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+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\n\n\n\\section{Introduction}\n\n\nIron--chromium alloys are not only scientifically interesting due to their peculiar material properties, but also play an important technological role as the base component for stainless steels \\cite{Vit2011.chapter,Wen2011.book}. \nFrom a basic-science perspective,\nFe--Cr alloys at varying relative compositions and structures exhibit intricate phenomena such as giant magnetoresistance~\\cite{Gru1986.PRL57.2442}, a spin-ice phase~\\cite{Bur1983.JPhysF13.451}, and the ``\\SI{475}{\\celsius} embrittlement'' effect~\\cite{Sah2009.MaterSciEngA508.1}. \nFrom an application point of view,\nprecision design of steels---beyond the traditional method of empirical trial and error---benefits from an atomic-level understanding of the ins and outs of Fe--Cr alloys.\nOf particular importance is to explore the mechanisms by which the Cr atoms render the open surfaces of Cr-containing steels \nresistant to corrosion \nat and above Cr concentrations of 9--10\\%~\\cite{Lev1989.Wear131.39}.\n\n\n\nThe static properties of the iron--chromium system have been extensively studied at the atomic level. For example, surface~\\cite{Rop2011.JPCM23.265004} and interface~\\cite{Lu2011.PhysStatusSolidiB248.2087} energies and surface segregation energies of chromium~\\cite{Lev2012.PRB85.064111} have been calculated for Fe--Cr alloys using \\emph{ab initio} methods.\nImportantly, such calculations have shown that the aforementioned critical Cr concentration for the onset of the corrosion resistance in stainless steels closely coincides with an onset of anomalous surface segregation of Cr in Fe--Cr alloys~\\cite{Rop2007.PRB76.220401, Lev2013.PRB87.075409}. This observation suggests that the segregation of Cr toward the surface plays a crucial role in facilitating the formation of the protective, self-healing layer of iron and chromium oxides that is known to be the reason for the corrosion resistivity of stainless steels~\\cite{Ole1980.MaterSciEng42.161,Lin1992.SurfSci277.43,Gon2007.InorgMater43.515,Don2009.CorrosSci51.827,Rop2021.SciRep11.6046}. \n\n\nThe current \\emph{ab initio} modeling methods give reasonably accurate predictions of the static properties of Fe--Cr alloys. The drawback of these methods is that they are computationally heavy, \nto the extent that the length scales in the modeling can be no larger than a few nanometers and that the costs of studying time-dependent phenomena are still largely prohibitive.\nIn an attempt to overcome these limitations, semi-empirical potential models have been developed for the Fe--Cr system.\nThe most recent of these \nare the concentration-dependent embedded-atom model~(CDEAM)~\\cite{Car2005.PRL95.075702,*Stu2009.ModelSimulMaterSc17.075005}, the two-band embedded-atom model~(2BEAM)~\\cite{Bon2011.PhilosMag91.1724}, and \nthe Tersoff potential~\\cite{Hen2013.JPCM25.445401}. \nThese potentials have been fitted to various basic material properties such as cohesion energies, lattice constants, and elastic properties, and, in general, they describe the point-defect energetics and the solubility of chromium in iron at low concentrations fairly well.\nAs such, the semi-empirical Fe--Cr potential models are well suited for modeling bulk Fe--Cr alloys and have been employed, in conjunction with Monte Carlo methods, to study \ndiffusion coefficients and precipitation kinetics~\\cite{Sen2014.ActaMater73.97}, vacancy migration near grain boundaries~\\cite{Cas2014.ComputMaterSci84.217}, and equilibrium configurations of phase-separated Fe--Cr alloys~\\cite{Zhu2011.JNuclMater417.1082} (using an earlier 2BEAM parametrization~\\cite{Ols2005.PRB72.214119}).