\n    \\PDF{\\rv{x}}{x}\n    =\n        \\Fox{1}{1}{2}{2}{\\left|x-\\mu}%{{\\mu_\\mathrm{s}}\\right|;\\BB{\\pDefine{P}}_{1,0,\\gamma}%{{\\gamma_\\mathrm{s}},\\mu}%{{\\mu_\\mathrm{s}}}}\n    \\end{align}\n    where $\\BB{\\pDefine{P}}_{1,0,\\gamma}%{{\\gamma_\\mathrm{s}},\\mu}%{{\\mu_\\mathrm{s}}}$ reduces to\n    \\begin{align}\n    \\BB{\\pDefine{P}}_{\\gamma}%{{\\gamma_\\mathrm{s}}}\n    =\n        \\left(\n                \\frac{1}{\\gamma}%{{\\gamma_\\mathrm{s}}},\\frac{1}{\\gamma}%{{\\gamma_\\mathrm{s}}},\\left(0,\\frac{1}{2}\\right),\\left(0,\\frac{1}{2}\\right),\\left(1,\\frac{1}{2}\\right),\\left(1,\\frac{1}{2}\\right)\n        \\right).\n    \\end{align}\nThis result shows that \\emph{only strictly} stable densities with $\\alpha}%{{\\alpha_\\mathrm{s}}= 1$ are known in terms of $H$-functions.\n\n\n\\item\n\nNonnegative Stable Laws: Since all the one-sided stable distributions are \\emph{extremal} (i.e., $\\beta}%{{\\beta_\\mathrm{s}}=\\pm 1$ with the support $\\mathbbmss{R}_\\pm$), $\\rv{x}\\sim\\Stable{\\alpha}%{{\\alpha_\\mathrm{s}}}{\\beta}%{{\\beta_\\mathrm{s}}}{\\gamma}%{{\\gamma_\\mathrm{s}}}{\\mu}%{{\\mu_\\mathrm{s}}}$ is nonnegative if and only if $0<\\alpha}%{{\\alpha_\\mathrm{s}}<1$, $\\beta}%{{\\beta_\\mathrm{s}}=1$, and $\\mu}%{{\\mu_\\mathrm{s}}=0$ (necessarily strictly stable). Hence,\n    \\begin{align}\n    \\PDF{\\rv{x}}{x}\n    =\n        \\Fox{0}{1}{1}{1}{x;\\BB{\\pDefine{P}}_\\mathrm{pdf}^\\star},   \\quad x\\geq0\n    \\end{align}\n    where\n    \\begin{align}\n    \\BB{\\pDefine{P}}_\\mathrm{pdf}^\\star\n    =\n        \\left(\n                \\frac{\\omega_{\\alpha}%{{\\alpha_\\mathrm{s}},1,\\gamma}%{{\\gamma_\\mathrm{s}}}}{\\alpha}%{{\\alpha_\\mathrm{s}}},\n                \\omega_{\\alpha}%{{\\alpha_\\mathrm{s}},1,\\gamma}%{{\\gamma_\\mathrm{s}}},\n                1-\\frac{1}{\\alpha}%{{\\alpha_\\mathrm{s}}},0,\\frac{1}{\\alpha}%{{\\alpha_\\mathrm{s}}},1\n        \\right).\n    \\end{align}\n    Using the language of $H$-functions, the characteristic function of $\\rv{x}$ can be expressed again in terms of $H$-function as follows:\n    \\begin{align}\n    \\CF{\\rv{x}}{\\omega}\n    =\n        \\Fox{1}{1}{1}{2}{-\\jmath\\omega;\\BB{\\pDefine{P}}_\\mathrm{cf}^\\star}\n    \\end{align}\n    where\n    \\begin{align}\n    \\BB{\\pDefine{P}}_\\mathrm{cf}^\\star\n    =\n        \\left(\n                \\frac{1}{\\alpha}%{{\\alpha_\\mathrm{s}}},\\frac{1}{\\omega_{\\alpha}%{{\\alpha_\\mathrm{s}},1,\\gamma}%{{\\gamma_\\mathrm{s}}}},0,\\B{0}_2,1,\\left(\\frac{1}{\\alpha}%{{\\alpha_\\mathrm{s}}},1\\right)\n        \\right).\n    \\end{align}\n\\emph{All nonnegative} stable random variables are $H$-variates. The aggregate interference power in a Poisson field belongs to this class of stable laws.\n\\end{itemize}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nPattern avoidance is a rich and interesting subject which has received much attention since Knuth first connected the notion of $[231]$-avoidance with stack sortability~\\cite{knuth.TAOCP1}.  Pattern avoidance has also appeared in the study of smoothness of Schubert varieties~\\cite{billeyLakshmibai.2000, Billey98patternavoidance}, the Temperley-Lieb algebra and the computation of Kazhdahn-Lusztig polynomials~\\cite{fan.1996, fanGreen.1999}.  There is also an extensive literature on enumeration of permutations avoiding a given pattern; for an introduction, see~\\cite{bona.permutations}.  \nPattern containment (the complementary problem to pattern avoidance) was previously known to be related to the strong Bruhat order; in particular, Tenner showed that a principal order ideal of a permutation is Boolean if and only if the permutation avoids the patterns $[321]$ and $[3412]$~\\cite{tenner.2007}.  \n\nWhile many have studied pattern avoidance for particular patterns, there are relatively few results on pattern avoidance as a general phenomenon.  Additionally, while there has been a great deal of combinatorial research on pattern avoidance, there have been few algebraic characterizations.  \nIn this paper, we first introduce an equivalence between pattern containment and a factorization problem for certain permutation patterns.  We then use these results directly in analysing the fibers certain quotients of the $0$-Hecke monoid.  Finally, we consider the question of pattern avoidance in the affine permutation group.\n\nWe begin by introducing the notion of a width system, which, in some cases, allows the factorization of a permutation $x$ containing a pattern $\\sigma$ as $x=y\\sigma'z$, where $\\sigma'$ is a `shift' of $\\sigma$, $y$ and $z$ satisfy certain compatibility requirements, and the $\\operatorname{len}(x)=\\operatorname{len}(y)+\\operatorname{len}(\\sigma)+\\operatorname{len}(z)$.  This factorization generalizes an important result of Billey, Jockusch, and Stanley~\\cite{BilleyJockuschStanley.1993}, which states that any permutation $x$ containing a $[321]$-pattern contains a braid; that is, some reduced word for $x$ in the simple transpositions contains a contiguous subword $s_i s_{i+1} s_i$.  (This subword, in our context, plays the role of the $\\sigma'$.)  Equivalently, a permutation that is $[321]$-avoiding is fully commutative, meaning that every reduced word may be obtained by commutation relations.  These permutations have been extensively studied, with major contributions by Fan and Green~\\cite{fan.1996, fanGreen.1999} and Stembridge~\\cite{stembridge.1996}, who associated a certain poset to each fully commutative element, where linear extensions of the poset are in bijection with reduced words for the permutation. \n\nWidth systems allow us to extend this notion of subword containment considerably, and give an algebraic condition for pattern containment for certain patterns.  The width system is simply a measure of various widths of a pattern occurrence within a permutation (called an `instance').  For certain width systems, an instance of minimal width implies a factorization of the form discussed above.  These width systems tend to exist for relatively long permutations.  The main results are contained in Propositions~\\ref{prop:s2widthSystems}, \\ref{prop:s3widthSystems}, \\ref{prop:widthSystemExtend}, \\ref{prop:widthSystemExtend2}, and Corollary~\\ref{cor:bountifulPerms}.  \n\nWe then apply these results directly, and study pattern avoidance of certain patterns (most interestingly  $[321]$-avoidance) in the context of quotients of the $0$-Hecke monoid.\nThe non-decreasing parking functions $\\operatorname{NDPF}_N$ may be realized as a quotient of the $0$-Hecke monoid for the symmetric group $S_N$, and coincide with the set of order-preserving regressive functions on a poset when the poset is a chain.  These functions are enumerated by the Catalan numbers; if one represents $f \\in \\operatorname{NDPF}_N$ as a step function, its graph will be a (rotated) Dyck path.  These functions form a $\\mathcal{J}$-trivial monoid under composition, and may be realized as a quotient of the $0$-Hecke monoid; the monoid $\\operatorname{NDPF}_n$ coincides with the Catalan monoid.  We show that the fibers of this quotient each contain a unique $[321]$-avoiding permutation of minimal length and a $[231]$-avoiding permutation of maximal length (Theorem ~\\ref{thm:ndpfFibers231}).  We then show that a slightly modified quotient has fibers containing a unique $[321]$-avoiding permutation of minimal length, and a $[312]$-avoiding permutation of maximal length (Theorem~\\ref{thm:ndpfFibers312}).\n\nThis provides a bijection between $[312]$ and $[321]$-avoiding permutations.  The bijection is equivalent to the bijection of Simion and Schmidt between $[132]$-avoiding permutations and $[123]$-avoiding permutations~\\cite{simion.schmidt.1985}, but here we have given an algebraic interpretation of the bijection.  (The patterns $[312]$ and $[123]$ are the respective ``complements'' of the patterns $[312]$ and $[321]$.)\n\nWe then combine these results to obtain a bijection between $[4321]$-avoiding permutations and elements of a submonoid of $\\operatorname{NDPF}_{2N}$ (Theorem~\\ref{thm:ndpfFibers4321}), which we consider as a parabolic submonoid of a type $B$ generalization of non-decreasing parking functions, which coincide with the double Catalan monoid~\\cite{mazorchukSteinberg2011}.\n\nWe then expand our discussion to the affine symmetric group and affine $0$-Hecke monoid.  The affine symmetric group was introduced originally by Lusztig~\\cite{lusztig.1983}, and questions concerning pattern avoidance in the affine symmetric group have recently been studied by Lam~\\cite{Lam06affinestanley}, Green~\\cite{Green.2002}, Billey and Crites~\\cite{billeyCrites.2011}.  Lam and Green separately showed that an affine permutation contains a $[321]$-pattern if and only if it contains a braid, in the same sense as in the finite case.  \n\nWe introduce a definition for affine non-decreasing parking functions $\\operatorname{NDPF^{(1)}}_N$, and demonstrate that this monoid of functions may be obtained as a quotient of the affine symmetric group.  We obtain a combinatorial map from affine permutations to $\\operatorname{NDPF^{(1)}}_N$ and demonstrate that this map coincides with the definition of $\\operatorname{NDPF^{(1)}}_N$ by generators and relations as a quotient of $\\tilde{S}_N$.  Finally, we prove that each fiber of this quotient contains a unique $[321]$-avoiding element of minimal length (Theorem~\\ref{thm:affNdpfFibers321}).  \n\n\\subsection{Overview.}\n\nIn Section~\\ref{sec:widthSystems} we introduce \\textbf{width systems} on permutation patterns as a potential system for understanding pattern containment algebraically.  The main results of this section describe a class of permutation patterns $\\sigma$ such that any permutation $x$ containing $\\sigma$ factors as $x=y\\sigma' z$, with $\\operatorname{len}(x)=\\operatorname{len}(y)+\\operatorname{len}(\\sigma)+\\operatorname{len}(z)$.  Here $\\sigma'$ is a ``shift'' of $\\sigma$, and some significant restrictions on $y$ and $z$ are established.  \nThe main results are contained in Propositions~\\ref{prop:s2widthSystems}, \\ref{prop:s3widthSystems}, \\ref{prop:widthSystemExtend},\\ref{prop:widthSystemExtend2}, and Corollary~\\ref{cor:bountifulPerms}.  \n\nWe apply these ideas directly in Section~\\ref{sec:ndpfPattAvoid} while analyzing the fiber of a certain quotient of the $0$-Hecke monoid of the symmetric group.  In Theorem~\\ref{thm:ndpfFibers231}, we show that each fiber of the quotient contains a unique $[321]$-avoiding permutation and a unique $[231]$-avoiding permutation.  We then apply an involution and study a slightly different quotient in which fibers contain a unique $[321]$-avoiding permutation and a unique $[312]$-avoiding permutation (Theorem~\\ref{thm:ndpfFibers312}).  In Section~\\ref{sec:bndpfPattAvoid}, we consider a different monoid-morphism of the $0$-Hecke monoid for which each fiber contains a unique $[4321]$-avoiding permutation (Theorem~\\ref{thm:ndpfFibers4321}).\n\nWe then define the Affine Nondecreasing Parking Functions in Section~\\ref{sec:affNdpfPattAvoid}, and establish these as a quotient of the $0$-Hecke monoid of the affine symmetric group.  We prove the existence of a unique $[321]$-avoiding affine permutation in each fiber of this quotient (Theorem~\\ref{thm:affNdpfFibers321}). \n\n\\subsection{Acknowledgements.}\n\nThis paper originally appeared as a chapter in the author's PhD thesis, awarded by the University of California, Davis.  As such thanks are due to my co-advisors, Prof. Anne Schilling and Nicolas M. Thi\\'ery, as well as my committee members, who provided useful feedback during the writing process.  Thanks are also due to the incredible math department at Davis, which provided a fertile ground for study for five years.  As I prepare this paper, I am a postdoctoral researcher at York University.  Additional support (and copious amounts of coffee) is provided by the Fields Institute.\n\n\\section{Background and Notation}\n\\label{sec:bgnot}\n\n\\subsection{Pattern Avoidance.}\n\nPattern avoidance phenomena have been studied extensively, originally by Knuth in his 1973 classic, The Art of Computer Programming~\\cite{knuth.TAOCP1}.  A thorough introduction to the subject may be found in the book ``Combinatorics of Permutations'' by Bona~\\cite{bona.permutations}.  A \\textbf{pattern} $\\sigma$ is a permutation in $S_k$ for some $k$; given a permutation $x \\in S_N$, we say that $x$ \\textbf{contains the pattern $\\sigma$} if, in the one-line notation for $x=[x_1, \\ldots, x_N]$, there exists a subsequence $[x_{i_1}, \\ldots, x_{i_k}]$ whose elements are in the same relative order as the elements in $p$.  If $x$ does not contain $\\sigma$, then we say that \n$x$ \\textbf{avoids $\\sigma$}, or that $x$ is \\textbf{$\\sigma$-avoiding.}  (Note that if $k>N$, $x$ must avoid $\\sigma$.)\n\nFor example, the pattern $[1, 2]$ appears in any $x$ such that there exists a $x_i < x_j$ for some $i<j$.  The only $[12]$-avoiding permutation in $S_N$, then, is the long element, which is strictly decreasing in one-line notation.  As a larger example, the permutation $[\\mathbf{3}, \\mathbf{4}, 5, \\mathbf{2}, 1, 6]$ contains the pattern $[231]$ at the bold positions.  In fact, this permutation contains six distinct instances of the pattern $[231]$.\n\nAn interesting and natural question is, given a pattern $\\sigma$, how many permutations in $S_N$ avoid $\\sigma$?  It has been known since Knuth's original work that for any pattern in $S_3$, there are Catalan-many permutations in $S_N$ avoiding $\\sigma$~\\cite{knuth.TAOCP1}.\n\nThe $[321]$-avoiding permutations are of particular importance.  It was shown in~\\cite{BilleyJockuschStanley.1993} that a permutation $x\\in S_N$ is $[321]$-avoiding if and only if $x$ is `braid free.'  In particular, this means that there is no reduced word for $x$ containing the consecutive subsequence of $s_i s_{i+1} s_i$ (or $s_{i+1} s_i s_{i+1}$, equivalently), where the $s_i$ are the simple transpositions generating $S_N$.  Such permutations are called \\textbf{fully commutative}. \n\nLam~\\cite{Lam06affinestanley} and Green~\\cite{Green.2002} separately showed that this result extends to the  affine symmetric group.  The affine symmetric group (see Definition~\\ref{def:affSn}) is a subset of the permutations of $\\mathbb{Z}$, satisfying some periodicity conditions.  Pattern avoidance for the affine symmetric group works exactly as in a finite symmetric group.  \nThe one-line notation for $x$ is the doubly infinite sequence $x=[\\ldots, x_{-1}, x_0, x_1, \\ldots, x_N, x_{N+1}, \\ldots]$.\nThen $x$ contains a pattern $\\sigma$ if any subsequence of $x$ in one-line notation has the same relative order as $\\sigma$.  \\textbf{Fully commutative elements} of the affine symmetric group are those which have no reduced word containing the consecutive subsequence $s_i s_{i+1} s_i$, where the indices are considered modulo $N$.\nGreen showed that the fully commutative elements of the affine symmetric group coincide with the $[321]$-avoiding affine permutations.\n\nFan and Green~\\cite{fan.1996, fanGreen.1999} previously studied the quotient of the full Hecke algebra $H_q(W)$ for $W$ simply-laced, by the ideal $I$ generated by $T_{sts} + T_{st} + T_{ts} + T_s + T_t + 1$ for $s$ and $t$ generators of $W$ satisfying a braid relation $sts=tst$.  This quotient $H\/I$ yields the \\textbf{Temperley-Lieb Algebra}.  Fan showed that this quotient has a basis indexed by fully commutative elements of $W$, and in further work with Richard Green derived information relating this quotient to the Kazhdan-Lusztig basis for $H_q(W)$.\n\nA further application of pattern avoidance occurs in the study of rational smoothness of Schubert varieties; an introduction to this topic may be found in~\\cite{billeyLakshmibai.2000}.  The Schubert varieties $X_w$ in Type $A$ are indexed by permutations; a result of Billey~\\cite{Billey98patternavoidance} shows that $X_w$ is smooth if and only if $w$ is simultaneously $[3412]$- and $[4231]$-avoiding.  More recently, Billey and Crites have extended this result to affine Schubert varieties (for affine Type A)~\\cite{billeyCrites.2011}, showing that an affine Schubert variety $X_w$ is rationally smooth if and only if $w$ is simultaneously $[3412]$- and $[4231]$-avoiding or is a special kind of affine permutation, called a ``twisted spiral.''  \n\n\\subsection{$0$-Hecke monoids}\n\\label{ssec:zeroHeckeDefinition}\n\nLet $W$ be a finite Coxeter group with index set $I=\\{1, \\ldots, N-1\\}$. It has a presentation\n\\begin{equation}\n  W = \\langle\\, s_i \\; \\text{for} \\; i\\in I\\ \\text{such that }\\ (s_is_j)^{m(s_i,s_j)},\\ \\forall i,j\\in I\\,\\rangle\\,,\n\\end{equation}\nwhere $I$ is a finite set, $m(s_i,s_j) \\in \\{1,2,\\dots,\\infty\\}$, and $m(s_i,s_i)=1$.\nThe elements $s_i$ with $i\\in I$ are called \\emph{simple reflections}, and the\nrelations can be rewritten as:\n\\begin{equation}\n  \\begin{alignedat}{2}\n    s_i^2 &=1 &\\quad& \\text{ for all $i\\in I$}\\,,\\\\\n    \\underbrace{s_is_js_is_js_i \\cdots}_{m(s_i,s_j)} &=\n    \\underbrace{s_js_is_js_is_j\\cdots}_{m(s_i,s_j)} && \\text{ for all $i,j\\in I$}\\, ,\n  \\end{alignedat}\n\\end{equation}\nwhere $1$ denotes the identity in $W$. An expression $w=s_{i_1}\\cdots s_{i_\\ell}$ for $w\\in W$ \nis called \\emph{reduced} if it is of minimal length $\\ell$.\nSee~\\cite{Bjorner_Brenti.2005, Humphreys.1990} for further details on Coxeter groups.\n\nThe Coxeter group of type $A_{N-1}$ is the symmetric group $S_N$ with generators\n$\\{s_1,\\dots,s_{N-1}\\}$ and relations:\n\\begin{equation}\n  \\begin{alignedat}{2}\n    s_i^2           & = 1                &     & \\text{ for } 1\\leq i\\leq n-1\\,,\\\\\n    s_i s_j         & = s_j s_i            &     & \\text{ for } |i-j|\\geq2\\,, \\\\\n    s_i s_{i+1} s_i & = s_{i+1} s_i s_{i+1} &\\quad& \\text{ for } 1\\leq i\\leq n-2\\,;\n  \\end{alignedat}\n\\end{equation}\nthe last two relations are called the \\emph{braid relations}.\n\n\\begin{definition}[\\textbf{$0$-Hecke monoid}]\nThe $0$-Hecke monoid $H_0(W) = \\langle \\pi_i \\mid i \\in I \\rangle$ of a Coxeter group $W$ \nis generated by the \\emph{simple projections} $\\pi_i$ with relations\n\\begin{equation}\n  \\begin{alignedat}{2}\n    \\pi_i^2 &=\\pi_i &\\quad& \\text{ for all $i\\in I$,}\\\\\n    \\underbrace{\\pi_i\\pi_j\\pi_i\\pi_j\\cdots}_{m(s_i,s_{j})} &=\n    \\underbrace{\\pi_j\\pi_i\\pi_j\\pi_i\\cdots}_{m(s_i,s_{j})} && \\text{ for all $i,j\\in I$}\\ .\n  \\end{alignedat}\n\\end{equation}\nThanks to these relations, the elements of $H_0(W)$ are canonically\nindexed by the elements of $W$ by setting $\\pi_w :=\n\\pi_{i_1}\\cdots\\pi_{i_k}$ for any reduced word $i_1 \\dots i_k$ of $w$.\n\\end{definition}\n\n\n\\subsection{Non-decreasing Parking Functions}\n\\label{ssec:ndpf}\n\nWe consider a collection of functions which form a monoid under composition.\n  Notice that we use the right action in this paper, so\nthat for $x\\in P$ and a function $f:P\\to P$ we write $x.f$ for the value of\n$x$ under $f$. \n\n\\begin{definition}[\\textbf{Monoid of Non-Decreasing Parking Functions}]\n  Let $P=\\{1, \\ldots, N+1\\}$ be a poset. The set $\\operatorname{NDPF}_{N+1}$ of functions $f: P \\to P$ which are\n  \\begin{itemize}\n  \\item \\emph{order preserving}, that is, for all $x,y\\in P,\\ x\\leq_P y$ implies\n    $x.f\\leq_P y.f$\n  \\item \\emph{regressive}, that is, for all $x\\in P$ one has $x.f \\leq_P x$\n  \\end{itemize}\n  is a monoid under composition.\n\\end{definition}\n\\begin{proof}\n  It is trivial that the identity function is order preserving and regressive and that\n  the composition of two order preserving and regressive functions is as well.\n\\end{proof}\n\nAccording to~\\cite[14.5.3]{Ganyushkin_Mazorchuk.2009}, not much is\nknown about these monoids.\n\nWhen $P$ is a chain on $N$ elements, we obtain the monoid $\\operatorname{NDPF}_N$ of\nnondecreasing parking functions on the set $\\{1, \\ldots, N\\}$ (see\ne.g.~\\cite{Solomon.1996}; it also is described under the notation\n$\\mathcal C_N$ in e.g.~\\cite[Chapter~XI.4]{Pin.2009} and, together\nwith many variants, in~\\cite[Chapter~14]{Ganyushkin_Mazorchuk.2009}).\nThe unique minimal set of generators for $\\operatorname{NDPF}_N$ is given by the\nfamily of idempotents $(\\pi_i)_{i\\in\\{1,\\dots,N-1\\}}$, where each\n$\\pi_i$ is defined by $(i+1).\\pi_i:=i$ and $j.\\pi_i:=j$ otherwise. The\nrelations between those generators are given by:\n\\begin{gather*}\n  \\pi_i\\pi_j = \\pi_j\\pi_i \\quad \\text{ for all $|i-j|>1$}\\,,\\\\\n  \\pi_i\\pi_{i-1}=\\pi_i\\pi_{i-1}\\pi_i=\\pi_{i-1}\\pi_i\\pi_{i-1}\\,.\n\\end{gather*}\nIt follows that $\\operatorname{NDPF}_N$ is the natural quotient of $H_0(S_N)$ by\nthe relation $\\pi_i\\pi_{i+1}\\pi_i = \\pi_{i+1}\\pi_i$, via the quotient\nmap $\\pi_i\\mapsto\n\\pi_i$~\\cite{Hivert.Thiery.HeckeSg.2006,Hivert.Thiery.HeckeGroup.2007,\n  Ganyushkin_Mazorchuk.2010}. Similarly, it is a natural quotient of\nKiselman's\nmonoid~\\cite{Ganyushkin_Mazorchuk.2010,Kudryavtseva_Mazorchuk.2009}.  \nIn~\\cite{dhst.2011}, this monoid was studied as an instance of the larger class of order-preserving regressive functions on monoids, and a set of explicit orthogonal idempotents in the algebra was described.\n\n\n\\section{Width Systems, Pattern Containment, and Factorizations.}\n\\label{sec:widthSystems}\n\nIn this section we introduce width systems on permutation patterns, which sometimes provide useful factorizations of a permutation containing a given pattern.  The results established here will be directly applied in Sections~\\ref{sec:ndpfPattAvoid} and~\\ref{sec:bndpfPattAvoid}. \n\n\\begin{definition}\nLet $x$ be a permutation and $\\sigma \\in S_k$ a pattern.  We say that $x$ \\textbf{factorizes over $\\sigma$} if there exist permutations $y$, $z$, and $\\sigma'$ such that:\n\\begin{enumerate}\n\\item $x = y \\sigma' z$,\n\\item $\\sigma'$ has a reduced word matching a reduced word for $\\sigma$ with indices shifted by some $j$,\n\\item The permutation $y$ satisfies $y^{-1}(j)<\\cdots < y^{-1}(j+k)$, \n\\item The permutation $z$ satisfies $z(j)<\\cdots < z(j+k)$,\n\\item $\\operatorname{len}(x) = \\operatorname{len}(y) + \\operatorname{len}(\\sigma') + \\operatorname{len}(z)$.\n\\end{enumerate}\n\\end{definition}\n\nSet $W=S_N$ and $J\\subset I$, with $I$ the generating set of $W$.  An element $x\\in W$ has a \\textbf{right descent} $i$ if $\\operatorname{len}(x s_i)<\\operatorname{len}(x)$, and has a \\textbf{left descent} $i$ if $\\operatorname{len}(s_i x)<\\operatorname{len}(x)$.  Equivalently, $x$ has a right (resp., left) descent at $i$ if and only if some reduced word for $x$ ends (resp., begins) with $i$.  Let $W^J$ be the set of elements in $W$ with no right descents in $J$.  Similarly, $\\leftexp{J}{W}$ consists of those elements with no left descents in $J$.  Finally, $W_J$ is the \\textbf{parabolic subgroup} of $W$ generated by $\\{ s_i \\mid i \\in J\\}$.\n\nRecall that a \\textbf{reduced word} or \\textbf{reduced expression} for a permutation $x$ is a minimal-length expression for $x$ as a product of the simple transpositions $s_i$.  Throughout this chapter, we will use double parentheses enclosing a sequence of indices to denote words.  For example, $((1,3,2))$ corresponds to the element  $s_1s_2s_3$ in $S_4$.  Note that same expression can also indicate an element of $H_0(S_4)$, with $((1,3,2))$ corresponding to the element $\\pi_1\\pi_2\\pi_3$.  Context should make usage clear.\n\n\\begin{definition}\nLet $\\sigma$ be a permutation pattern in $S_k$, with reduced word $((i_1, \\ldots, i_m))$.  Let $J=\\{j, j+1, \\ldots, j+l\\}$ for some $l\\geq k-1$ and $\\sigma' \\in W_J$ with reduced word $((i_1+j, \\ldots, i_m+j))$.  Then we call $\\sigma'$ a \\textbf{$J$-shift} or \\textbf{shift} of $\\sigma$.\n\\end{definition}\n\n\n\\begin{proposition}\nA permutation $x\\in S_N$ factorizes over $\\sigma$ if and only if $x$ admits a factorization $x=y\\sigma' z$ with $y \\in W^J, \\sigma' \\in W_J$, and $z\\in \\leftexp{J}{W}$, and $\\operatorname{len}(x) = \\operatorname{len}(y) + \\operatorname{len}(\\sigma') + \\operatorname{len}(z)$.\n\\end{proposition}\n\\begin{proof}\nThis is simply a restatement of the definition of factorization over $\\sigma$.  In particular, $y\\in W^J$ and $z\\in \\leftexp{J}{W}$.\n\\end{proof}\nThis condition is illustrated diagrammatically in Figure~\\ref{fig.patternContainment} using a string-diagram for the permutation $x$ factorized as $y \\sigma' z$.  In the string diagram of a permutation $x$, a vertical string connects each $j$ to $x(j)$, with strings arranged so as to have as few crossings as possible.\nComposition of permutations is accomplished by vertical concatenation of string diagrams.    In the diagram, $x$ is the vertical concatenation (and product of) of $y$, $\\sigma'$ and $z$.\n\nThe permutation $y^{-1}$ preserves the order of $\\{j, j+1, \\ldots, j+k\\}$, and thus the strings leading into the elements $\\{j, j+1, \\ldots, j+k\\}$ do not cross.  Likewise, $z$ preserves the order of $\\{j, j+1, \\ldots, j+k\\}$, and thus the strings leading out of $\\{j, j+1, \\ldots, j+k\\}$ in $z$ do not cross.  In between, $\\sigma'$ rearranges $\\{j, j+1, \\ldots, j+k\\}$ according to the pattern $\\sigma$.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=1]{patternContainment.pdf}\n  \\end{center}\n  \\caption{Diagrammatic representation of a permutation $x$ factorizing over a pattern $\\sigma$ as $x=y\\sigma z$ by composition of string diagrams. }\n  \\label{fig.patternContainment}\n\\end{figure}\n\nBy the above discussion, it is clear that if $x$ admits a factorization $y\\sigma' z$ with $y \\in W^J, \\sigma' \\in W_J$, and $z\\in \\leftexp{J}{W}$ then $x$ contains $\\sigma$.  The question, then, is when this condition is sharp.  This question is interesting because it provides an algebraic description of pattern containment.  For example, a permutation $x$ which contains a $[321]$-pattern is guaranteed to have a reduced expression which contains a braid.  Braid containment can be re-stated as a factorization over $[321]$.  When the factorization question is sharp, (ie, $x$ contains $\\sigma$ if and only if $x$ factorizes over $\\sigma$) one obtains an algebraic description of $\\sigma$-containment.  The class of patterns with this property is rather larger than just $[321]$, as we will see in Propositions~\\ref{prop:s2widthSystems}, \\ref{prop:s3widthSystems}, and~\\ref{prop:widthSystemExtend}.\n\n\\begin{problem}\nFor which patterns $\\sigma$ does $x$ contain $\\sigma$ if and only if $x\\in W^J \\sigma' \\leftexp{J}{W}$, where $\\sigma'$ is a $J$-shift of $\\sigma$ for some $J$?\n\\end{problem}\n\nAs a tool for attacking this problem, we introduce the notion of a width system for a pattern.\n\n\\begin{definition}\nSuppose $x$ contains $\\sigma$ at positions $(i_1, \\ldots, i_k)$; the tuple $P=(P_1, \\ldots, P_k)$ is called an \\textbf{instance} of the pattern $\\sigma$, and we denote the set of all instances of $\\sigma$ in $x$ by $P_x$.\n\\end{definition}\n\n\\begin{definition}\nA \\textbf{width} on an instance $P$ of $\\sigma$ is a difference $P_j-P_i$ with $j>i$.  \nA \\textbf{width system} $w$ for a permutation pattern $\\sigma \\in S_k$ is a function assigning a tuple of widths to each instance of $\\sigma$ in $x$.  An instance $P$ of a pattern in $x$ is \\textbf{minimal} (with respect to $\\sigma$ and $w$) if $w(P)$ is lexicographically minimal amongst all instances of $\\sigma$ in $x$.  Finally, an instance $P=(P_1, \\ldots, P_k)$ is \\textbf{locally minimal} if $P$ is the minimal instance of $\\sigma$ in the partial permutation $[x_{P_1}, x_{P_1+1}, \\ldots, x_{P_k-1}, x_{P_k}]$.\n\\end{definition}\n\n\\begin{example}\nConsider the pattern $[231]$ and let $P=(p, q, r)$ be an arbitrary instance of $\\sigma$ in a permutation $x$.  We choose to consider the width system $w(P)=(r-p, q-p)$.  (Other width systems include $u(P)=(r-q, q-p)$ and $v(P)=(r-q)$, for example.)\n\nThe permutation $x=[3, 4, 5, 2, 1, 6]$ contains six $[231]$ patterns.  The following table records each $[231]$-instance $P$ and the width of the instance $w(P)$:\n\\begin{equation*}\n\\begin{array}[b]{|c c c|}\n \\hline           &         P     &  w(P)  \\\\ \\hline\n \\left[\\mathbf{3}, \\mathbf{4}, 5, \\mathbf{2}, 1, 6\\right]       & (1, 2, 4) & (3, 1) \\\\\n \\left[\\mathbf{3}, \\mathbf{4}, 5, 2, \\mathbf{1}, 6\\right]       & (1, 2, 5) & (4, 1) \\\\\n \\left[\\mathbf{3}, 4, \\mathbf{5}, \\mathbf{2}, 1, 6\\right]       & (1, 3, 4) & (3, 2) \\\\\n \\left[\\mathbf{3}, 4, \\mathbf{5}, 2, \\mathbf{1}, 6\\right]       & (1, 3, 5) & (4, 2) \\\\\n \\left[3, \\mathbf{4}, \\mathbf{5}, \\mathbf{2}, 1, 6\\right]       & (2, 3, 4) & (2, 1) \\\\\n \\left[3, \\mathbf{4}, \\mathbf{5}, 2, \\mathbf{1}, 6\\right]       & (2, 3, 5) & (3, 1) \\\\  \\hline\n\\end{array}\n\\end{equation*}\nThus, under the width system $w$ the instance $(2, 3, 4)$ is the minimal $[231]$-instance; it is also the only locally minimal $[231]$-instance.