\n\n\n\nThe aforementioned semi-empirical potential models~\\cite{Bon2011.PhilosMag91.1724,Car2005.PRL95.075702,Stu2009.ModelSimulMaterSc17.075005,Hen2013.JPCM25.445401} turn out to be, however, much less successful in predicting\nproperties of Fe--Cr surfaces~\\cite{Kur2015.PRB92.214113}. \nTheir disagreement with \n\\emph{ab initio} calculations is succinctly demonstrated by inspecting the segregation of a Cr atom from the bulk to the surface of a bcc Fe crystal: all three semi-empirical models predict the Cr atom to segregate to the second layer of Fe atoms, whereas according to \\emph{ab initio} calculations the first layer (i.e., the surface itself) should be preferred~\\cite{Kur2015.PRB92.214113}. In addition, all three models fail to reproduce the \\emph{ab initio} results of Fe--Cr interface energies~\\cite{Lu2011.PhysStatusSolidiB248.2087}. As a consequence of these shortcomings, no Monte Carlo simulations of atom kinetics \nnear Fe--Cr surfaces\nhave been performed, and the current atomic-level knowledge of surface physics in Fe--Cr alloys is far from satisfactory, \ndespite the practical importance of the topic in relation to the\ncorrosion of steel surfaces.\n\n\n\n\nIn this paper, we develop a new semi-empirical potential model that is suitable for investigating surface physics in Fe--Cr alloys at the atomic level,\nyet offers performance on par with the existing models in describing the bulk alloy.\nWe choose to work in the Tersoff formalism and use previously developed potentials for the homonuclear Fe--Fe~\\cite{Mul2007.JPCM19.326220,Bjo2007.NIMB259.853} and Cr--Cr~\\cite{Hen2013.JPCM25.445401} interactions while constructing a new one for the heteronuclear Fe--Cr interaction.\nThis choice \nof homonuclear potentials \nmakes the new potential model directly compatible with the previously developed Fe--C~\\cite{Hen2009.PRB79.144107,Hen2013.JPCM25.445401} and Cr--C~\\cite{Hen2013.JPCM25.445401} models~\\footnote{The Fe--C and Cr--C models both use the same C--C potential from Refs.~\\cite{Bre1990.PRB42.9458,Jus2005.JApplPhys98.123520}, and are therefore mutually compatible.}, so that they can all be combined into an Fe--Cr--C potential model for stainless steel.  \n\n\n\n\nThe remainder of this paper is organized as follows. In Sec.~\\ref{sec:methods}, we \nintroduce \nthe Tersoff potential formalism, describe the fitting procedure by which we develop the new Fe--Cr potential, and \noutline \nthe DFT methods \nthat we use to generate the target data for the fitting procedure.\nSection~\\ref{sec:results} is devoted to presenting the new potential and benchmarking it against the pre-existing potential models and DFT calculations, \nwith both bulk and surface properties considered. \n\\softpekcom{Removed a false sentence about the contents. The results mentioned could possibly be added into the paper, though.}\nFinally, we summarize our main results and discuss their implications in Sec.~\\ref{sec:discussion}.\n\n\\section{Theory and methods}\\label{sec:methods}\n\n\\subsection{Potential formalism}\\label{subsec:potential_formalism}\n\nThe reactive Tersoff formalism~\\cite{Ter1988.PRB37.6991,Bre1990.PRB42.9458,Alb2002.PRB65.195124} adopted in this work originates from Pauling's concept of bond order~\\cite{Pau1960.book,Abe1985.PRB31.6184};\nit can also be formally linked~\\cite{Alb2002.PRB65.195124} to\nboth the tight-binding scheme~\\cite{Cle93.PRB48.22} and the embedded-atom method~\\cite{Daw1984.PRB29.6443,Bre1989.PRL63.1022}. Since the same formalism has already been described extensively elsewhere~\\cite{Alb2002.PRB65.195124,Alb2002.PRB66.035205,Nor2003.JPCM15.5649,Jus2005.JApplPhys98.123520}, we will give here only a brief overview.\nIn the molecular dynamics code {\\footnotesize{LAMMPS}}~\\cite{Pli1995.JCompPhys117.1}, this formalism is available as the potential style \\texttt{tersoff\/zbl}.\nAlthough the only atomic types considered in this work are the elements Fe and Cr, we present the potential formalism below for a general system with an arbitrary number of atomic types.\n\nLet each atom in the system (regardless of its type) be assigned a unique ordinal number, which we will denote using the roman indices $\\inda,\\indb,\\indc\\in\\mathbb{N}$. Furthermore, let $\\left(\\elemindex{\\inda}\\right)_{\\inda}$ be a finite sequence such that $\\elemindex{\\inda}$ gives the type of the $\\inda$th atom, with $\\inda \\in \\left\\{1,\\dots,N\\right\\}$ and $N$ denoting the total number of atoms in the system~\\footnote{In our case, $\\elemindex{\\inda}\\in \\left\\{ \\mathrm{Fe},\\mathrm{Cr} \\right\\}\\ \\forall \\inda\\left\\{1,\\dots,N\\right\\}$ and $N=N_\\mathrm{Fe}+ N_\\mathrm{Cr}$.