\n\nIn the permutation $y=[1, 4, 8, 5, 2, 7, 6, 3]$, we have the following instances and widths of the pattern $[231]$:\n\\begin{equation*}\n\\begin{array}[b]{|c c c|}\n  \\hline          &         P     &  w(P)  \\\\ \\hline\n \\left[ 1, \\mathbf{4}, \\mathbf{8}, 5, \\mathbf{2}, 7, 6, 3 \\right]  & (2, 3, 5) & (3, 1) \\\\\n \\left[ 1, \\mathbf{4}, \\mathbf{8}, 5, 2, 7, 6, \\mathbf{3} \\right]  & (2, 3, 8) & (6, 1) \\\\\n \\left[ 1, \\mathbf{4}, 8, \\mathbf{5}, \\mathbf{2}, 7, 6, 3 \\right]  & (2, 4, 5) & (3, 2) \\\\\n \\left[ 1, \\mathbf{4}, 8, \\mathbf{5}, 2, 7, 6, \\mathbf{3} \\right]  & (2, 4, 8) & (6, 2) \\\\\n \\left[ 1, \\mathbf{4}, 8, 5, 2, \\mathbf{7}, 6, \\mathbf{3} \\right]  & (2, 6, 8) & (6, 4) \\\\\n \\left[ 1, \\mathbf{4}, 8, 5, 2, 7, \\mathbf{6}, \\mathbf{3} \\right]  & (2, 7, 8) & (6, 5) \\\\\n \\left[ 1, 4, 8, \\mathbf{5}, 2, \\mathbf{7}, 6, \\mathbf{3} \\right]  & (4, 6, 8) & (4, 2) \\\\\n \\left[ 1, 4, 8, \\mathbf{5}, 2, 7, \\mathbf{6}, \\mathbf{3} \\right]  & (4, 7, 8) & (4, 3) \\\\  \\hline\n\\end{array}\n\\end{equation*}\nHere, the instance $(2, 3, 5)$ is minimal under $w$.  Additionally, the instance $(4, 6, 8)$ is locally minimal, since it is the minimal instance of $[231]$ in the partial permutation \n$\\left[ \\mathbf{5}, 2, \\mathbf{7}, 6, \\mathbf{3} \\right]$.\n\\end{example}\n\nFor certain width systems, minimality provides a natural factorization of $x$ over $\\sigma$.\n\n\\begin{example}\n\\label{ex:231bountiful}\n\nWe consider the width system for the pattern $[231]$ depicted in Figure~\\ref{fig.minimal231}.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=1]{minimal231.pdf}\n  \\end{center}\n  \\caption{A diagram of a minimal $[231]$ pattern.  The circled numbers represent elements $(x_p, x_q, x_r)$ filling the roles of the pattern; the widths are denoted $a$ and $b$, and the restrictions on $x_t$ with $p<t<q$ and $x_s$ with $s<q<r$ implied by minimality of the pair $(a,b)$ are also recorded.  The red arrows record the fact that shifting the end elements towards the center using a sequence of simple transpositions reduces the length of the permutation.}\n  \\label{fig.minimal231}\n\\end{figure}\n\nLet $x=[x_1,x_2,\\ldots,x_N] \\in S_N$ containing a $[231]$-pattern, and let $(p<q<r)$ be the indices of a minimal-width $[231]$-pattern in $x$ under the width system $w=(r-p, q-p)$.  (So $x_r<x_p<x_q$.)\n\nMinimality of the total width $(r-p)$ implies that for every $s$ with $q<s<r$, we have $x_s>x_p(>x_r)$, as otherwise $(x_p, x_q, x_s)$ would be a $[231]$-pattern of smaller width.  Then multiplying $x$ on the right by $u_1=s_{r-1}s_{r-2}\\ldots s_{q+1}$ yields a permutation of length $\\operatorname{len}(x)-(r-q-1)$, with \n\\[\nxu_1=[x_1,\\ldots,x_p,\\ldots,x_q,x_r,x_{q+1}\\ldots,x_N].\n\\]\n\nMinimality of the inner width $(q-p)$ implies that for every $t$ with $p<t<q$, then $x_t<x_r$.  (If $x_p<x_t<x_q$, then $(x_t, x_q, x_r)$ would form a $[231]$-pattern of lower width.  If $x_p>x_t$, then $q$ was not chosen minimally.)  Then multiplying $xu_1$ on the right by $u_2=s_ps_{p+1}\\ldots s_{q-2}$ yields a permutation of length $\\operatorname{len}(xu_1)-(q-p-1) = \\operatorname{len}(x) - r + p + 2)$.  This permutation is:\n\\[\nxu_1u_2=[x_1,\\ldots,x_{q-1},x_p,x_q,x_r,x_{q+1}\\ldots,x_N].\n\\]\n\nSince $[x_p, x_q, x_r]$ form a $[231]$-pattern, we may further reduce the length of this permutation by multiplying on the right by $s_{q}s_{q-1}$.  The resulting permutation has no right descents in the set $J:=\\{q-1, q\\}$.\n\nWe then set $y=xu_1u_2s_{q}s_{q-1}$, $\\sigma'=s_{q-1}s_{q}$, and $z=(u_1u_2)^{-1}$.  Notice that $z$ has no left descents in $\\{q-1, q\\}$ by construction, since it preserved the left-to-right order of $x_p, x_q$ and $x_r$.   Then $x=y \\sigma' z$ is a factorization of $x$ over $\\sigma$.\n\\end{example}\n\nOne may use a similar system of minimal widths to show that any permutation containing a $[321]$-pattern contains a braid, replicating a result of Billey, Jockusch, and Stanley~\\cite{BilleyJockuschStanley.1993}.  The corresponding system of widths is depicted in Figure~\\ref{fig.minimal321}.\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=1]{minimal321.pdf}\n  \\end{center}\n  \\caption{A diagram of a left-minimal $[321]$ pattern, labeled analogously to the labeling in Figure~\\ref{fig.minimal231}.}\n  \\label{fig.minimal321}\n\\end{figure}\n\n\\begin{definition}\nLet $\\sigma$ be a permutation with a width system.  The width system is \\textbf{bountiful} if for any $x$ containing a locally minimal $\\sigma$ at positions $(p_1, \\ldots, p_k)$, any $x_t$ with $p_i<t<p_{i+1}$ has either $x_t<x_{p_k}$ for all $p_k<t$ or $x_t>x_{p_k}$ for all $p_k>t$.\n\\end{definition}\n\n\\begin{proposition}\nIf a pattern $\\sigma$ admits a bountiful width system, then any $x$ containing $\\sigma$ factorizes over $\\sigma$.\n\\end{proposition}\n\\begin{proof}\nBy definition, any $x_t$ with $p_i<t<p_{i+1}$ has either $x_t<x_{p_k}$ for all $p_k<t$ or $x_t>x_{p_k}$ for all $p_k>t$.  Then using methods exactly as in Example~\\ref{ex:231bountiful}, we may vacate the elements $x_t$ by multiplying on the right by simple transpositions, moving ``small'' $x_t$ out to the left and moving ``large'' $x_t$ out to the right.  This brings the minimal instance of the pattern $\\sigma$ together into adjacent positions $(j, j+1, \\ldots, j+k)$, while simultaneously creating a reduced word for the right factor $z$ in the factorization.  Then we set $J=\\{j, j+1, \\ldots, j+k-1\\}$, and let $\\sigma'$ be the $J$-shift of $\\sigma$.  Set $y=x z^{-1} \\sigma'^{-1}$.  Then by construction $x=y \\sigma' z$ is a factorization of $x$ over $\\sigma$.\n\\end{proof}\n\nThus, establishing bountiful width systems allows the direct factorization of $x$ containing $\\sigma$ as an element of $W^J \\sigma' \\leftexp{J}{W}$.\n\n\\begin{problem}\nCharacterize the patterns which admit bountiful width systems.\n\\end{problem}\n\n\\begin{example}\n\\label{ex:123broken}\nThe permutation $x = [1324] = s_2$ contains a $[123]$-pattern, but does not factor over $[123]$.  To factor over $[123]$, we have $x\\in W^J 1_J \\leftexp{J}{W}$, with $J=\\{1,2\\}$ or $J=\\{2,3\\}$.  Both choices for $J$ contain $2$, so it is impossible to write $x$ as such a product.\n\\end{example}\n\n\n\\begin{proposition}\n\\label{prop:s2widthSystems}\nBoth patterns in $S_2$ admit bountiful width systems.\n\\end{proposition}\n\\begin{proof}\nAny minimal $[12]$- or $[21]$-pattern must be adjacent, and so the conditions for a bountiful width system hold vacuously.\n\\end{proof}\n\n\\begin{proposition}\n\\label{prop:s3widthSystems}\nAll of the patterns in $S_3$ except $[123]$ admit a bountiful width system, as depicted in Figure~\\ref{fig.s3widthSystems}.\n\\end{proposition}\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=1]{s3widthSystems.pdf}\n  \\end{center}\n  \\caption{Diagrams of bountiful width systems for the five patterns in $S_3$ which admit bountiful width systems.}\n  \\label{fig.s3widthSystems}\n\\end{figure}\n\n\\begin{proof}\nA bountiful width systems has already been provided for the pattern $[231]$.  We only provide the details of the proof that the $[213]$ pattern is bountiful, as the proofs that the width systems for the patterns $[132]$, $[312]$ and $[321]$ are bountiful are analogous.\n\nLet $x\\in S_N$ contain a $[213]$ pattern at positions $(x_p, x_q, x_r)$, and choose the width system $(a,b)=(r-q, q-p)$.  \n\nSuppose that $(x_p, x_q, x_r)$ is lexicographically minimal in this width system, and consider $x_t$ with $p<t<q$ and $x_s$ with $q<s<r$. Then $a=1$:\n\\begin{itemize}\n\\item If $x_s<x_p$, then $(x_p, x_s, x_r)$ is a $[213]$ pattern with $a$ smaller.\n\\item If $x_p<x_s$, then $(x_p, x_q, x_s)$ is a $[213]$ pattern with $a$ smaller.\n\\end{itemize}\nThus, we must have $r-q=1$.\n\nSince $b$ is minimal, we must also have that $x_t>x_q$ or $x_t<x_p$ for every $t$ with $p<t<q$.  This completes the proof that the width system is bountiful.\n\\end{proof}\n\n\n\n\\begin{proposition}\n\\label{prop:widthSystemExtend}\nLet $\\sigma$ be a pattern in $S_{K-1}$ with a bountiful width system, and let $\\sigma_+ = [K, \\sigma_1, \\ldots, \\sigma_{K-1}]$. Then $\\sigma_+$ admits a bountiful width system.  \n\nSimilarly, let $\\sigma_-= [\\sigma_1+1, \\ldots, \\sigma_{K-1}+1, 1]$.  Then $\\sigma_-$ admits a bountiful width system.\n\\end{proposition}\n\\begin{proof}\nLet $w=(w_1, w_2, \\ldots, w_{k-2})$ be a bountiful width system on $\\sigma$ (so $w_i$ is the difference between indices of an instance of $\\sigma$ in a given permutation).  Let $x$ contain $\\sigma_+$ in positions \n$( x_p, \\ldots, x_q )$.  For $\\sigma_+$, we show that the width system \n$w_+=(w_1, w_2, \\ldots, w_{k-2}, q-p)$ is bountiful, where $w_i$ measures widths of elements in $\\sigma$ as in $w$.\n\nConsider a $\\sigma_+$-pattern in a permutation $x$ that is minimal under the width system $w_+$, appearing at indices given by the tuple $p:=(i_1, \\ldots, i_{k+1})$.  Then $x$ contains a $\\sigma$-pattern at positions $(i_2, \\ldots, i_{k+1})$.  This pattern may not be minimal under $w$ but, by the choice of width system, is as close as possible to being $w$-minimal, in the following sense.\n\nWe examine two cases.\n\\begin{itemize}\n\\item If there are no indices $t$ with $i_2<t<i_{k+1}$ such that $x_t>x_{i_1}$, then $\\sigma$ must be $w$-minimal on the range $i_2, \\ldots, i_{k+1}$.  (Otherwise, a $w$-minimal $\\sigma$-pattern in that space would extend to a pattern that was less than $p$ in the $w_+$ width system.)  Then bountifulness of the $\\sigma$ pattern ensures that for any $t$ with $i_j<t<i_{j+1}$ with $j\\geq 2$; then $x_t<x_{i_k}$ for all $i_k<t$ or $x_t>x_{i_k}$ for all $i_k>t$.  (The ``small'' elements are still smaller than the ``large'' element $x_{i_1}$.)\n\n\n\n\\item On the other hand, if there exist some $t$ with $i_2<t<i_{k+1}$ such that $x_t>x_{i_1}$, we may move these $x_t$ out of the $\\sigma$ pattern to the right by a sequence of simple transpositions, each decreasing the length of the permutation by one.  Let $u$ be the product of this sequence of simple transpositions.  Then $xu$ fulfills the previous case.  Each of the $x_t$ were larger than all pattern elements to the right, so we see that $\\sigma_+$ fulfills the requirements of a bountiful pattern.\n\n\\end{itemize}\n\nThe proof that $\\sigma_-$ admits a bountiful width system is similar.\n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:bountifulPerms}\nLet $\\sigma \\in S_K$ be a permutation pattern, where the length of $\\sigma$ is at most one less than the length of the long element in $S_K$.  Then $\\sigma$ admits a bountiful width system.\n\\end{corollary}\n\\begin{proof}\nThis follows inductively from Proposition~\\ref{prop:widthSystemExtend}, and the fact that the patterns $[12]$ and $[21]$ both admit bountiful width systems.\n\\end{proof}\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=1]{widthSystemExtend.pdf}\n  \\end{center}\n  \\caption{Diagram of extensions of a bountiful width system $w$ by the additional widths $a$ or $(a, b)$, as described in the proofs of Propositions~\\ref{prop:widthSystemExtend} and~\\ref{prop:widthSystemExtend2}. }\n  \\label{fig.widthSystemExtend}\n\\end{figure}\n\n\\begin{proposition}\n\\label{prop:widthSystemExtend2}\nLet $\\sigma$ be a pattern in $S_{K-2}$ with a bountiful width system, and let $\\sigma_{++} = [K-1, K, \\sigma_1, \\ldots, \\sigma_{K-1}]$. Then $\\sigma_{++}$ admits a bountiful width system.  \n\nSimilarly, let $\\sigma_{--}= [\\sigma_1+2, \\ldots, \\sigma_{K-2}+2, 1, 2]$.  Then $\\sigma_{--}$ admits a bountiful width system.\n\\end{proposition}\n\\begin{proof}\nThe proof of this proposition closely mirrors the proof of Proposition~\\ref{prop:widthSystemExtend}.  Let $w=(w_1, w_2, \\ldots, w_{k-2})$ a bountiful width system on $\\sigma$.  Let $x$ contain $\\sigma_{++}$ in positions \n$( x_p, x_r, x_s, \\ldots, x_q )$.  For $\\sigma_{++}$, we claim that the width system \n$w_{++}=(w_1, w_2, \\ldots, w_{k-2}, q-p, s-r)$ is bountiful, where $w_i$ measures widths of elements in $\\sigma$ as in $w$.  (The width system $w_{++}$ is depicted in Figure~\\ref{fig.widthSystemExtend}.)\n\nAgain, local minimality of $\\sigma$ ensures that all $x_t$ with $s<t<q$ with $x_t$ not in the instance of $\\sigma_{++}$ are either smaller than all pattern elements to the left of $x_t$, or larger than all pattern elements to the right of $x_t$.  The choice of $w_{++}$ ensures that all $x_t$ with $p<t<r$ are either less than $x_p$ or larger than $x_r$, and that all $x_t$ with $r<t<s$ are less than $x_p$.  Then $w_{++}$ is bountiful.\n\nThe proof that $\\sigma_{--}$ is bountiful is analogous.\n\\end{proof}\n\n\\subsection{Further Directions}\nPreliminary investigation suggests that patterns admitting a bountiful width system are somewhat rare, though there are more than those described by Corollary~\\ref{cor:bountifulPerms}.  Weakening the definition of a factorization over a permutation may provide an additional avenue of investigation, though.\n\n\\begin{definition}\nA permutation $x\\in W=S_N$ \\textbf{left-factorizes} over a pattern $\\sigma\\in S_K$ if $x=y\\sigma' z$ with:\n\\begin{itemize}\n\\item $\\sigma' \\in W_J$, with $J=\\{j, j+1, \\ldots, j+k\\}$ and $\\sigma'$ containing a $\\sigma$-pattern,\n\\item $y \\in W^J$,\n\\item $\\operatorname{len}(x)=\\operatorname{len}(y) + \\operatorname{len}(\\sigma) + \\operatorname{len}(z)$.\n\\end{itemize}\n\\end{definition}\n\nThis definition drops the requirement that $z \\in \\leftexp{J}{W}$.  This definition may be too weak, though, since one can show that any permutation containing the pattern $[K, K-1, \\ldots, 1]$ left-factors over every pattern in $S_K$.\n\nOn the other hand, consider Example~\\ref{ex:123broken}.  The permutation $x=[1, 3, 2, 4]=s_2$ admits a factorization $S^{\\{1,3\\}} 1_{\\{1,3\\}} \\leftexp{\\{1,3\\}}{S}$, and the element $1_{\\{1,3\\}}$ contains a $[123]$-pattern.  Allowing factorizations over arbitrary subgroups -- and obtaining a combinatorial characterization of these factorizations -- may provide a way forward.\n\n\\begin{problem}\nFind a general characterization of pattern containment in terms of factorizations of a permutation.\n\\end{problem}\n\n\n\\section{Pattern Avoidance and the $\\operatorname{NDPF}$ Quotient}\n\\label{sec:ndpfPattAvoid}\n\nIn this section, we consider certain quotients of the $0$-Hecke monoid of the symmetric group, and relate the fibers of the quotient to pattern-avoidance.  The $0$-Hecke monoid $H_0(S_N)$ is defined in Definition~\\ref{ssec:zeroHeckeDefinition}, and the Non-decreasing Parking Function $\\operatorname{NDPF}_N$ quotient is discussed in Section~\\ref{ssec:ndpf}, in its guise as the the monoid of order-preserving regressive functions on a chain.