}. \nIn the Tersoff formalism, the total potential energy $E_\\mathrm{tot}$ of the system can then be written as a sum of individual bond energies:\n\\begin{equation}\\label{eq:total_potential}\nE_\\mathrm{tot}=\\sum_{\\inda=1}^N \\sum_{\\indb = \\inda+1}^N \\left\\{ \\VZBL{\\elemindex{\\inda}}{\\elemindex{\\indb}} \\bigl(r_{\\inda\\indb}\\bigr)\\left[1-F_{\\elemindex{\\inda}\\elemindex{\\indb}} \\bigl(r_{\\inda\\indb}\\bigr)\\right]+\\VABOP{\\elemindex{\\inda}}{\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr)F_{\\elemindex{\\inda}\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr)\\right\\},\n\\end{equation}\nwhere $r_{\\inda\\indb}$ is the distance between atoms $\\inda$ and $\\indb$, $\\VZBL{\\elemindex{\\inda}}{\\elemindex{\\indb}}$ is the universal Ziegler--Biersack--Littmark (ZBL) potential~\\cite{Zie1985_book,*Zie1985.incollection} for elements $\\elemindex{\\inda}$ and $\\elemindex{\\indb}$, \\(\\VABOP{\\elemindex{\\inda}}{\\elemindex{\\indb}}\\) is the pure Tersoff potential for these elements,\nand\n\\begin{equation}\\label{eq:fermi}\nF_{\\elemindex{\\inda}\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr)=\\left\\{1+\\exp\\left[-b_{\\mathrm{F},\\elemindex{\\inda}\\elemindex{\\indb}}\\left(r_{\\inda\\indb}-r_{\\mathrm{F},\\elemindex{\\inda}\\elemindex{\\indb}}\\right)\\right]\\right\\}^{-1}\n\\end{equation}\nis a Fermi function used to join the short-range ZBL and longer-range Tersoff parts smoothly together.  The values of the parameters $b_\\mathrm{F}$ and $r_\\mathrm{F}$ are chosen manually such that the potential is essentially the unmodified Tersoff potential at and past equilibrium bonding distances and that a smooth transition to the ZBL potential at short separations is obtained for all realistic coordination numbers. \\softpekcom{Refer to other papers utilizing this same interpolation approach?}\n\nIncorporation of the ZBL potential [as done in Eq.~\\eqref{eq:total_potential}] is needed to make the potential formalism suitable for modeling nonequilibrium phenomena such as melting or high-energy particle irradiation processes, which typically involve repulsive short-distance interactions originating mainly from the screened Coulomb repulsion between the positively charged nuclei. The ZBL potential is written as\n\\begin{equation}\\label{eq:ZBL}\n\\VZBL{\\elemindex{\\inda}}{\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr)=\\frac{e^2}{4\\pi \\varepsilon_0}\\frac{Z_{\\elemindex{\\inda}}Z_{\\elemindex{\\indb}}}{r_{\\inda\\indb}} \\phi\\bigl(r_{\\inda\\indb}\/a_{\\elemindex{\\inda}\\elemindex{\\indb}} \\bigr),\n\\end{equation}\nwhere $Z_\\elemindex{\\inda}$ is the atomic number of element $\\elemindex{\\inda}$,\n\\begin{equation}\na_{\\elemindex{\\inda}\\elemindex{\\indb}}= \\frac{\\SI{0.8854}{\\bohr}}{Z_{\\elemindex{\\inda}}^{0.23}+Z_{\\elemindex{\\indb}}^{0.23}}\n\\end{equation}\nwith $\\si{\\bohr}$ denoting the Bohr radius, and $\\phi$ is the universal screening function\n\\begin{equation}\n\\begin{split}\n\\phi\\left(x\\right)&=0.1818\\,e^{-3.2\\, x}+ 0.5099\\, e^{-0.9423\\,x} \\\\ & \\quad + 0.2802\\,  e^{-0.4028\\,x} + 0.02817\\,e^{-0.2016\\,x}.\n\\end{split}\n\\end{equation}\nThis screening function has been fitted to the interaction energy between ions, and its accuracy is $\\sim\\mkern-6mu 10\\%$~\\cite{Zie1985_book,*Zie1985.incollection}.\n\nThe Tersoff part $\\VABOP{}{}$ is what chiefly determines the equilibrium properties of the system. It is written as\n\\begin{equation}\\label{eq:ABOP}\n\\VABOP{\\elemindex{\\inda}}{\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr) = f^\\mathrm{c}_{\\elemindex{\\inda}\\elemindex{\\indb}} \\bigl(r_{\\inda\\indb}\\bigr)\\left[V_{\\elemindex{\\inda}\\elemindex{\\indb}}^\\mathrm{R}\\bigl(r_{\\inda\\indb}\\bigr) - \\frac{b_{\\elemindex{\\inda}\\elemindex{\\indb}}+b_{\\elemindex{\\indb}\\elemindex{\\inda}}}{2} V_{\\elemindex{\\inda}\\elemindex{\\indb}}^\\mathrm{A}\\bigl(r_{\\inda\\indb}\\bigr) \\right],\n\\end{equation} \nwhere $f^\\mathrm{c}$ is a cutoff function for the pair interaction, $V^\\mathrm{R}$ is a repulsive and $V^\\mathrm{A}$ an attractive pair potential, and $b$ is a bond-order term that describes three-body interactions and angularity. The pair potentials are of the Morse-like form\n\\begin{subequations}\\label{eq:pair_potentials}\n\\begin{align}\n\\label{eq:pair_potentialsR}\nV^\\mathrm{R}_{\\elemindex{\\inda}\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr) &=\\frac{D_{0,\\elemindex{\\inda}\\elemindex{\\indb}}}{S_{\\elemindex{\\inda}\\elemindex{\\indb}}-1} \\exp\\left[- \\sqrt{2S_{\\elemindex{\\inda}\\elemindex{\\indb}}}\\beta_{\\elemindex{\\inda}\\elemindex{\\indb}}\\left(r_{\\inda\\indb}-r_{0,\\elemindex{\\inda}\\elemindex{\\indb}}\\right)\\right], \\\\ \n\\label{eq:pair_potentialsA}\nV^\\mathrm{A}_{\\elemindex{\\inda}\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr) &=\\frac{S_{\\elemindex{\\inda}\\elemindex{\\indb}}D_{0,\\elemindex{\\inda}\\elemindex{\\indb}}}{S_{\\elemindex{\\inda}\\elemindex{\\indb}}-1} \\exp\\left[-\\frac{\\sqrt{2}\\beta_{\\elemindex{\\inda}\\elemindex{\\indb}}}{\\sqrt{S_{\\elemindex{\\inda}\\elemindex{\\indb}}}}\\left(r_{\\inda\\indb}-r_{0,\\elemindex{\\inda}\\elemindex{\\indb}}\\right)\\right],\n\\end{align}\n\\end{subequations}\nwhere $D_0$ and $r_0$ are the bond energy and length of the dimer molecule, respectively, and $S>1$ is \na dimensionless parameter that adjusts the relative strengths of the repulsive and attractive terms. \nThe parameter $\\beta$ is related to the ground-state vibrational frequency $\\omega$ and the reduced mass $\\mu$ of the dimer according to \n\\begin{equation} \n\\beta_{\\elemindex{\\inda}\\elemindex{\\indb}}=\\frac{\\sqrt{2\\mu_{\\elemindex{\\inda}\\elemindex{\\indb}}}\\pi\\omega_{\\elemindex{\\inda}\\elemindex{\\indb}}}{\\sqrt{D_{0,\\elemindex{\\inda}\\elemindex{\\indb}}}}.\n\\end{equation}\nThe bond-order term is given by \n\\begin{equation}\nb_{\\elemindex{\\inda}\\elemindex{\\indb}} = \\frac{1}{\\sqrt{1+\\chi_{\\elemindex{\\inda}\\elemindex{\\indb}}}},\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:chi}\n\\chi_{\\elemindex{\\inda}\\elemindex{\\indb}}=\\sum_{\\indc\\left(\\neq \\inda,\\indb\\right)} f^\\mathrm{c}_{\\elemindex{\\inda}\\elemindex{\\indc}}\\bigl(r_{\\inda\\indc}\\bigr) g_{\\elemindex{\\inda}\\elemindex{\\indc}}\\bigl(\\theta_{\\inda\\indb\\indc}\\bigr)\\exp\\bigl[\\alpha_{\\elemindex{\\inda}\\elemindex{\\indb}\\elemindex{\\indc}}\\bigl(r_{\\inda\\indb}-r_{\\inda\\indc} \\bigr)\\bigr].\n\\end{equation} \nIn Eq.~\\eqref{eq:chi}, $\\theta_{\\inda\\indb\\indc}$ is the angle between the vectors $\\mathbf{r}_{\\inda\\indb}= \\mathbf{r}_{\\indb} - \\mathbf{r}_{\\inda}$ and $\\mathbf{r}_{\\inda\\indc}$, and $g$ is the angular function \n\\begin{equation}\\label{eq:g}\ng_{\\elemindex{\\inda}\\elemindex{\\indc}}\\bigl(\\theta_{\\inda\\indb\\indc}\\bigr)=\\gamma_{\\elemindex{\\inda}\\elemindex{\\indc}}\\left[1+\\frac{c_{\\elemindex{\\inda}\\elemindex{\\indc}}^2}{d_{\\elemindex{\\inda}\\elemindex{\\indc}}^2}-\\frac{c_{\\elemindex{\\inda}\\elemindex{\\indc}}^2}{d_{\\elemindex{\\inda}\\elemindex{\\indc}}^2+\\left(h_{\\elemindex{\\inda}\\elemindex{\\indc}}+\\cos\\theta_{\\inda\\indb\\indc}\\right)^2}\\right],\n\\end{equation}\nwhere $\\gamma$, $c$, $d$, and $h$ are adjustable parameters. \n\nThe cutoff function $f^\\mathrm{c}$ appearing in Eqs.~\\eqref{eq:ABOP} and~\\eqref{eq:chi} is continuously differentiable and is defined piecewise as\n\\begin{equation}\\label{eq:cutoff}\nf^\\mathrm{c}_{\\elemindex{\\inda}\\elemindex{\\indb}}\\bigl(r_{\\inda\\indb}\\bigr) =\n\\begin{cases}\n1,&  r_{\\inda\\indb} < R_{\\elemindex{\\inda}\\elemindex{\\indb}}-D_{\\elemindex{\\inda}\\elemindex{\\indb}}, \\\\\n\\frac{1}{2}-\\frac{1}{2}\\sin\\frac{\\pi\\left(r_{\\inda\\indb}-R_{\\elemindex{\\inda}\\elemindex{\\indb}}\\right)}{2D_{\\elemindex{\\inda}\\elemindex{\\indb}}},& |r_{\\inda\\indb}-R_{\\elemindex{\\inda}\\elemindex{\\indb}}|\\leq D_{\\elemindex{\\inda}\\elemindex{\\indb}}, \\\\\n0,& r_{\\inda\\indb} > R_{\\elemindex{\\inda}\\elemindex{\\indb}}+D_{\\elemindex{\\inda}\\elemindex{\\indb}},\n\\end{cases}\n\\end{equation} \nwhere $R$ and $D$ determine, respectively, the center and width of the cutoff interval. Typically, $R$ is chosen to lie midway between the second- and third-nearest neighbors in the relevant equilibrium crystal.\n\n\\subsection{Fitting procedure}\\label{subsec:fitting}\n\nIn order to devise a well-performing Fe--Cr potential in the Tersoff formalism, we use the following approach: \nThe parameters for the Cr--Cr interaction are all taken completely unchanged from Ref.~\\cite{Hen2013.JPCM25.445401}, while the parameters for the Fe--Fe interaction are taken unchanged from Refs.