\n\n\\begin{definition}\nFor $x\\in H_0(S_N)$, we say $x$ {\\bf contains a braid} if some reduced word for $x$ contains a contiguous subword $\\pi_i \\pi_{i+1} \\pi_i$.  \n\nThe permutation $x$ contains an {\\bf unmatched ascent} if some reduced word for $x$ contains a contiguous subword $\\pi_i \\pi_{i+1}$ that is not part of a braid.  More precisely, if inserting a $\\pi_i$ directly after the $\\pi_i \\pi_{i+1}$ increases the length of $x$, then $x$ contains an unmatched ascent.  Equivalently, $x$ may be factorized as $x=y \\pi_i \\pi_{i+1} z$, where $y$ has no right descents in $\\{i, i+1\\}$, and $z$ has no left descents in $\\{i, i+1\\}$, and $\\operatorname{len}(x)=\\operatorname{len}(y)+2+\\operatorname{len}(z)$.\n\nAn {\\bf unmatched descent} is analogously defined as a contiguous subword $\\pi_{i+1}\\pi_i$ such that insertion of a $\\pi_i$ immediately before this subword increases the length of $x$.  Equivalently, $x$ may be factorized as $x=y \\pi_{i+1}\\pi_i z$, where $y$ has no right descents in $\\{i, i+1\\}$, and $z$ has no left descents in $\\{i, i+1\\}$, and $\\operatorname{len}(x)=\\operatorname{len}(y)+2+\\operatorname{len}(z)$.\n\\end{definition}\n\n\\begin{lemma}\n\\label{lemma:231unmatched}\nFor $x\\in S_N$, $x$ contains a $[231]$-pattern if and only if $x$ has an unmatched ascent.  Likewise, $x$ contains a $[312]$-pattern if and only if $x$ has an unmatched descent.\n\\end{lemma}\n\\begin{proof}\nThis is a straightforward application of the bountiful width system for the patterns $[231]$ and $[312]$.  The resulting factorization contains an unmatched ascent (resp., descent).\n\\end{proof}\n\nThis process of inserting an $s_i$ can be made more precise in the symmetric group setting: suppose $s_{j_1}\\ldots s_{i}s_{i+1}\\ldots s_{j_k}$ is a reduced expression for $x \\in S_N$.  Then write $x=x_1 s_{i}s_{i+1} x_2$.  To insert $s_i$, multiply $x$ on the right by $x_2^{-1}s_ix_2$.  As such, this insertion can be realized as multiplication by some reflection.\n\nThis insertion is generally not a valid operation in $H_0(S_N)$, since inverses do not exist.  However, the operation does make sense in the $\\operatorname{NDPF}$ setting: the $\\operatorname{NDPF}$ relation simply allows one to exchange a braid for an unmatched ascent or vice-versa.\n\n\\begin{theorem}\n\\label{thm:ndpfFibers231}\nEach fiber of the map $\\phi: H_0(S_N)\\to \\operatorname{NDPF}_N$ contains a unique $[321]$-avoiding element of minimal length and a unique $[231]$-avoiding element of maximal length.\n\\end{theorem}\n\n\\begin{proof}\nThe first part of the theorem follows directly from a result of Billey, Jockusch, and Stanley~\\cite{BilleyJockuschStanley.1993}, which states that a symmetric group element contains a braid if and only if the corresponding permutation contains a $[321]$.  Alternatively, one can use the width system for $[321]$ established in Proposition~\\ref{prop:s3widthSystems} to obtain a factorization including a braid.  Then for any $x$ in the fiber of $\\phi$, one can remove braids obtained from minimal-width $[321]$-patterns using the $\\operatorname{NDPF}$ relation and obtain a $[321]$-avoiding element.  Each application of the $\\operatorname{NDPF}$-relation reduces the length of the permutation by one, so this process must eventually terminate in a $[321]$-avoiding element.  Uniqueness follows since there are exactly $C_N$ $[321]$-avoiding elements in $S_N$, where $C_N$ is the $N$th Catalan number, and are thus in bijection with elements of $\\operatorname{NDPF}_N$.\n\nFor the second part, we use the bountiful $[231]$ width system established in Example~\\ref{ex:231bountiful}.  Let $x$ contain a $[231]$-pattern.  The width system allows us to write a factorization $x=y \\pi_i \\pi_{i+1} z$, where $y$ has no right descents in $\\{i, i+1\\}$ and $z$ has no left descents in $\\{i, i+1\\}$.  Then we may apply the $\\operatorname{NDPF}$ relation to insert a $\\pi_i$, turning the $[231]$-pattern into a $[321]$ pattern, and increasing the length of $x$ by one.  Since we are in a finite symmetric group, there is an upper bound on the length one may obtain by this process, and so the process must terminate with a $[231]$-avoiding element.  Recall that $[231]$-avoiding permutations are also counted by the Catalan numbers~\\cite{knuth.TAOCP1}, and apply the same reasoning as above to complete the theorem.\n\\end{proof}\n\nRecall that the \\emph{right action} of $S_N$ acts on positions.  A permutation $y$ has a \\emph{right descent at position i} if the two consecutive elements $y_i,y_{i+1}$ are out of order in one-line notation.  Then multiplying on the right by $s_i$ puts these two positions back in order and reduces the length of $y$ by one.  Likewise, if $y$ does not have a right descent at $i$, multiplying by $s_i$ increases the length by one.\n\n\\begin{example}[Fibers of the $\\operatorname{NDPF}$ quotient]\nFor $S_4$, the fibers of the $\\operatorname{NDPF}$ quotient can be found in Figure~\\ref{fig.s4ndpfFibers}.  \n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=1]{s4ndpfFibers.pdf}\n  \\end{center}\n  \\caption{Fibers of the $\\operatorname{NDPF}$ quotient for $H_0(S_4)$.}\n  \\label{fig.s4ndpfFibers}\n\\end{figure}\nAs a larger example, let $\\sigma=[3,6,4,5,7,2,1]\\in S_7$.  For Lemma~\\ref{lemma:231unmatched}, we find minimal-width $[231]$-patterns, with the element corresponding to the $3$ chosen as far to the left as possible.  (The subsequence $(5, 7, 2)$ of $\\sigma$ is such a minimal $[231]$-pattern.)  Then applying the transformation $[231]\\rightarrow [321]$ on that instance of the pattern preserves the fiber of the $\\operatorname{NDPF}$ quotient, and increases the length of the permutation by $1$.  By sequentially removing eight such minimal $[231]$-patterns, one obtains the long element in $S_7$, which is $[231]$-avoiding.  The fiber containing the long element also contains a $[321]$-avoiding element $[2,3,4,5,6,7,1]$, which has length $6$, and is the shortest element in its fiber.\n\\end{example}\n\n\n\nWe now fix bountiful width system for $[231]$- and $[321]$-patterns, which we will use for the remainder of this section.\n\\begin{definition}\nLet $x\\in S_N$, $x=[x_1, \\ldots, x_N]$ in one-line notation, and consider all $[231]$-patterns $(x_p,x_q,x_r)$ in $x$.  The \\textbf{width} of a $[231]$-pattern $(x_p,x_q,x_r)$ is the pair $(r-p, q-p)$.  The pattern is a \\textbf{minimally chosen $[231]$-pattern} if the width is lexicographically minimal amongst all $[231]$-patterns in $x$.\n\nOn the other hand, call a $[321]$-pattern $(x_p,x_q,x_r)$ \\textbf{left minimal} if for all $t$ with $p<t<q$, $x_t<x_r$, and for all $s$ with $q<s<r$, $x_s>x_q$.\n\\end{definition}\n\nThe following is a direct result of the proof of Lemma~\\ref{lemma:231unmatched}.\n\\begin{corollary}\n\\label{cor:fiber231}\nLet $x\\in S_N$.  Let $(x_p,x_q,x_r)$ be a minimally chosen $[231]$-pattern in $x$.  Then the permutation \n\\[\n[x_1, \\ldots, x_{p-1}, x_q, x_{p+1}, \\ldots, x_{q-1}, x_p, x_{q+1}, \\ldots, x_r, \\ldots, x_N],\n\\]\nobtained by applying the transposition $t_{p,q}$, is in the same $\\operatorname{NDPF}$-fiber as $x$.  The result of applying this transposition is a left-minimal $[321]$-pattern.\n\\end{corollary}\n\n\n\n\\subsection{Involution}\n\\label{subsec:involution}\n\nLet $\\Psi$ be the involution on the symmetric group induced by conjugation by the longest word.  Then $\\Psi$ acts on the generators by sending $s_i \\to s_{N-i}$.  This descends to an isomorphism of $H_0(S_N)$ by exchanging the generators in the same way: $\\pi_i \\to \\pi_{N-i}$.\n\nWe can thus obtain a second map from $H_0(S_N)\\to \\operatorname{NDPF}_N$ by pre-composing with $\\Psi$.  This has the effect of changing the $\\operatorname{NDPF}$ relation to a statement about unmatched \\emph{descents} instead of unmatched ascents.  Then applying the $\\operatorname{NDPF}$ relation allows one to exchange braids for unmatched descents and vice-versa, giving the following theorem.\n\n\\begin{theorem}\n\\label{thm:ndpfFibers312}\nEach fiber of the map $\\phi \\circ \\Psi: H_0(S_N)\\to \\operatorname{NDPF}_N$ contains a unique $[321]$-avoiding element for minimal length and a unique $[312]$-avoiding element of maximal length.\n\\end{theorem}\n\nThe proof is exactly the mirror of the proof in previous section.\n\nWe fix bountiful width system for $[312]$-patterns, and a second bountiful width system for $[321]$-patterns, which we will use for the remainder of this section.\n\\begin{definition}\nLet $x\\in S_N$, $x=[x_1, \\ldots, x_N]$ in one-line notation, and consider all $[312]$-patterns $(x_p,x_q,x_r)$ in $x$.  The \\textbf{width} of a $[312]$-pattern $(x_p,x_q,x_r)$ is the pair $(r-p, r-q)$.  The pattern is a \\textbf{minimally chosen $[312]$-pattern} if the width is lexicographically minimal amongst all $[312]$-patterns in $x$.\n\nLikewise, call a $[321]$-pattern $(x_p,x_q,x_r)$ \\textbf{right minimal} if the \\textbf{right width} $(p-r, r-q)$ is lexicographically minimal amongst all $[321]$-patterns in $x$.\nOn the other hand, call a $[321]$-pattern $(x_p,x_q,x_r)$ \\textbf{right minimal} if for all $t$ with $p<t<q$, $x_t<x_q$, and for all $s$ with $q<s<r$, $x_s>x_p$.\n\\end{definition}\n\n\\begin{corollary}\n\\label{cor:fiber312}\nLet $x\\in S_N$.  Let $(x_p,x_q,x_r)$ be a minimally chosen $[312]$-pattern in $x$.  Then the permutation \n\\[\n[x_1, \\ldots, x_{p-1}, x_q, x_{p+1}, \\ldots, x_{q-1}, x_p, x_{q+1}, \\ldots, x_r, \\ldots, x_N],\n\\]\nobtained by applying the transposition $t_{p,q}$, is in the same $\\operatorname{NDPF}\\circ \\Psi$-fiber as $x$.  The result of applying this transposition is a right-minimal $[321]$-pattern.\n\\end{corollary}\n\n\n\\section{Type B $\\operatorname{NDPF}$ and $[4321]$-Avoidance}\n\\label{sec:bndpfPattAvoid}\n\nIn this section, we establish a monoid morphism of $H_0(S_N)$ whose fibers each contain a unique $[4321]$-avoiding permutation.  To motivate this map, we begin with a discussion of Non-Decreasing Parking Functions of Type $B$.\n\nThe Weyl Group of Type $B$ may be identified with the \\textbf{signed symmetric group} $S_N^B$, which is discussed (for example) in~\\cite{Bjorner_Brenti.2005}.  Combinatorially, $S_N^B$ may be understood as a group permuting a collection of $N$ labeled coins, each of which can be flipped to heads or tails.  The size of $S_N^B$ is thus $2^NN!$.   A minimal set of generators of this group are exactly the simple transpositions $\\{t_i\\mid i \\in \\{1, \\ldots, N-1\\}\\}$ interchanging the coins labeled $i$ and $i+1$, along with an extra generator $t_N$ which flips the last coin.  \n\nThe group $S_N^B$ can be embedded into $S_{2N}$ by identifying the $t_i$ with $s_is_{2N-i}$ for each $i \\in \\{1, \\ldots, N-1\\}$, and $t_N$ with $s_N$.\n\n\\begin{definition}\nThe {\\bf Type B Non-Decreasing Parking Functions} $\\operatorname{BNDPF}_N$ are the elements of the submonoid of $\\operatorname{NDPF}_{2N}$ generated by the collection $\\mu_i := \\pi_i\\pi_{2N-i}$ for $i$ in the set $\\{1, \\ldots, N\\}$.\n\\end{definition}\n\nNote that $\\mu_N = \\pi_N^2 = \\pi_N $.\n\nThe number of $\\operatorname{BNDPF}_N$ has been explicitly computed up to $N=9$, though a proof for a general enumeration has proven elusive, in the absence of a more conceptual description of the full set of functions generated thusly.  The sequence obtained (starting with the $0$-th term) is\n\\[\n\t( 1, 2, 7, 33, 183, 1118, 7281, 49626, 349999, 253507, \\ldots ),\n\\]\nwhich agrees with the sequence \n\\[\n\\sum_{j=0}^N \\binom{N}{j}^2 C_j \n\\]\nso far as it has been computed.  This appears in Sloane's On-Line Encyclopedia of Integer Sequences as sequence $A086618$~\\cite{Sloane}, and was first noticed by Hivert and Thi\\'ery~\\cite{Hivert.Thiery.HeckeGroup.2007}.\n\n\\begin{conjecture}\n\\[ \n| \\operatorname{BNDPF}_N | = \\sum_{j=0}^N \\binom{N}{j}^2 C_j.\n\\]\n\\end{conjecture}\n\nLet $X$ be some object (group, monoid, algebra) defined by generators $S$ and relations $R$.  Recall that a \\emph{parabolic subobject} $X_J$ is generated by a subset $J$ of the set $S$ of simple generators, retaining the same relations $R$ as the original object.  Let $\\operatorname{BNDPF}_{N,\\hat{N}}$ denote the parabolic submonoid of of $\\operatorname{BNDPF}_N$ retaining all generators but $\\mu_N$.\n\nConsider the embedding of $\\operatorname{BNDPF}_{N,\\hat{N}}$ in $\\operatorname{NDPF}_{2N}$.  Then a reduced word for an element of $\\operatorname{BNDPF}_{N,\\hat{N}}$ can be separated into a pairing of $\\operatorname{NDPF}_N$ elements as follows:\n\\begin{align}\n\\mu_{i_1}\\mu_{i_2}\\ldots\\mu_{i_k} &=& \\pi_{i_1}\\pi_{2N-i_1}\\pi_{2N-i_2}\\pi_{i_2}\\ldots\\pi_{i_k}\\pi_{2N-i_k} \\\\\n&=& \\pi_{i_1}\\pi_{i_2}\\ldots\\pi_{i_k}\\pi_{2N-i_1}\\pi_{2N-i_2}\\ldots\\pi_{2N-i_k}\n\\end{align}\n\nIn particular, one can take any element $x\\in H_0(S_N)$ and associate it to the pair:\n\\[\n\\omega(x):=(\\phi(x), \\phi\\circ\\Psi(x)),\n\\] \nrecalling that $\\Psi$ is the Dynkin automorphism on $H_0(S_N)$, described in Section~\\ref{subsec:involution}.  \n\n\nGiven the results of the earlier section, one naturally asks about the fiber of $\\omega$.  It is easy to do some computations and see that the situation is not quite so nice as before.  In $H_0(S_4)$ the only fiber with order greater than one contains the elements $[4321]$ and $[4231]$.  Notice what happens here: $[4231]$ contains both a $[231]$-pattern and a $[312]$-pattern, which is straightened into two $[321]$-patterns.   On the level of reduced words, two reduced words for $[4231]$ are $((3,2,1,2,3))=((1,2,3,2,1))$, one of which ends with the unmatched ascent $[2,3]$ while the other ends with the unmatched descent $[2,1]$.  Multiplying on the right by the simple transposition $s_2$ matches both of these simultaneously.\n\nIn fact, this is a perfectly general operation.  Let $x\\in H_0(S_N)$.  For any minimally-chosen $[231]$-pattern in $x$, one can locate an unmatched ascent in $x$ that corresponds to the pattern.  Here the smaller element to the right remains fixed while the two ascending elements to the left are exchanged.  Then applying the $\\operatorname{NDPF}$ relation to turn the $[231]$ into a $[321]$ preserves the fiber of $\\phi$.  Likewise, one can turn a minimal $[312]$ into a $[321]$ and preserve the fiber of $\\phi\\circ\\Psi(x)$.  Here the larger element to the left is fixed while the two ascending elements to the right are exchanged.  Hence, to preserve the fiber of $\\omega$, one must find a pair of ascending elements with a large element to the left and a small element to the right: this is exactly a $[4231]$-pattern.  \n\nOne may make this more precise by defining a system of widths under which minimal $[4231]$-patterns contain a locally minimal $[231]$-pattern and a locally-minimal $[321]$-pattern.  The results of Section~\\ref{sec:widthSystems} imply that this is possible.  Applying the $\\operatorname{NDPF}$ relation, this becomes a $[4321]$. \n\nOn the other hand, we can define a minimal $[4321]$-pattern by a tuple of widths analogous to the constructions of minimal $[231]$-patterns.  The construction of this tuple, and the constraints implied when the tuple is minimal, is depicted in Figure~\\ref{fig.minimal4321}.  Such a minimal pattern may always be turned into a $[4231]$-pattern while preserving the fiber of $\\omega$.\n\nLet $x\\in S_N$ and $P=(x_p, x_q, x_r, x_s)$ a $[4321]$-pattern in $x$.  For the remainder of this section, we fix the width system $(q-p, r-q, s-r)$, and use the same width system for $[4231]$-patterns.  One may check directly that this is a bountiful width system in both cases.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=1]{minimal4321.pdf}\n  \\end{center}\n  \\caption{A diagram of a minimal $[4321]$ pattern, labeled analogously to the labeling in Figure~\\ref{fig.minimal231}.}\n  \\label{fig.minimal4321}\n\\end{figure}\n\n\\begin{lemma}\nLet $x$ contain a minimal $[4321]$-pattern $P=(x_p, x_q, x_r, x_s)$, and let $x' = x t_{r,s}$, where $t_{r,s}$ is the transposition exchanging $x_r$ and $x_s$.  Then $\\omega(x')=\\omega(x)$.\n\\end{lemma}\n\\begin{proof}\nSince the width system on $[4321]$-patterns is bountiful, we can factor $x = y x_J z$, with $\\operatorname{len}(x)=\\operatorname{len}(y)+ \\operatorname{len}(x_J)+\\operatorname{len}(z)$ where \n\\[x_J = s_{s-2}s_{s-1} s_s s_{s-1} s_{s-2}s_{s-1}.\\]\nBy the discussion above, the trailing $s_{s-1}$ in $x_J$ may be removed to simultaneously yield an unmatched ascent and an unmatched descent.  Then this removal preserves the fiber of both $\\phi$ and $\\Psi\\circ \\phi$, and thus also preserves the fiber of $\\omega$.\n\\end{proof}\n\n\nNote that there need not be a unique $[4231]$-avoiding element in a given fiber of $\\omega$.  The first example of this behavior occurs in $N=7$, where there is a fiber consisting of $[5274163], [5472163],$ and $[5276143]$.  In this list, the first element is $[4321]$-avoiding, and the two latter elements are $[4231]$-avoiding.  In the first element, there are $[4231]$ patterns $[5241]$ and $[7463]$ which can be respectively straightened to yield the other two elements.  Notice that either transposition moves the 4 past the bounding element of the other $[4231]$-pattern, thus obstructing the second transposition.\n\n\\begin{theorem}\n\\label{thm:ndpfFibers4321}\nEach fiber of $\\omega$ contains a unique $[4321]$-avoiding element.\n\\end{theorem}\n\n\\begin{proof}\nGiven any element of $H_0(S_N)$, we have seen that we can preserve the fiber of $\\omega$ by turning locally minimal $[4321]$-patterns into $[4231]$-patterns.  Each such operation reduces the length of the element being acted upon, and thus this can only be done so many times.  Furthermore, any minimal-length element in the fiber of $\\omega$ will be $[4321]$-avoiding.  We claim that this element is unique.  \n\nFirst, note that one can impose a partial order on the fiber of $\\omega$ with $x$ covering $y$ if $x$ is obtained from $y$ by turning a locally minimal $[4321]$-pattern into a $[4231]$-pattern.  Then the partial order is obtained by taking the transitive closure of the covering relation.  Note that if $x$ covers $y$ then $x$ is longer than $y$.  The Hasse diagram of this poset is connected, since any element of the fiber can be obtained from another by a sequence of $\\operatorname{NDPF}$ relations respecting both the fiber of $\\phi$ and $\\phi\\circ\\Psi(x)$.  \n\nLet $x$ be an element of $H_0(S_N)$ containing (at least) two locally minimal $[4321]$-patterns, in positions $(x_a,x_b,x_c,x_d)$ and $(x_p,x_q,x_r,x_s)$, with $a<b<c<d, p<q<r<s$.  Then one can exchange $x_b$ with $x_c$ or $x_q$ with $x_r$ and preserve the fiber of $\\omega$.  Let $y$ be the element obtained from exchanging $x_b$ with $x_c$, and $z$ obtained by exchanging $x_q$ with $x_r$.  Then we claim that there exists $w$ covered by both $y$ and $z$.  (In other words, the poset structure on each fiber is a meet semilattice.)\n\nIf the tuples $(a,b,c,d)$ and $(p,q,r,s)$ are disjoint, then the claim is clearly true.  Likewise, if $a=p$ and\/or $d=s$ the claim holds.  A complete but perhaps unenlightening proof of the claim can be accomplished by showing that it holds for all $\\operatorname{BNDPF}_{N,\\hat{N}}$ with $N<8$, where every possible intermingling of the tuples with every possible ordering of the entries $x_.$ occurs at least once.  It is best to perform this check with a computer, given that there are 2761 elements in $\\operatorname{BNDPF}_{7,\\hat{7}}$, with $7! = 5040$ elements in the fibers, and indeed a computer check shows that the claim holds.  The code accomplishing this is provided below.\n\nLet's look at a couple cases, though, to get a feeling for why this should be true.  Refer to the extremal elements at the edge of the $[4321]$ pattern as the ``boundary,'' and the elements to be transposed as the ``interior.''  The main cases are the following:\n\nCase $c=r$: Just take the smaller of $x_s$ and $x_d$ to be the common right boundary for both patterns.\n\nCase $c=q$: The problem for $[4231]$ patterns was that one could apply a transposition that obstructed the other transposition by moving one of the interior elements past its boundary.  But here, we have $x_d<x_c$ and $x_s<x_q=x_d$, so we can use $s$ as the boundary for both patterns, and the obstruction is averted.  In this case, though, the two transpositions generate six elements in the fiber, instead of four.  We can still find a common meet, though.  $[x_ax_bx_qx_rx_s]$ becomes $[x_ax_qx_bx_rx_s]$ and $[x_ax_bx_rx_qx_s]$, which both cover $[x_ax_rx_bx_qx_s]$, for example.\n\nCase $r=d$ or $q=d$: Again, just take $s$ as a common boundary for the two patterns.\n\nAnd so on.  Many cases are symmetric to the three considered above, and every interesting case is solved by changing the boundary of one of the patterns.\n\nNow that every pair of elements have a common meet, we are almost done.  Suppose there exist two different $[4321]$-avoiding elements $A_1$ and $A_2$ in some fiber.  Then since the fiber is connected, we can find a minimal element $x$ where a branching occurred, so that $x$ covers both $y>A_1$ and $z>A_2$, and $x$ is of minimal length.  But if both $y$ and $z$ were obtainable from $x$, then there exists a $w$ of shorter length below them both.  Now $w$ sits above some $[4321]$-avoiding element, as well.  If $w>A_1$ but not $A_2$, then in fact a branching occurred at $z$, contradicting the minimality of $x$.  The same reasoning holds if $w>A_2$ but not $A_1$.  If $w$ is above both $A_1$ and $A_2$, then in fact $y$ was comparable to $A_2$ and $z$ was comparable to $A_1$, and there was not a branching at $x$ at all.\n\\end{proof}\n\n\\subsection{Code for Theorem~\\ref{thm:ndpfFibers4321}.}\nHere we provide code for checking the claim of Theorem~\\ref{thm:ndpfFibers4321} that each fiber of $\\omega$ contains a unique $[4321]$-avoiding element.  The code is written for the Sage computer algebra system, which has extensive built-in functions for combinatorics of permutations, including detecting the presence of permutation patterns.  \n\nThe code below constructs a directed graph (see the function \\textbf{omegaFibers}) whose connected components are fibers of $\\omega$.  The vertices of this graph are permutations, and the edges correspond to straightening locally-minimal $[4231]$-patterns into $[4321]$ patterns.  A component is `bad' if it does not contain exactly one $[4321]$-avoiding permutation.\n\\begin{verbatim}\ndef width4231(p):\n    \"\"\"\n    This function returns the width of a [4231]-instance p.\n    \"\"\"\n    return (p[1]-p[0], p[2]-p[1], p[3]-p[2]) \n\ndef min4231(x):\n    \"\"\"\n    This function takes a permutation x and finds all minimal-width\n    4231-patterns in x, and returns them as a list.\n    \"\"\"\n    P=x.pattern_positions([4,2,3,1])\n    if P==[]:\n        return None\n    minimal=[P[0]]\n    for i in [1..len(P)-1]:\n        if width4231(P[i])<width4231(minimal[0]):\n            minimal = [ P[i] ]\n        else:\n            if width4231(P[i])==width4231(minimal[0]):\n                minimal.append(P[i])\n    return minimal\n    \ndef localMin4231(x):\n    \"\"\"\n    This function finds all locally-minimal 4231-patterns in a \n    permutation x, and returns them as a list.\n    \"\"\"\n    P=x.pattern_positions([4,2,3,1])\n    if P==[]:\n        return None\n    localMin=[]\n    for p in P:\n        xp=Permutation(x[ p[0]:p[3]+1 ])\n        qp=[i - p[0] for i in p]\n        qmin=min4231(xp)\n        if qp in qmin: localMin.append(p)\n    return localMin\n\ndef omegaFibers(N):\n    \"\"\"\n    Given N, this function builds a digraph whose vertices are given by\n    permutations of N, and with an edge a->b whenever b is obtained\n    from a by straightening a locally minimal 4231-pattern into a \n    4321-pattern. \n    The connected components of G are the fibers of the map omega.\n    \"\"\"\n    S=Permutations(N)\n    G=DiGraph()\n    G.add_vertices(S.list())\n    for x in S:\n        if x.has_pattern([4,2,3,1]):\n            # print x, localMin4231(x)\n            #add edges to G for each locally minimal 4231.\n            Q=localMin4231(x)\n            for q in Q:\n                y=Permutation((q[1]+1,q[2]+1))*x\n                G.add_edge(x,y)\n    return G\n    \ndef headCount(G):    \n    \"\"\"\n    This function takes the diGraph G produced by the omegaFibers \n    function, and finds any connected components with more than one\n    4321-pattern.  It returns a list of all such connected components.\n    \"\"\"\n    bad=[]\n    for H in G.connected_components_subgraphs():\n        total=0\n        for a in H:\n            if not a.has_pattern([4,3,2,1]): total+=1\n        if total != 1:\n            #prints if any fiber has more than one 4321-av elt\n            print H, total\n            bad.append(H)\n    print \"N =\", N \n    print \"\\tTotal connected components: \\t\", count\n    print \"\\tBad connected components: \\t\", len(bad), '\\n'\n    return bad\n\\end{verbatim}\n\nAs explained in Theorem~\\ref{thm:ndpfFibers4321}, we should check that each fiber of $\\omega$ contains a unique $[4321]$-avoiding element for each $N\\leq 7$.  This is accomplished by running the following commands:\n\\begin{verbatim}\nsage: for N in [1..7]:\nsage:     G=omegaFibers(N)\nsage:     HH=headCount(G)\n\\end{verbatim}\nThe output of this loop is as follows:\n\\begin{verbatim}\nN = 1\n\tTotal connected components: \t1\n\tBad connected components: \t0 \n\nN = 2\n\tTotal connected components: \t2\n\tBad connected components: \t0 \n\nN = 3\n\tTotal connected components: \t6\n\tBad connected components: \t0 \n\nN = 4\n\tTotal connected components: \t23\n\tBad connected components: \t0 \n\nN = 5\n\tTotal connected components: \t103\n\tBad connected components: \t0 \n\nN = 6\n\tTotal connected components: \t513\n\tBad connected components: \t0 \n\nN = 7\n\tTotal connected components: \t2761\n\tBad connected components: \t0 \n\\end{verbatim} \nThere are no bad components, and thus the theorem holds.  \n\nThe sequence $(1, 2, 6, 23, 103, 513, 2761)$ is the beginning of the sequence counting $[4321]$-avoiding permutations.  This sequence also counts $[1234]$-avoiding permutations (reversing a $[1234]$-avoiding permutation yields a $[4321]$-avoiding permutation, and \\textit{vice versa}), and is listed in that context in Sloane's On-Line Encyclopedia of Integer Sequences (sequence $A005802$)~\\cite{Sloane}.\n\nThe author executed this code on a computer with a 900-mhz Intel Celeron processor (blazingly fast by 1995 standards) and 2 gigabytes of RAM.  On this machine, the $N=6$ case took 3.86 seconds of CPU time, and the $N=7$ case took just over one minute (62.06s) of CPU time.  The $N=8$ case (which is unnecessary to the proof) correctly returns 15767 connected components, none of which are bad, in 1117.24 seconds (or 18.6 minutes).\n\n\n\\section{Affine $\\operatorname{NDPF}$ and Affine $[321]$-Avoidance}\n\\label{sec:affNdpfPattAvoid}\n\nThe affine symmetric group is the Weyl group of type $A_N^{(1)}$, whose Dynkin diagram is given by a cycle with $N$ nodes.  All subscripts on generators for type $A_N^{(1)}$ in this section will be considered $(\\text{mod } N)$.  A combinatorial realization of this Weyl group is given below.\n\n\\begin{definition}\n\\label{def:affSn}\nThe \\textbf{affine symmetric group} $\\tilde{S}_N$ is the set of bijections $\\sigma: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfying:\n\\begin{itemize}\n    \\item Skew-Periodicity: $\\sigma(i+N) = \\sigma(i) + N$, and\n    \\item Sum Rule: $\\sum_{i=1}^N \\sigma(i) = \\binom{N+1}{2}$.\n\\end{itemize}\n\\end{definition}\n\nWe will often denote elements of $\\tilde{S}_N$ in the \\textbf{window notation}, which is a one-line notation where we only write $(\\sigma(1), \\sigma(2), \\ldots, \\sigma(N))$.  Due to the skew-periodicity restriction, writing the window notation for $\\sigma$ specifies $\\sigma$ on all of $\\mathbb{Z}$.  \n\nThe generators $s_i$ of $\\tilde{S}_N$ are indexed by the set $I=\\{0, 1, \\ldots, N-1\\}$, and $s_i$ acts by exchanging $j$ and $j+1$ for all $j \\equiv i (\\text{mod } N)$.  These satisfy the relations:\n\\begin{itemize}\n    \\item Reflection: $s_i^2 = 1$,\n    \\item Commutation: $s_j s_i = s_i s_j$ when $|i-j| > 1$, and\n    \\item Braid Relations: $s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}$. \\\\\n\\end{itemize}\nIn these relations, all indices should be considered mod $N$.\n\nSince the Dynkin diagram is a cycle, it admits a dihedral group's worth of automorphisms.  One can implement a ``flip'' automorphism $\\Phi$ by fixing $s_0$ and sending $s_i \\rightarrow s_{N-i}$ for all $i \\neq 0$, extending the automorphism used in the finite case.  A ``rotation'' automorphism $\\rho$ can be implemented by simply sending each generator $s_i \\rightarrow s_{i+1}$.  Combinatorially, this corresponds to the following operation.  Given the window notation $(\\sigma_1, \\sigma_2, \\ldots, \\sigma_N)$, we have:\n\\[\n\\rho(\\sigma) = (\\sigma_N-N+1, \\sigma_1 +1, \\sigma_2 +1, \\ldots, \\sigma_{N-1}+1).\n\\]\nThis can be thought of as shifting the base window one place to the left, and then adding one to every entry.  It is clear that this operation preserves the skew periodicity and sum rules for affine permutations, and it is also easy to see that $\\rho^N = 1$.  \n\n\nAs before, we can define the Hecke algebra of $\\tilde{S}_N$, and the $0$-Hecke algebra, generated by $\\pi_i$ with $\\pi_i$ idempotent anti-sorting operators, exactly mirroring the case for the finite symmetric group.  As in the finite case, elements of the $0$-Hecke algebra are in bijection with affine permutations.  We can also define the $\\operatorname{NDPF}$ quotient of $H_0(\\tilde{S}_N)$, by introducing the relation \n\\[\n\\pi_{i+1}\\pi_i \\pi_{i+1} = \\pi_{i+1} \\pi_i.\n\\]  \nThis allows us to give combinatorial definition for the affine $\\operatorname{NDPF}$, which we will prove to be equivalent to the quotient.\n\n\\begin{definition}\nThe extended affine non-decreasing parking functions are the functions $f: \\mathbb{Z}\\rightarrow \\mathbb{Z}$ which are:\n\\begin{itemize}\n    \\item Regressive: $f(i)\\leq i$,\n    \\item Order Preserving: $i\\leq j \\Rightarrow f(i)\\leq f(j)$, and\n    \\item Skew Periodic: $f(i+N) = f(i)+N$.\n\\end{itemize}\nDefine the \\textbf{shift functions} $\\operatorname{sh}_t$ as the functions sending $i \\rightarrow i-t$ for every $i$.\n\nThe \\textbf{affine non-decreasing parking functions} $\\operatorname{NDPF^{(1)}}_N$ are obtained from the extended affine non-decreasing parking functions by removing the shift functions for all $t\\neq 0$.  \n\\end{definition}\n\nNotice that the definition implies that\n\\[\nf(N)-f(1) \\leq N.\n\\]\nFurthermore, since the shift functions are not in $\\operatorname{NDPF^{(1)}}_N$, there is always some $j \\in \\{0, 1, \\ldots, N\\}$ such that $f(j)\\neq f(j+1)$ unless $f$ is the identity.\n\nWe now state the main result of this section, which will be proved in pieces throughout the remainder of the chapter.\n\\begin{theorem}\n\\label{andpfMainThm}\nThe affine non-decreasing parking functions $\\operatorname{NDPF^{(1)}}_N$ are a $\\mathcal{J}$-trivial monoid which can be obtained as a quotient of the $0$-Hecke monoid of the affine symmetric group by the relations $\\pi_j\\pi_{j+1}\\pi_j = \\pi_j \\pi_{j+1}$, where the subscripts are interpreted modulo $N$.  Each fiber of this quotient contains a unique $[321]$-avoiding affine permutation.\n\\end{theorem}\n\n\\begin{proposition}\n\\label{andpfgens}\nAs a monoid, $\\operatorname{NDPF^{(1)}}_N$ is generated by the functions $f_i$ defined by:\n\\begin{displaymath}\n   f_i(j) = \\left\\{\n     \\begin{array}{lr}\n       j-1 & : j\\equiv i+1 (\\text{mod }N)\\\\\n       j   & : j\\not \\equiv i+1 (\\text{mod }N).\\\\\n     \\end{array}\n   \\right.\n\\end{displaymath} \nThese functions satisfy the relations:\n\\begin{align*}\nf_i^2 &=& f_i \\\\\nf_if_j &=& f_jf_i \\text{ when $|i-j|>1$, and} \\\\\nf_if_{i+1}f_i = f_{i+1}f_if_{i+1} &=& f_{i+1}f_i \\text{ when $|i-j|=1$,}\n\\end{align*}\nwhere the indices are understood to be taken $(\\text{mod }N)$.\n\\end{proposition}\n\\begin{proof}\nOne can easily check that these functions $f_i$ satisfy the given relations.  We then check that any $f\\in \\operatorname{NDPF^{(1)}}_N$ maybe written as a composition of the $f_i$.\n\nLet $f \\in \\operatorname{NDPF^{(1)}}_N$.  If there is no $j \\in \\{0, \\ldots, N\\}$ such that $f(j)=f(j+1)$, then $f$ is a shift function, and is thus the identity.\n\nOtherwise, we have some $j$ such that $f(j)=f(j+1)$.  We can then build $f$ using $f_i$'s by the following procedure.  Notice that, if any $g \\in \\operatorname{NDPF^{(1)}}$ has $g(j)=g(j+1)$ for some $j$, we can emulate a shift function by concatenating $g$ with $f_j f_{j+1} \\cdots f_{j+N-1}$, where the subscripts are understood to be taken $(\\text{mod }N)$.  In other words, we have:\n\\[\ng \\operatorname{sh}_1 = g f_j f_{j+1} \\cdots f_{j+N-1}.\n\\]\n\nSuppose, without loss of generality, that $f(N)\\neq f(N+1)$, so that $N$ and $N+1$ are in different fibers of $f$, and $N$ is maximal in its fiber.  (If the ``break'' occurs elsewhere, we simply use that break as the `top' element for the purposes of our algorithm.  Alternately, we can apply the Dynkin automorphism to $f$ until $\\rho^k f(N)\\neq \\rho^k f(N+1)$. for some $k$.  We can use this algorithm to construct $\\rho^k f$, and then apply $\\rho$ $N-k$ times to obtain $f$.)  Begin with $g = 1$, and construct $g$ algorithmically as follows.\n\n\\begin{itemize}\n\\item Collect together the fibers.  Set $g'$ to be the shortest element in $\\operatorname{NDPF}_N$ such that the fibers of $g'$ match the fibers of $f$ in the base window.  Let $g_0$ be the affine function obtained from a reduced word for $g'$.  This is the pointwise maximal function in $\\operatorname{NDPF^{(1)}}_N$ with fibers equal to the fibers of $f$.\n\n\\item Now that the fibers are collected, post-compose $g_0$ with $f_i$'s to move the images into place.  We begin with $g:=g_0$ and apply the following loop:\\\\\n\\begin{align*}\n&&\\text{while } g\\neq f: \\\\\n&&\\phantom{aaaa}\\text{for } i \\text{ in } \\{1, \\ldots, N \\}: \\\\\n&&\\phantom{aaaaaaaa}\\text{if } g(i+1)>f(i+1) \\text{ and } g^{-1}(g(i+1)-1) = \\emptyset: \\\\\n&&\\phantom{aaaaaaaaaaaa}g := g.f_i.\n\\end{align*}\n\nThis process clearly preserves the fibers of $g_0$ (which coincide with the fibers of $f$),  and terminates only if $g=f$.  We need to show that the algorithm eventually halts.\n\nRecall that $g_0(i)\\geq f(i)$ for all $i$, and then notice that it is impossible to obtain any $g$ in the evaluation of the algorithm with $g(i)<f(i)$, so that we always have $g(i)-f(i)>0$.  With each application of a $f_j$, the sum $\\sum_{i=1}^N (g(i)-f(i))$ decreases by one.\n\nSuppose the loop becomes stuck; then for every $i$ either $f(i+1)=g(i+1)$ or $g^{-1}(g(i+1)-1) \\neq \\emptyset$.  If there is no $i$ with $f(i+1)=g(i+1)$, then there must be some $i$ with $g^{-1}(g(i+1)-1) = \\emptyset$, since $g(N)-g(1)\\leq N$ and $g \\neq 1$.  Then we can find a minimal $i \\in \\{1, \\ldots, N\\}$ with $f(i+1)=g(i+1)$.  \n\nNow, find $j$ minimal such that $f(i+j) \\neq g(i+j)$, so that $f(i+j-1) = g(i+j-1)$.  In particular, notice that $i+j-1$ and $i+j$ must be in different fibers for both $f$ and $g$.  If $g^{-1}(g(i+j)-1) = \\emptyset$, then the loop would apply a $f_{i+j-1}$ to $g$, but the loop is stuck, so this does not occur and we have that $f(i+j-1)=g(i+j-1)=g(i+j)-1< f(i+j)\\leq g(i+j) = g(i+j-1)+1$.  This then forces $g(i+j)=f(i+j)$, contradicting the condition on $j$.\n\nThus, the loop must eventually terminate, with $g=f$.\n\\end{itemize}\n\nWe have not yet shown that these relations are all of the relations in the monoid; this must wait until we have developed more of the combinatorics of $\\operatorname{NDPF^{(1)}}_N$.  In fact, $\\operatorname{NDPF^{(1)}}_N$ is a quotient of the $0$-Hecke monoid of $\\tilde{S}_N$ by the relations $\\pi_i \\pi_{i+1} \\pi_i = \\pi_i\\pi_{i+1}$ for each $i \\in I$, where subscripts are understood to be taken $\\text{mod } N$.  To prove this (and simultaneously prove that we have in fact written all the relations in $\\operatorname{NDPF^{(1)}}_N$), we will define three maps, $P, Q$, and $R$ (illustrated in Figure~\\ref{fig.affineNDPFMaps}).  The map $P: H_0(\\tilde{S}_N)\\rightarrow \\operatorname{NDPF^{(1)}}_N$ is the algebraic quotient on generators sending $\\pi_i \\rightarrow f_i$.  The map $Q: H_0(\\tilde{S}_N)\\rightarrow \\operatorname{NDPF^{(1)}}_N$ is a combinatorial algorithm that assigns an element of $\\operatorname{NDPF^{(1)}}_N$ to any affine permutation.  In Lemma~\\ref{lem.combQuotient} we show that $P=Q$.  Additionally, we have already shown that $P$ is onto (since the $f_i$ generate $\\operatorname{NDPF^{(1)}}_N$), so $Q$ is onto as well.\n\nThe third map $R: \\operatorname{NDPF^{(1)}}_N \\rightarrow H_0(\\tilde{S}_N)$ assigns a $[321]$-avoiding affine permutation to an $f\\in \\operatorname{NDPF^{(1)}}_N$.  In fact, $R \\circ P$ is the identity on the set of $[321]$-avoiding affine permutations, and $P\\circ R$ is the identity on $\\operatorname{NDPF^{(1)}}_N$.  This then implies that there are no additional relations in $\\operatorname{NDPF^{(1)}}_N$.\n\\end{proof}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=.75]{andpfMaps.pdf}\n  \\end{center}\n  \\caption{Maps between $H_0(\\tilde{S}_N)$ and $\\operatorname{NDPF^{(1)}}_N$.}\n  \\label{fig.affineNDPFMaps}\n\\end{figure}\n\n\\begin{corollary}\nThe map $P: H_0(\\tilde{S}_N)\\rightarrow \\operatorname{NDPF^{(1)}}_N$, defined by sending $\\pi_i \\rightarrow f_i$ and extending multiplicatively, is a monoid morphism.\n\\end{corollary}\n\\begin{proof}\nThe generators $f_i$ satisfy all relations in the $0$-Hecke algebra, so $P$ is a quotient of $H_0(\\tilde{S}_N)$ by whatever additional relations exist in $\\operatorname{NDPF^{(1)}}_N$.\n\\end{proof}\n\n\\begin{lemma}\nAny function $f \\in \\operatorname{NDPF^{(1)}}_N$ is entirely determined by its set of fibers, set of images, and one valuation $f(i)$ for some $i\\in \\mathbb{Z}$.\n\\end{lemma}\n\\begin{proof}\nThis follows immediately from the fact that $f$ is regressive and order preserving.\n\\end{proof}\n\n\\begin{lemma}\nLet $f \\in \\operatorname{NDPF^{(1)}}_N$, and $F_f=\\{m_j\\}$ be the set of maximal elements of the fibers of $f$.  Each pair of distinct elements $m_j, m_k$ of the set $F_f \\cap \\{1, 2, \\ldots, N\\}$ has $f(m_j) \\not \\equiv f(m_k) (\\text{mod } N)$.\n\\end{lemma}\n\\begin{proof}\nSuppose not.  Then $f(m_j)-f(m_i) = kN$ for some $k \\in \\mathbb{Z}$, implying that \n$f(m_j)=f(m_i+kN)$.  Since $f(m_j)-f(m_i)\\leq N$, we must have $k=0$.  But then $m_j$ and $m_i$ are in the same fiber, providing a contradiction.\n\\end{proof}\n\n\n\\begin{theorem}\n$\\operatorname{NDPF^{(1)}}_N$ is $\\mathcal{J}$-trivial.\n\\end{theorem}\n\\begin{proof}\nThi is a direct consequence of the regressiveness of functions in $\\operatorname{NDPF^{(1)}}_N$.  Let $M:= \\operatorname{NDPF^{(1)}}_N$, and $f\\in M$.  Then each $g\\in MfM$ has $g(i)\\leq f(i)$ for all $i \\in \\mathbb{Z}$.  Thus, if $MgM=MfM$, we must have $f=g$.  Then the $\\mathcal{J}$-equivalence classes of $M$ are trivial, so $\\operatorname{NDPF^{(1)}}_N$ is $\\mathcal{J}$-trivial.\n\\end{proof}\n\nNote that $\\operatorname{NDPF^{(1)}}_N$ is not aperiodic in the sense of a finite monoid.  (Aperiodicity was defined in Section~\\ref{sec:bgnot}.)  Take the function $f$ where $f(i)=0$ for all $i \\in \\{1, \\ldots, N\\}$.  Then $f^k(1) = (1-k)N$, so there is no $k$ such that $f^k = f^{k+1}$.\n\n\\subsection{Combinatorial Quotient}\n\\label{subsec:combintorialQuotient}\n\nA direct combinatorial map from affine permutations to $\\operatorname{NDPF^{(1)}}_N$ is now discussed.  This map directly constructs a function $f$ from an arbitrary affine permutation $x$, with the same effect as applying the algebraic $\\operatorname{NDPF^{(1)}}$ quotient to the $0$-Hecke monoid element indexed by $x$.  We first define the combinatorial quotient in the finite case and provide an example (Figure~\\ref{fig.combQuotient}).\n\n\\begin{definition}\nThe combinatorial quotient $Q_{cl}: H_0(S_N) \\rightarrow \\operatorname{NDPF}_N$ is given by the following algorithm, which assigns a function $f$ to a permutation $x$.\n\\begin{enumerate}\n\\item Set $f(N):=x(N)$.\n\\item Suppose $i$ is maximal such that $f(i)$ is not yet defined.  If $x(i)<x(i+1)$, set $f(i):=f(i+1)$.  Otherwise, set $f(i):=x(i)$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=1]{combQuotient.pdf}\n  \\end{center}\n  \\caption{Example of the combinatorial quotient $Q_{cl}: H_0(S_5) \\rightarrow \\operatorname{NDPF}_5$.  The string diagram is read left-to-right, with the permutation illustrated with black strings and the image function drawn in red.  The permutation in the left diagram, then, is $x=[2,4,1,5,3]$ and $Q_{cl}(x)$ is the function $f=[1,1,1,3,3]$.  For the permutation on the right, we have $y=[4,2,1,5,3]$ and $Q_{cl}(y)=Q_{cl}(x)=[1,1,1,3,3]$.  Notice that these two permutations $x$ and $y$ are related by turning the $[321]$-pattern in $y$ into a $[231]$-pattern in $x$, preserving the fiber of $Q$.}\n  \\label{fig.combQuotient}\n\\end{figure}\n\nNote that the map $Q_{cl}$ is closely related to bijection of Simion and Schmidt between $[132]$-avoiding permutations and $[123]$-avoiding permutations~\\cite{simion.schmidt.1985}.  (The bijection is also covered very nicely in~\\cite{bona.permutations})  This bijection operates by marking all left-to-right minima (ie, elements smaller than all elements to their left) of a $[132]$-avoiding permutation, and then reverse-sorting all elements which are not marked. The resulting permutation is $[123]$-avoiding.  For example, the permutation\n$[\\textbf{5}, 6, \\textbf{4}, 7, \\textbf{1}, 2, 3]$ avoids the pattern $[132]$; the bold entries are the left-to-right minima.  Sorting the non-bold entries, one obtains the permutation\n$[\\textbf{5}, 7, \\textbf{4}, 6, \\textbf{1}, 3, 2]$, which avoids the permutation $[123]$.  Notice that the bold entries are still left-to-right minima after anti-sorting the other entries.\n\nThe patterns $[231]$ and $[123]$ are the respective ``reverses'' of the patterns $[132]$ and $[321]$, obtained by simply reversing the one-line notation.  It is trivial to observe that $x$ avoids $p$ if and only if the reverse of $x$ avoids the reverse of $p$.  Then the ``reverse'' of the Simion-Schmidt algorithm (which marks right-to-left minima, and sorts the other entries) gives a bijection between $[231]$- and $[321]$-avoiding permutations; in fact, this is the same bijection given by the fibers of the $\\operatorname{NDPF}$ quotient of the $0$-Hecke monoid.\n\nA similar combinatorial quotient may be defined from $\\tilde{S}_N\\rightarrow \\operatorname{NDPF^{(1)}}_N$, generalizing the map $Q_{cl}$.  This map will assign a function $f$ to an affine permutation $x$.\n\nBelow, we will show that each fiber of the map $Q$ contains a unique $[321]$-avoiding affine permutation (Theorem~\\ref{thm:affNdpfFibers321}).  However, it is too much to expect a bijection between affine $[231]$- and $[321]$-avoiding permutations.  By a result of Crites, there are infinitely many affine permutations that avoid a pattern $\\sigma$ if and only if $\\sigma$ contains the pattern $[321]$~\\cite{Crites.2010}.  Thus, there are infinitely many $[321]$-avoiding affine permutations, but only finitely many $[231]$-avoiding affine permutations.\n\nWe first identify some $k \\in \\{1, 2, \\ldots, N\\}$ such that for every $j>k$, $x(j)>x(k)$.  \n\n\\begin{lemma}\nLet $k_0 \\in \\{1, 2, \\ldots, N\\}$ have $x(k_0)\\leq x(m)$ for every $m \\in \\{1, 2, \\ldots, N\\}$.  Then for every $j>k_0$, $x(j)>x(k_0)$.\n\\end{lemma}\n\\begin{proof}\n Suppose $j>k_0$ with $x(j)<x(k_0)$.  