~\\cite{Mul2007.JPCM19.326220,Bjo2007.NIMB259.853} with the exception that the value of the cutoff distance $R_{\\mathrm{Fe}\\,\\mathrm{Fe}}$ is increased to \\SI{3.5}{\\angstrom} from the original \\SI{3.15}{\\angstrom} to avoid an unphysical increase in Young's modulus of elasticity at elevated temperatures~\\cite{[{This same cutoff adjustment has been previously made by, e.g., }]Kuo2016.CMS111.525}. \nFurthermore, the two parameters $b_{\\mathrm{F},\\,\\mathrm{Fe}\\,\\mathrm{Cr}}$ and $r_{\\mathrm{F},\\,\\mathrm{Fe}\\,\\mathrm{Cr}}$ appearing in the Fermi function [Eq.~\\eqref{eq:fermi}] of the Fe--Cr potential are chosen to be the same as in Ref.~\\cite{Hen2013.JPCM25.445401}. \nAfter making these initial choices, we are left with \na total of 17 Fe--Cr potential parameters \nthat we then determine by numerical optimization.\n\n\nWe implement the numerical optimization in {\\footnotesize MATLAB}~\\cite{matlab}. To this end, the fitting of the 17 non-predetermined potential parameters is formulated as the nonlinear constrained least-squares minimization problem\n\\begin{equation}\\label{eq:minimization_problem}\n\\min_{\\xi_i\\in[0,1]}T(\\xi_1,\\dots,\\xi_{17}),\n\\end{equation}\nwhere $\\xi_i\\in[0,1]$  is a normalized optimization variable that maps to the closed interval between the minimum and maximum values we allow for the $i$th potential parameter. \nThe minimum (maximum) allowed values are chosen small (large) enough to have no significant impact on the optimal solution.\nThe target function $T$ is the weighted square sum of the differences between the target values $t_j$ and the potential model's predictions $p_j$,\n\\begin{equation}\\label{eq:target_function}\nT(\\xi_1,\\dots,\\xi_{17})=\\sum_{j=1}^{N_{\\mathrm{data}}}w_j \\left[ t_j - p_j\\left(\\xi_1,\\dots,\\xi_{17}\\right)\\right]^2,\n\\end{equation}\nwhere $w_j \\geq 0$ and $N_{\\mathrm{data}}$ is the number of evaluated quantities in the fitting database. The constrained minimization problem \\eqref{eq:minimization_problem} is solved using the trust-region-reflective algorithm~\\cite{Mor1983.SJSSC3.553,Byr1988.MaPr40.247,Bra1999.SJSC21.1} implemented in the {\\footnotesize MATLAB} function \\texttt{lsqnonlin}. The potential predictions $p_j$ are evaluated by calling the \\texttt{minimize} command in {\\footnotesize LAMMPS}~\\cite{Pli1995.JCompPhys117.1}.\nThe target values in our fitting database are determined beforehand by DFT calculations, with the database consisting of the following quantities ($N_{\\mathrm{data}}=40$):\n\\begin{itemize}\n    \\item FeCr dimer bond length $r_0$ (target value \\SI{2.1428}{\\angstrom});\n   \n    \\item formation energies of nine Cr point defects in bcc Fe (see Table~\\ref{table:defect_energies} for the specific defects and the target formation energies);\n    \\item mixing energy of Fe\\textsubscript{52}Cr\\textsubscript{2} (7 different configurations);\n    \\item mixing energy of Fe\\textsubscript{51}Cr\\textsubscript{3} (16 different configurations);\n    \\item mixing energy of Fe\\textsubscript{40}Cr\\textsubscript{14} (SQS cell);\n    \\item mixing energy of Fe\\textsubscript{27}Cr\\textsubscript{27} (SQS cell);\n    \\item mixing energy of Fe\\textsubscript{14}Cr\\textsubscript{40} (SQS cell);\n    \\item Cr segregation energies to the first four layers of a (100) surface of bcc Fe.\n\\end{itemize}\nHere the alloy mixing energies are calculated for a periodically repeating 54-atom bcc \ncell; the subscripts indicate the numbers of Fe and Cr atoms per cell. \nFor computational efficiency, the mixing energies corresponding to the combinatorially challenging intermediate Cr concentrations from 25.9\\% to 74.1\\% are estimated by means of special quasirandom structures (SQSs)~\\cite{Zun1990.PRL65.353}. For a given Cr concentration $c_{\\mathrm{Cr}} \\coloneqq N_\\mathrm{Cr}\/\\left(N_\\mathrm{Fe}+N_\\mathrm{Cr}\\right)$, the SQS cell is a single, computationally constructed 54-atom cell whose lattice sites are occupied by Fe and Cr atoms in such a way that the resulting periodic lattice mimics as closely as possible the first few, physically most relevant radial correlation functions of a perfectly random lattice with the same $c_{\\mathrm{Cr}}$. We generate the SQS cells using the \\texttt{mcsqs} code~\\cite{Wal2013.Calphad42.13} of the Alloy Theoretic Automated Toolkit~\\cite{Wal2002.Calphad26.539}.\n\n\n\n\n\\subsection{DFT calculations}\\label{subsec:dft_methods}\n\nBefore solving the optimization problem~\\eqref{eq:minimization_problem}, we carry out \\emph{ab initio} DFT calculations to determine the target values $t_j$ in Eq.~\\eqref{eq:target_function}.\nAll our DFT calculations are performed using the GPAW code~\\cite{Mor2005.PRB71.035109,gpaw2} (version 1.1.0) and the Atomic Simulation Environment (ASE)~\\cite{ase} (version 3.11). Valence--core interactions are modeled with the projector augmented-wave method (GPAW\/PAW version 0.8). \nThe generalized-gradient approximation is used in the form of the Perdew--Burke--Ernzerhof exchange\u2013correlation functional~\\cite{pbe}.\n\nAll the bulk DFT calculations are carried out using a $3\\times 3\\times 3$ cubic simulation cell of 54 atoms. \nWave functions are represented on a real-space grid of $48\\times 48 \\times 48$ points, and Brillouin-zone integrations are performed on \na Monkhorst--Pack grid~\\cite{Mon1976.PRB13.5188} of $4\\times 4 \\times 4$ $k$ points.\nAll atomic coordinates are relaxed using the Broyden--Fletcher--Goldfarb--Shanno algorithm \nuntil the Hellmann--Feynman forces are less than \\SI[per-mode=symbol]{0.05}{\\electronvolt\\per\\angstrom}. \n\nThe DFT calculations involving an Fe(100) surface are performed for a $2\\times 2\\times 5$ slab geometry with a distance of \\SI{24}{\\angstrom} between the two surfaces. \nA real-space grid of $48\\times 48 \\times 288$ points and a Monkhorst--Pack grid of $4 \\times 4 \\times 1$ $k$ points are used. \nThe two centermost atomic layers of the slab are fixed to their bulk positions, while all other atoms are relaxed using the {\\footnotesize{FIRE}} algorithm~\\cite{Bit2006.PRL97.170201} with a force tolerance of \\SI[per-mode=symbol]{0.05}{\\electronvolt\\per\\angstrom}. \n\n\n\n\n\n\n\\section{Results}\\label{sec:results}\n\n\\subsection{New Fe--Cr potential}\n\n\\softpekcom{If needed, discuss the values of the cutoff parameters here.}\n\nThe numerically optimized parameter values for the new Fe--Cr Tersoff potential are shown in Tables~\\ref{table:abop_parameters} and~\\ref{table:alpha_parameters}. The corresponding potential input file for the LAMMPS potential style \\texttt{tersoff\/zbl} is available at [URL].\n\nIt is worth noting from Table~\\ref{table:alpha_parameters} that the values of the six three-body parameters $\\alpha$ pertaining to the heteronuclear Fe--Cr part have apparently not been numerically optimized in the old Tersoff parametrization of Ref.~\\cite{Hen2013.JPCM25.445401}, on the grounds of them all having the same exact value of 1. \nThis should be contrasted with \nthe new parametrization, where all six heteronuclear $\\alpha$ parameters have been fully incorporated into the optimization. \nThe resulting increase in the dimensionality of the optimization space may in part explain why we have succeeded in significantly improving the Tersoff potential's agreement with the \\emph{ab initio} target data \nfor both bulk and surface properties, as demonstrated below.\n\n\n\n\n\n\n\n\\begin{table}[htb]\n\\caption{\\label{table:abop_parameters}Parameters for the new Tersoff potential of the Fe--Cr system. \nThe Fe--Fe part is the same as originally introduced in Ref.~\\cite{Mul2007.JPCM19.326220} and subsequently augmented with the additional parameters $b_\\mathrm{F}$ and $r_\\mathrm{F}$ [Eq.~\\eqref{eq:fermi}] in Ref.~\\cite{Bjo2007.NIMB259.853}, except for a slightly larger cutoff $R$ that we adopt to avoid an unphysical increase in Young's modulus of elasticity at elevated temperatures; \nthe Cr--Cr part is from Ref.~\\cite{Hen2013.JPCM25.445401}; the Fe--Cr part given in the last column is derived in this work. \nNote that the three-body parameter $\\alpha$ has only one value associated with each of the two homonuclear potentials ($\\alpha_{\\mathrm{Fe}\\Fe\\mathrm{Fe}}$ and $\\alpha_{\\mathrm{Cr}\\Cr\\mathrm{Cr}}$, as shown here) but assumes six distinct values for the heteronuclear Fe--Cr part (as listed separately in Table~\\ref{table:alpha_parameters}). \nAll the two-body Fe--Cr parameters  \nare symmetric with respect to interchange of the atomic types (i.e., $\\gamma_{\\mathrm{Fe}\\,\\mathrm{Cr}}=\\gamma_{\\mathrm{Cr}\\,\\mathrm{Fe}}$, and similarly for the others).