Then there exists $p\\in \\mathbb{N}$ such that $j-pN \\in \\{1, 2, \\ldots, N\\}$, so that $x(j-pN) = x(j)-pN < x(k_0)$, contradicting the minimality of $x(k_0)$.\n\\end{proof}\n\nNow the affine combinatorial quotient is defined by the following algorithm.\n\n\\begin{definition}\nThe combinatorial quotient $Q: H_0(\\tilde{S}_N) \\rightarrow \\operatorname{NDPF^{(1)}}_N$ is given by the following algorithm, which assigns a function $f$ to an affine permutation $x$.\n\\begin{enumerate}\n\\item Let $k_0 \\in \\{1, 2, \\ldots, N\\}$ have $x(k_0)\\leq x(m)$ for every $m \\in \\{1, 2, \\ldots, N\\}$.  Set $f(k_0)=x(k_0)$.\n\\item Choose $i \\in \\{1, 2, \\ldots, N-1\\}$ minimal such that $f(k_0 - i)$ is not yet defined.  If $x(k_0-i+1)<x(k_0-i)$, set $f(k_0-i) := f(k_0-i+1)$.  Otherwise, set $f(k_0-i) := x(k_0-i)$.\n\\item Define $f$ on all other $i$ using skew periodicity.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{lemma}\n\\label{lem.combQuotient}\nThe affine combinatorial quotient $Q$ agrees with the algebraic $\\operatorname{NDPF^{(1)}}$ quotient $P$. \n\\end{lemma}\n\\begin{proof}\nWe denote the combinatorial quotient by $Q$ and the algebraic quotient by $P$.  \n\nOne can easily check that $Q(1)=P(1)=1$, and $Q(\\pi_i)=P(\\pi_i)=f_i$.  Since $P$ is a monoid morphism, we have that $P(x\\pi_i)=P(x)P(\\pi_i)= ff_i$.   We then assume that $Q(x)=P(x)=f$, and consider $Q(x\\pi_i)$.  We will show that $Q(x\\pi_i)= Q(x)f_i = ff_i = P(x\\pi_i)$.  \n\nIf $\\pi_i$ is a right descent of $x$ then $Q(x\\pi_i)=Q(x)=f=P(x\\pi_i)$, and we are done.\n\nIf $\\pi_i$ is not a right descent of $x$, we have $x(kN+i) < x(kN+i+1)$ for all $k\\in \\mathbb{Z}$, and \n\\[\nx\\pi_i(j) = \n    \\left\\{\n     \\begin{array}{lr}\n       x(j)   \\text{ for all } j\\not \\equiv i, i+1 (\\text{mod } N)\\\\\n       x(j+1) \\text{ for all } j     \\equiv i      (\\text{mod } N)\\\\\n       x(j-1) \\text{ for all } j     \\equiv    i+1 (\\text{mod } N)\\\\\n     \\end{array}\n   \\right.\n\\]  \nWe examine the functions $Q(x\\pi_i)$ and $ff_i$ on $i$ and $i+1$, since these functions are equal on $j\\not \\equiv i, i+1 (\\text{mod } N)$, and the actions on $i$ and $i+1$ then determine the functions on all $j \\equiv i, i+1 (\\text{mod } N)$. \n\nWe consider two cases, depending on whether $i$ and $i+1$ are in the same fiber of $f$.  \n\n\\begin{itemize}\n\\item If $i$ and $i+1$ are in the same fiber of $f$ and $i+1$ is maximal in this fiber, we must (by construction of $Q$) have $x(i+1)<x(i)$, contradicting the assumption that $\\pi_i$ was not a right descent of $x$.\n\n\\item If $i$ and $i+1$ are in the same fiber of $f$ and $i+1$ is not maximal in this fiber, then there exists some (minimal) $m> i+1>i$ with $x(m)<x(i)$ and $x(m)<x(i+1)$, maximal in the fiber of $i$ and $i+1$.  Then $x(m)<x(i+1)=x\\pi_i(i)$ and $x(m)<x(i)=x\\pi_i(i+1)$.  Since the maximal size of a fiber of $f$ is $N$, we have that $m-i\\leq N$.  Then (since $i+1$ not maximal in the fiber of $f$) $m\\not \\equiv i+1 (\\text{mod } N)$.\n\nIf $m \\equiv i (\\text{mod } N)$, then $i$ is maximal in its fiber, and we must have $i$ and $i+1$ in different fibers, contrary to assumption.\n\nIf $m\\not \\equiv i (\\text{mod } N)$, we have $x(m)=x\\pi_i(m) < x\\pi_i(i), x\\pi_i(i+1)$, and so by the construction of $Q$, we have \n$Q(x\\pi_i)(i)=Q(x\\pi_i)(i+1)=Q(x\\pi_i)(m)=Q(x)(m) = x(m)$.  Then in this case, $Q(x\\pi_i)=f$.\n\nOn the other hand, $ff_i(i)=f(i)=f(m)=ff_i(m)$, and $ff_i(i+1)=f(i)=f(m)=ff_i(m)$, so $ff_i=f$.\n\n\\item If $i$ and $i+1$ are in different fibers of $f$, then we have $i$ maximal in its fiber, and take $m$ (possibly equal to $i+1$) to be the maximal element of the fiber in which $i+1$ sits.  We note that if $m\\equiv i+1 (\\text{mod } N)$, then we must have $i$ and $i+1$ in the same fiber, reducing to the previous case.\n\nOtherwise, applying the construction of $Q$, we find that $Q(x\\pi_i)(i+1) = x(i)$, and that $Q(x\\pi_i)(i) = x(i)$; thus $i+1$ is removed from its fiber and merged into the fiber with $i$.  The resulting function is equal to $ff_i$.\n\\end{itemize}\nThis exhausts all cases, completing the proof.\n\\end{proof}\n\n\\begin{corollary}\nThe finite type combinatorial quotient agrees with the $\\operatorname{NDPF}_N$ quotient of $H_0(S_N)$ obtained by introducing the relations $\\pi_i \\pi_{i+1}\\pi_i = \\pi_{i+1} \\pi_i$ for $i\\in \\{1, \\ldots, N-2\\}$. \n\\end{corollary}\n\\begin{proof}\nThis follows immediately from Lemma~\\ref{lem.combQuotient} by parabolic restriction to the finite case.  In the finite case, the index set is $\\{1, 2, \\ldots, N-1\\}$, so we must have $i\\in \\{1, \\ldots, N-2\\}$.\n\\end{proof}\n\n\n\n\\subsection{Affine $[321]$-Avoidance}\n\nAn affine permutation $x$ avoids a pattern $\\sigma \\in S_k$ if there is no subsequence of $x$ in the same relative order as $\\sigma$.  This ostensibly means that an infinite check is necessary, however one may show that only a finite number of comparisons is necessary to determine if $x$ contains a $[321]$-pattern.  The following lemma is equivalent to~\\cite[Lemma 2.6]{Green.2002}.\n\n\\begin{lemma}\nLet $x$ contain at least one $[321]$-pattern, with $x_i> x_j> x_k$ and $i<j<k$.  Then $x$ contains a $[321]$-pattern $x_{i'}> x_j> x_{k'}$ such that $i\\leq i'<j<k'\\leq k$, $j-i'< N$, and $k'-j < N$.\n\\end{lemma}\n\\begin{proof}\nWe have $x_j>x_k>x_{k-aN} = x_k - aN$ for $a \\in \\mathbb{N}$, so if $k-j>N$, we can find a $[321]$ pattern replacing $x_k$ with $x_{k-aN}$.  A similar argument allows us to replace $i$ with $i+bN$ for the maximal $b\\in \\mathbb{N}$ such that $j-(i+bN) < N$.\n\\end{proof}\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[scale=1]{minimalAffine321.pdf}\n  \\end{center}\n  \\caption{Diagram of a bountiful width system for the pattern $[321]$ for affine permutations.  The pattern occurs at positions $(x_p, x_q, x_r)$, with width system given by $(r-p, q-p)$.  In the case where $r-p>N$, there is an `overlap' of $j=r-N-p$.  Bountifulness of the width system ensures that the elements in the overlap may be moved moved out of the interior of the pattern instance by a sequence of simple transpositions, each decreasing the length of the permutation by one, just as in the non-affine case.}\n  \\label{fig.minimalAffine321}\n\\end{figure}\n\nAs noted by Green, one can then check whether an affine permutation contains a $[321]$-pattern using at most $\\binom{N}{3}$ comparisons.  Green also showed that any affine permutation containing a $[321]$-pattern contains a braid; we can actually replicate this result using a width system on the affine permutation, as depicted in Figure~\\ref{fig.minimalAffine321}.  The Lemma ensures that the width of a minimal $[321]$-pattern under this width system has a total width of at most $2N-2$.  One must consider the case when the total width of a minimal $[321]$-instance is greater than $N$, but nothing untoward occurs in this case: the width system is bountiful and allows a factorization of $x$ over $[321]$.\n\nWe now prove the main result of this section.\n\n\\begin{theorem}\n\\label{thm:affNdpfFibers321}\nEach fiber of the $\\operatorname{NDPF^{(1)}}_N$ quotient of $\\tilde{S}_N$ contains a unique $[321]$-avoiding affine permutation.\n\\end{theorem}\n\\begin{proof}\nWe first establish that each fiber contains a $[321]$-avoiding affine permutation, and then show that this permutation is unique.\n\nRecall the algebraic quotient map $P: H_0(\\tilde{S}_N) \\rightarrow \\operatorname{NDPF^{(1)}}_N$, which introduces the relation $\\pi_i \\pi_{i+1}\\pi_i = \\pi_{i+1} \\pi_i$.  \n\nChoose an arbitrary affine permutation $x$; we show that the fiber $Q^{-1}\\circ Q(x)$ contains a $[321]$-avoiding permutation.  If $x$ is itself $[321]$-avoiding, we are already done.  So assume $x$ contains a $[321]$-pattern.  As shown by Green~\\cite{Green.2002}, an affine permutation $x$ contains a $[321]$-pattern if and only if $x$ has a reduced word containing a braid; thus, $x=y \\pi_i \\pi_{i+1} \\pi_i z$ for some permutations $y$ and $z$ with $\\operatorname{len}(x)=\\operatorname{len}(y)+3+\\operatorname{len}(z)$.  Applying the $\\operatorname{NDPF^{(1)}}_N$ relations, we may set $x'= y \\pi_{i+1} \\pi_i z$, and have $Q(x)=Q(x')$, with $\\operatorname{len}(x')=\\operatorname{len}(x)-1$.  If $x'$ contains a $[321]$, we apply this trick again, reducing the length by one.  Since $x$ is of finite length, this process must eventually terminate; the permutation at which the process terminates must then be $[321]$-avoiding.  Then the fiber $Q^{-1}\\circ Q(x)$ contains a $[321]$-avoiding permutation.\n\nWe now show that each fiber contains a unique $[321]$-avoiding affine permutation, using the combinatorial quotient map.\n\nLet $x$ be $[321]$-avoiding, and let $Q(x)=f$ an affine non-decreasing parking function; we use information from $f$ to reconstruct $x$.  Let $\\{m_i\\}$ be the set of elements of $\\mathbb{Z}$ that are maximal in their fibers under $f$.  By the construction of the combinatorial quotient map, we have $x(m_i)=f(m_i)$ for every $i$.  Since $f$ is in $\\operatorname{NDPF^{(1)}}_N$, we have $x(m_i)<x(m_{i'})$ whenever $i<i'$; thus $\\{x(m_i)\\}$ is a strictly increasing sequence.\n\nLet $\\{m_{i,j}\\} = f^{-1}\\circ f(m_i) \\setminus \\{m_i\\}$, with $m_{i,j}<m_{i,j+1}$ for every $j$.  Notice that if $i<i'$ and $j<j'$ then $m_{i,j}<m_{i',j'}$.  \n\nWe claim that if $i<i'$ and $j<j'$, then $x(m_{i,j})<x(m_{i',j'})$.  If not, then we have \n\\[\nx(m_{i'}) < x(m_{i',j'}) < x(m_{i,j}), \\text{ with }  m_{i,j} < m_{i',j'} < m_{i'},\n\\]\nin which case $x$ contains a $[321]$-pattern, contrary to assumption.  Thus, the sequence $\\{x(m_{i,j})\\}$ with $i$ and $j$ arbitrary is a strictly increasing sequence.\n\nNow $\\{f(m_i) = x(m_{i})\\}$ and $\\{x(m_{i,j})\\}$ are two increasing sequences.  Since $x$ is a bijection, and every $z\\in \\mathbb{Z}$ is either an $m_i$ or an $m_{i,j}$, $x$ is determined by the choice of $x(m_{1,1})$.  A valid choice for $x(m_{1,1})$ exists, since every $f$ arises as the image of some affine permutation under $Q$, and every fiber contains some $[321]$-avoiding element.  \n\nOne can show that the choice of $x(m_{1,1})$ is uniquely determined by the following argument.  Suppose two valid possibilities exist for $x(m_{1,1})$, giving rise to two different $[321]$-avoiding affine permutations $x$ and $x'$.  Suppose without loss of generality that $1\\leq m_{1,1}\\leq N$, and that $x(m_{1,1})<x'(m_{1,1})$.  Then: \n\\begin{align*}\n\\binom{N+2}{2} &=&  \\sum_{k=1}^{N} x(k) \\\\\n &=& \\sum ( x(m_i) + \\sum x(m_{i,j}) ) \\text{ where $m_i$, $m_{i,j} \\in \\{1, \\ldots, N\\}$ }\\\\\n &<& \\sum ( x'(m_i) + \\sum x'(m_{i,j}) ) \\text{ where $m_i$, $m_{i,j} \\in \\{1, \\ldots, N\\}$ }\\\\\n &=& \\sum_{k=1}^{N} x'(k) \\\\\n &=& \\binom{N+2}{2},\n\\end{align*}\nproviding a contradiction.  Hence $x(m_{1,1})$ is uniquely determined, and thus each fiber of $Q$ contains a unique $[321]$-avoiding permutation.\n\\end{proof}\n\n\\bibliographystyle{amsalpha}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSepsis, a kind of generalized infection leading to organ failure, is a major concern for health providers.  Sepsis causes 20\\%-30\\% of deaths in hospitals and consumes \\$15.4 billion annually in healthcare costs [Henry 2015].  If severe sepsis progresses to septic shock, mortality can run as high as 50\\% [Bhattacharjee 2017], with delay in diagnosis and treatment increasing mortality by 7.6\\% for every hour of delay.  \n\n\n\n\\subsection{Goals}\n\nAlthough the clinical definition of severe sepsis involves only \\emph{structured data} measurements, the Electronic Health Record also contains potentially useful \\emph{unstructured text}:  patient history, progress notes, lab reports, etc.  We would like to use machine learning to automate the understanding of these text notes to make decisions from them, such as ``\\emph{Is this patient likely  to satisfy the definition for severe sepsis within the next 24 hours?}''  However, in general, it is difficult for computer software to learn to make decisions from text.  The result is that unstructured notes in the EHR are severely under-utilized for computational purposes [Ohno-Machado 2011].\n\nWorking with data from {\\BaystateHealth} as part of a QI initiative, we performed a retrospective study using Electronic Health Record information for adult inpatients from 2012-2016.  We sought to answer three questions:\n\n\\begin{enumerate}\n\\item Can we predict, directly from the EHR, which patients will satisfy a clinical definition of severe sepsis at various future times?\n\n\\item Can we use only unstructured text notes in the EHR for predictions?\n\n\\item How does prediction accuracy compare when using only unstructured data, only structured data, or both types of data from the EHR?\n\n\\end{enumerate}\n\n\\subsection{Previous Research}\n\nResearchers use ``predict'' in  two different contexts.  Most are {screening} applications to determine whether patients \\emph{currently} have sepsis, typically as judged by a gold standard team of physicians.   Others look at \\emph{predicting ahead in time}, where sufficient structured data is not yet available to fulfill clinical criteria for severe sepsis.  Our focus here   is using {currently} available data to predict a {future} severe sepsis diagnosis, which is made using additional data available at that future time.\n\nBhattacharjee et al.\\ [2017] describes  10 automated  tools using  structured data to predict patients with sepsis, severe sepsis, or septic shock.\nNone  make significant usage of free text notes, and most of these models are screening models.\n\nA more recent paper by Horng et al.\\ [2017] built Emergency Department triage screening models for sepsis (including severe sepsis and septic shock).  They make significant use of text notes and show about 25\\%\\ improvement when notes are added to structured data.  Text modeling involved term frequencies, bi-grams, and topic models, resulting in vectors with 15,000 dimensions.  (Reported results did not include  using unstructured data {alone}.)  \n\nAn earlier use of text notes to predict hospital mortality [Lehman 2012] employed Hierarchical Dirichlet Processes on UMLS clinical concepts extracted from the text.  Their mortality predictions showed text-based analysis was better than using structured data alone, and the combination of both produced best results (paralleling our results with severe sepsis).\n\nIn other work predicting sepsis, Desautels et al.\\ [2016] reported on a machine learning approach using structured variables that showed improved predictive performance over a number of other methods for predicting sepsis, up to 4 hours in advance.  