\n}\n\\begin{ruledtabular}\n\\begin{tabular}{llddd}\n& & \\multicolumn{3}{c}{Interaction}\\\\\n\\cline{3-5}\n\\multicolumn{2}{c}{Parameter}&\\multicolumn{1}{c}{Fe--Fe} &\\multicolumn{1}{c}{Cr--Cr}&\\multicolumn{1}{c}{Fe--Cr}\\\\\n\\hline\n$D_0$ &(eV) & 1.5 & 4.0422 & 1.2277 \\\\\n$r_0$ &(\\AA) & 2.29 & 2.1301 & 2.2320 \\\\ \n$\\beta$ &(\\AA$^{-1}$)& 1.4 & 1.6215 &0.8957\\\\\n$S$ & & 2.0693 & 3.3679 & 3.1743\\\\  \n$\\gamma$ & & 0.01158 &  0.1577 & 0.0996 \\\\ \n$c$ & & 1.2899 & 1.0329 & 0.0794\\\\ \n$d$ & &  0.3413 & 0.1381 & 5.9464 \\\\  \n$h$ & &  -0.26 & -0.2857 &0.2952 \\\\\n$R$ &(\\AA) & 3.5 & 3.2 & 3.15 \\\\\n$D$ &(\\AA) & 0.2 & 0.2  & 0.15\\\\\n$\\alpha$ & & 0 & 1.3966 & - \\\\\n$b_\\mathrm{F}$ &($\\mathrm{\\AA}^{-1}$) & 2.9 & 12.0 & 10.0 \\\\\n$r_\\mathrm{F}$ &(\\AA) & 0.95 & 1.7 & 1.0\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table}[htb]\n\\caption{\\label{table:alpha_parameters} Three-body coefficients $\\alpha$ pertaining to the Fe--Cr part of the potential for both the old and the new Tersoff parametrizations. The rest of the parameter values for the new Tersoff potential are given in Table~\\ref{table:abop_parameters}, \nwhile the old Tersoff potential is presented in its entirety in Ref.~\\cite{Hen2013.JPCM25.445401}.}\n\\begin{ruledtabular}\n\\begin{tabular}{ldd}\n& \\multicolumn{2}{c}{Fe--Cr parametrization}\\\\\n\\cline{2-3}\n\\multicolumn{1}{c}{Parameter}&\\multicolumn{1}{c}{Old Tersoff} &\\multicolumn{1}{c}{New Tersoff}\\\\\n\\hline\n$\\alpha_{\\mathrm{Fe}\\,\\mathrm{Fe}\\,\\mathrm{Cr}}$ & 1.0 & -0.356010 \\\\\n$\\alpha_{\\mathrm{Fe}\\,\\mathrm{Cr}\\,\\mathrm{Fe}}$ & 1.0 & -3.282748 \\\\\n$\\alpha_{\\mathrm{Cr}\\,\\mathrm{Fe}\\,\\mathrm{Fe}}$ & 1.0 & -1.181222 \\\\\n$\\alpha_{\\mathrm{Fe}\\,\\mathrm{Cr}\\,\\mathrm{Cr}}$ & 1.0 & -1.340769 \\\\\n$\\alpha_{\\mathrm{Cr}\\,\\mathrm{Fe}\\,\\mathrm{Cr}}$ & 1.0 & -0.233643 \\\\\n$\\alpha_{\\mathrm{Cr}\\,\\mathrm{Cr}\\,\\mathrm{Fe}}$ & 1.0 & -1.455751 \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsection{Comparison of the potential models in regard to bulk Fe--Cr}\\label{subsec:bulk_results}\n\n\\softpekcom{Although it might be beneficial in some sense to start with the surface properties, I think it is conceptually more straightforward to start with the bulk properties first. If you think otherwise, I'm open to changing this. One might also question if the bulk\/surface division is the best way to go for us. As for now, there are quite a lot more bulk than surface results---regardless of our project aims.}\n\n\n\\softpekcom{Start the bulk results with introducing Fig.~\\ref{fig:mixing_energy} and defining the mixing energy and formation energy.}\n\nLet us start with the properties of bulk Fe--Cr alloys and compare the predictions of the new Tersoff potential to those of our DFT calculations, the CDEAM potential, the 2BEAM potential, and the old Tersoff potential. \nIn particular, Figs.~\\ref{fig:mixing_energy} and~\\ref{fig:sro_700K} show, respectively, the mixing energy and the Cowley short-range order parameter of the bcc Fe--Cr alloy as a function of the chromium concentration, and Table~\\ref{table:defect_energies} lists the formation energies of various isolated Cr defects in bcc iron.\nIn Fig.~\\ref{fig:mixing_energy}, the alloy mixing energy is determined as the formation energy per atom averaged over different random-alloy configurations \nat a given chromium concentration $c_{\\mathrm{Cr}} \\coloneqq N_{\\mathrm{Cr}}\/\\left(N_{\\mathrm{Fe}}+N_{\\mathrm{Cr}}\\right)$. \nThe formation energy of a given structure, in turn, is defined in this work as\n\\begin{equation}\\label{eq:formation_energy}\nE_{\\mathrm{f}} = E_{\\mathrm{tot}}\\left(\\mathrm{Fe}_{1-c_{\\mathrm{Cr}}}\\mathrm{Cr}_{c_{\\mathrm{Cr}}}\\right) - N_{\\mathrm{Fe}}E_{\\mathrm{coh}} \\left(\\mathrm{Fe}\\right) - N_{\\mathrm{Cr}}E_{\\mathrm{coh}}\\left(\\mathrm{Cr}\\right),\n\\end{equation}\nwhere $E_{\\mathrm{tot}}$ is the total potential energy of the computational cell, $N_\\elemindex{}$ is the number of atoms of element $\\elemindex{}$ in the cell, and $E_{\\mathrm{coh}}\\left(\\elemindex{}\\right)$ is the cohesive energy of element $\\elemindex{}$. \nThe cohesive energy $E_{\\mathrm{coh}}\\left(\\elemindex{}\\right)$ is determined as the total potential energy of a cell containing only atoms of element $\\elemindex{}$ divided by the number of atoms in that cell.