They use a recent definition of sepsis [Singer 2016] that differs from the traditional classifications of sepsis\/severe~sepsis\/septic~shock.\nPaxton et al.\\ [2013] built models to detect septic shock using 1011 structured data features.  They also discussed issues around using the Electronic Health Record for modeling in situations where some treatments have already started.  \n\n\n\n\\section{Methods}\n\nWe analyzed 203,000 adult inpatient admissions (encounters) within {\\Baystate} hospitals over 5 years from 2012 through 2016.  Severe sepsis patients were identified using a modified version of {\\Baystate}'s clinical definition for severe sepsis, involving 8 structured variables.  Additionally, encounters were marked as severe sepsis if they had  corresponding ICD codes.  (Simply using severe sepsis ICD codes was not sufficiently reliable, as was previously found by Rhee et al.\\ [2017].)  For each patient who satisfied this definition of severe sepsis, we used time stamps for the individual structured variables (or ICD codes) to compute the earliest time the severe sepsis definition was satisfied, which we term the \\emph{Severe Sepsis Definition Time}.  \n\n\\subsection{Unstructured Data Models}\n\nWe collected textual  notes for patient encounters.  There were over 100 types of notes, with most being progress notes or history-and-physical notes.  When predicting severe sepsis 24 hours ahead for positive targets (severe sepsis patients), we removed all of the notes that occurred later than 24 hours prior to the Severe Sepsis Definition Time. Remaining data is referred to as the \\emph{Modeling Data Window, MDW}. These encounters constituted positive examples for our modeling. \n\nFor non-severe-sepsis patients, we chose a random time during their hospital stay and removed all notes that occurred later than 24 hours before that time.  These patients are negative examples.  Our intent was to simulate regular application of our models in a clinical setting, to suggest patients   for  further attention who are in danger of severe sepsis in the next 24 hours.\n\nWe eliminated encounters having no unstructured notes remaining in the MDW, because it makes no sense to try to predict with no information at all.  This left 68,482 total encounters, of which 1,427 (2.1\\%)  satisfied the definition of severe sepsis during their stay.\n\nWe then concatenated all text information for an encounter\\footnote{Not all types of text were used.} into a single text block which, along with the severe sepsis target flag, comprised our unstructured data.  Encounters were randomly divided into 3 groups of patients, stratifying them to maintain the global ratio of severe sepsis targets within each sample set.  For modeling and testing we used 3-fold cross validation, modeling on each set of 2 groups and using the remaining group to measure performance.  Final performance was computed over all 3 holdout sets.  We also built models using 2012--2015 data for model construction, and 2016 data for testing.\n\nModels to predict 4 and 8 hours ahead used corresponding data preprocessing, with revised Modeling Data  Windows.  Note that models for 24-hour-ahead predictions had fewer encounters than 4 or 8 hour predictions, because the former had fewer non-empty Modeling Data Windows.  In particular, patients admitted with pre-existing severe sepsis were usually eliminated from modeling data for 24-hour models, because their diagnosis occurred in the first 24 hours of their encounter, leaving an empty MDW.\n\nWe represented unstructured data using 300-dimensional  GloVe vector embeddings for terms [Pennington 2014], and then summing the vectors for terms in the text.  Training was by ridge regression.\n\n\\subsection{Structured Data Models}\n\nFor models using only structured data, we selected 12 variables thought to be helpful for predicting Severe Sepsis and gathered their data, with timestamps, from the \nEHR. Values were computed for mean and standard deviation, plus counts of abnormal high, abnormal low, and normal readings, resulting in 29 modeling variables.  Once again, we discarded encounters that lacked data for all 29 variables, leaving no data in the Modeling Data Window.  \n\nFor combination models using both unstructured and structured data, we required that both types of data be non-empty in the MDW.\n\n\\section{Results}\n\nTable~\\ref{unstruct} gives summary results across all 3 cross-validation folds of the data using only unstructured EHR text.  To emphasize actionable results where there is a practical need to avoid over-alarming, we focused upon the most likely predicted 1\\%, 5\\%, and 10\\% encounters for severe sepsis among hold-out data.\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Predicting Severe Sepsis Using Only Text from the EHR}\n  \\label{unstruct}\n    \\begin{tabular}{rlrrrr}\n    \\multicolumn{1}{p{4.215em}}{\\textbf{Predict}} &       & \\multicolumn{1}{c}{\\textbf{Encounters with}} & \\multicolumn{1}{p{4.215em}}{\\textbf{Top 1\\% }} & \\multicolumn{1}{p{4.215em}}{\\textbf{Top 5\\%  }} & \\multicolumn{1}{p{4.215em}}{\\textbf{Top 10\\%  }} \\\\\n\t\\multicolumn{1}{p{4.215em}}{\\textbf{Ahead}} &       & \\multicolumn{1}{c}{\\textbf{Usable Data in}} &  \\multicolumn{1}{p{4.215em}}{\\textbf{Predicted }} & \\multicolumn{1}{p{4.215em}}{\\textbf{Predicted }} & \\multicolumn{1}{p{4.215em}}{\\textbf{Predicted }} \\\\\n\t &       & \\multicolumn{1}{c}{\\textbf{Modeling Window}} & & & \\\\\n          &       &       &       &       &  \\\\\n    \\multicolumn{1}{l}{\\textbf{4 hours}} &       &       &       &       &  \\\\\n          & Sample Size &                    129,421  &         1,294  &         6,471  &      12,942  \\\\\n          & Targets found &                         2,527  & 521   & 801   & 952  \\\\\n          & \\% of Sample &       & \\textbf{40\\%} & \\textbf{12\\%} & \\textbf{7\\%} \\\\\n          & \\% of All Targets &       & \\textbf{21\\%} & \\textbf{32\\%} & \\textbf{38\\%} \\\\\n          & AUC      & \\textbf{0.636} &  &         &  \\\\\n          &       &       &       &       &  \\\\\n    \\multicolumn{1}{l}{\\textbf{8 hours}} &       &       &       &       &  \\\\\n          & Sample Size &                    117,768  &         1,178  &         5,888  &      11,777  \\\\\n          & Targets found &                         2,158  & 503   & 769   & 916  \\\\\n          & \\% of Sample &       & \\textbf{43\\%} & \\textbf{13\\%} & \\textbf{8\\%} \\\\\n          & \\% of All Targets &       & \\textbf{23\\%} & \\textbf{36\\%} & \\textbf{42\\%} \\\\\n          & AUC   & \\textbf{0.660} &       &       &  \\\\\n          &       &       &       &       &  \\\\\n    \\multicolumn{1}{l}{\\textbf{24 hours}} &       &       &       &       &  \\\\\n          & Sample Size &                       68,482  &            685  &         3,424  &         6,848  \\\\\n          & Targets found &                         1,427  & 412   & 707   & 829  \\\\\n          & \\% of Sample &       & \\textbf{60\\%} & \\textbf{21\\%} & \\textbf{12\\%} \\\\\n          & \\% of All Targets &       & \\textbf{29\\%} & \\textbf{50\\%} & \\textbf{58\\%} \\\\\n          & AUC   & \\textbf{0.727} &       &       &  \\\\\n    \\end{tabular\n  \\label{tab:addlabel\n\\end{table\n\n\n\nSurprisingly, predicting ahead 24 hours gives better results than predicting ahead 4 or 8 hours.  This is a consequence of the former set of encounters being smaller in number, and having a higher percentage of longer-term patients who have more information available in their modeling windows.  This explanation was supported by building 4, 8, and 24~hour predictive models using the same 24-hour structured data, which showed improved performance for 4 and 8-hour predictions vs.\\ 24-hour predictions.\n\nWe manually verified that unstructured text in the MDW did not already have sufficient information to \\emph{diagnose} severe sepsis 24 hours prior to the Severe Sepsis Definition Time.  Looking at the top 1\\% predicted encounters, we found very few (1\\%-3\\%) where narratives were sufficient for a prior diagnosis. We judged this to be an acceptable rate.  We similarly checked that Vasopressor drugs (an indication of septic shock) were not yet administered  during the Modeling Data Window.\n\nTable~\\ref{comparison} gives comparisons with unstructured, structured, or combination data when predicting 24 hours into the future.  Here we built severe sepsis models using encounters from 2012--2015, and testing with encounters from 2016.  Only encounters that have non-empty modeling windows for both unstructured and structured data were used.  The 2016 test set contained 13,603 usable encounters, for which 425 patients  (3.1\\%) satisfied the severe sepsis criteria in the next 24 hours.  Using out-of-time test data further supported results from Table~\\ref{unstruct} which used cross-validation.  To briefly summarize the comparison:  unstructured-only models performed comparably or slightly better than structured-only models, and the combination of unstructured and structured data gave a 5\\%-10\\% improvement over unstructured-only models.\n\n\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Predicting Severe Sepsis 24 Hours in the Future Using Unstructured, Structured, or Combination EHR Data}\n  \\label{comparison}\n    \\begin{tabular}{p{10.0em}ccc|cc|cc}\n    \\textbf{Type of EHR Data} & \\multicolumn{1}{p{2.0em}}{\\textbf{AUC}} & \\multicolumn{2}{c}{\\textbf{Predicted Top 1\\%}} & \\multicolumn{2}{c}{\\textbf{Predicted Top 5\\%}} & \\multicolumn{2}{c}{\\textbf{Predicted Top 10\\%}} \\\\\n    \\multicolumn{1}{c}{} &       & \\multicolumn{1}{p{4.0em}}{{Number In Set}} & \\multicolumn{1}{p{3.43em}|}{{Severe Sepsis}} & \\multicolumn{1}{p{4.0em}}{{Number In Set}} & \\multicolumn{1}{p{3.43em}|}{{Severe Sepsis}} & \\multicolumn{1}{p{4.0em}}{{Number In Set}} & \\multicolumn{1}{p{3.43em}}{{Severe Sepsis}} \\\\\n    Unstructured Text only & 0.81  & 136   & 115   & 680   & 217   & 1,360 & 247 \\\\\n    Structured Data only & 0.80  & 136   & 112   & 680   & 206   & 1,360 & 248 \\\\\n    Both Unstructured Text   \\\\\n\t\\multicolumn{1}{r}{And Structured Data} & 0.85  & 136   & 125   & 680   & 239   & 1,360 & 272 \\\\\n    \\end{tabular\n  \\label{tab:addlabel\n\\end{table\n\n\n\\section{Discussion}\t\n\nOur results indicate that we can construct models, directly from the Electronic Health Record, that are highly predictive  for severe sepsis diagnoses over the next 4, 8, and 24 hours,  and using only text notes.  This is the first result using text to \\emph{predict ahead in time} a sepsis diagnosis, as well as the first sepsis predictions \\emph{using only unstructured data}.\n\nThese severe sepsis models appear practically actionable and valuable.  For example when predicting ahead 24 hours, if we have 1,000 patients and take the model's top-scoring 10 patients (top 1\\%), then we expect 6 of them to satisfy the severe sepsis definition in the next 24 hours.\n\nIt was somewhat surprising that such results were possible using merely our baseline ``bag of words'' distributed  representation for the text.  Future work will explore more advanced representations that explicitly represent negation [Gallant 2013], which is helpful in different modeling contexts (such as computer assisted coding of ICD-10 medical codes).\n\nA possible source of noise in our data are the timestamps associated with both unstructured and structured items.  A planned prospective study will address this issue.\n\n\\subsection{The Case for Using Unstructured Text}\n\nThere are several advantages with using text notes for modeling, as compared with structured values.   The medical notes  tend to repeat lab results and structured variables, \\emph{but only the important ones},  as judged by skilled clinicians.  This expert judgment constitutes a quite   valuable resource, and it appears only in the unstructured data.\n\nSurprisingly, unstructured text requires less manual preprocessing effort than using structured data.  For example, with structured data there is not a {single} value for ``blood pressure'' to be extracted from the EHR; there are \\emph{multiple} blood pressure readings, and these need to be rolled up using max, min, mean, deltas, etc.  Also, structured data presents more of a problem with missing data.\n\nAnother consideration is that we want to be able to predict many additional health-related targets, such as  re-admission risk, over sedation, CHF, etc.  Yet we cannot practically include \\emph{all} structured data from the EHR --- a huge task --- so a separate subset of relevant structured variables needs to be manually defined and extracted for each different prediction target.  By contrast, we can  reuse \\emph{the same set} of unstructured data for multiple prediction models.\\footnote{An exception is that notes on patient history may be useful for targets like re-admission risk, but need to be excluded for  targets such as ICD-10  coding for current conditions.}\n\nSepsis may be an especially good candidate for using text notes because it has a complex definition involving many structured data values.  Structured data may work comparatively better for simpler targets that are defined by only a few structured data numbers.\n\n\n\\subsection{Concluding Remarks}\n\nWe will report additional details elsewhere, including results using a different prediction target (Over-Sedation), as well as results predicting severe sepsis using the MIMIC dataset [Johnson 2016].  \n\n\n\nOur findings strongly support the inclusion of unstructured notes when building predictive models.  Findings also support an implementation in a clinical setting to prospectively confirm accuracy and  usefulness of model predictions.\n\n\\subsubsection*{Acknowledgments}\n\nResearch supported by a grant from the National Science Foundation to MultiModel Research.  Baystate contributions were part of a QI initiative.  Thanks to Florin Brasoveanu, Patricia Humiston RN, Anees Hameed, Katie Devlin, Christian Lagier, Dan Greenberg and TechSpring for contributions to this work.\n\n\\section*{References}\n\n\\medskip\n\n\\small\n\nBhattacharjee P, Edelson DP, Churpek MM.  (2017) \nIdentifying Patients With Sepsis on the Hospital Wards.\n\\emph{Chest}. 2017 Apr;151(4):898-907. \n\nDesautels, T., Calvert, J., Hoffman, J., Jay, M., Kerem, Y., Shieh, L.,  Shimabukuro D, Chettipally U, Feldman MD, Barton C, Wales DJ, Das, R. (2016). 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