\nFor the Fe--Fe and Cr--Cr potentials in Table~\\ref{table:abop_parameters}, \n$E_{\\mathrm{coh}}\\left(\\mathrm{Fe}\\right)=\\SI{-4.179}{\\electronvolt}$ (bcc lattice constant $a_\\mathrm{Fe}=\\SI{2.889}{\\angstrom}$) and $E_{\\mathrm{coh}}\\left(\\mathrm{Cr}\\right)=\\SI{-4.099}{\\electronvolt}$ ($a_\\mathrm{Cr}=\\SI{2.872}{\\angstrom}$). \n\n\\begin{figure}[htb]\n\\includegraphics[width=0.99\\columnwidth,keepaspectratio]{mixing_energy_full_scale_panel_a_20200818_cropped}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.99\\columnwidth,keepaspectratio]{mixing_energy_small_scale_panel_b_20200818_cropped}\n\\caption{\\label{fig:mixing_energy}(a)~Mixing energy of the bcc Fe--Cr alloy as a function of the chromium concentration as given by our DFT calculations and the four Fe--Cr potential models. (b)~Close-up of the region below 15\\% Cr concentration. The calculations are performed using a 54-atom computational cell. A total of 131 different 54-atom configurations are used for the whole Cr concentration range from 0 to 1, with the error bars corresponding to the standard error of the mean at the given Cr concentration. The lines drawn through the data points are guides to the eye.\n\\softpekcom{I should say something about the configurations used either here or in the body text.} \n}\n\\end{figure}\n\nAs reported by Olsson, Abrikosov, Vitos, and Wallenius~\\cite{Ols2003.JNuclMater321.84}, \\emph{ab initio} calculations predict a negative (positive) mixing energy of Fe--Cr alloys for chromium concentrations below (above) $\\sim 6\\%$. As can be seen from Fig.~\\ref{fig:mixing_energy}(b), our DFT calculations corroborate this result. Although the zero-crossing behavior is qualitatively reproduced by all four potential models under consideration, \nthere are quantitative differences: the new Tersoff potential yields the best fit to the \\emph{ab initio} mixing energies for Cr concentrations $<6$\\%, closely followed by the CDEAM potential, while the 2BEAM and old Tersoff potentials predict the zero crossing to lie at a higher Cr concentration [Fig.~\\ref{fig:mixing_energy}(b)]. For chromium-rich alloys, which are included in Fig.~\\ref{fig:mixing_energy}(a), the new Tersoff potential matches the DFT values significantly better than the other three potentials.\nFor example, at 50\\% Cr concentration, where the alloy mixing energy has been calculated with the SQS cell~\\cite{Zun1990.PRL65.353}, the mixing energy given by the new Tersoff potential is only 0.2\\% larger than the DFT value of \\SI{4.663}{\\electronvolt}, whereas the CDEAM, 2BEAM and old Tersoff potentials differ, respectively, by 15\\%, $-9.7$\\%, and $-65$\\% from the DFT result.\n\\softpekcom{Let me know if you think I should invert the order of the panels in Fig.~\\ref{fig:mixing_energy}. \\akcom{Ok for me.}}\n\n\\softpekcom{As the second set of bulk data, present the point defect energies of Table \\ref{table:defect_energies}. Check also if increasing the computational cell size changes the formation energies appreciably, i.e., whether there is interactions between the defects.}\n\n\n\\begin{table*}[tb]\n\\caption{\\label{table:defect_energies}Energies of Cr point defects in bcc Fe, as per our DFT calculations and the CDEAM, 2BEAM, the old Tersoff, and the new Tersoff potentials. For consistency with the DFT treatment, all the formation energies are calculated using a 54-atom cell. The last line is for the Fe substitutional defect in bcc Cr.}\n\\begin{ruledtabular}\n\\begin{tabular}{lddddd}\n& \\multicolumn{5}{c}{Formation energy (eV)} \\\\\n\\cline{2-6}\n\\multicolumn{1}{c}{Defect} & \\multicolumn{1}{c}{DFT (GPAW)}&  \\multicolumn{1}{c}{CDEAM} & \\multicolumn{1}{c}{2BEAM} & \\multicolumn{1}{c}{Old Tersoff} & \\multicolumn{1}{c}{New Tersoff} \\\\\n\\hline\n$\\langle 100 \\rangle$ Fe--Cr & 5.327 & 3.56 & 3.66 &4.96 &5.15  \\\\\n$\\langle 110 \\rangle$ Fe--Cr & 4.090 & 3.20 & 3.18 & 4.06 &4.22 \\\\\n$\\langle 111 \\rangle$ Fe--Cr & 4.596 & 3.19 & 3.54 & 4.66 &4.88 \\\\\nOctahedral Cr & 5.339 & 3.56 & 3.33 & 6.01 & 5.30 \\\\\nTetrahderal Cr & 4.611 & 3.50 & 3.32 & 4.44 & 4.83\\\\\nSubstitutional Cr & -0.1444 & -0.112 & -0.233 & -0.0629  & -0.136\\\\\n$\\langle 100 \\rangle$ Cr--Cr & 5.43 &4.49  & 3.98 & 4.88 & 5.45 \\\\\n$\\langle 110 \\rangle$ Cr--Cr & 4.35 & 3.81 & 3.26 & 3.80 & 4.09\\\\\n$\\langle 111 \\rangle$ Cr--Cr & 4.699 & 4.63 & 3.51 & 4.58 & 5.00 \\\\\n\\hline\nSubstitutional Fe in Cr & 0.4825 & 0.498 & 0.357 & 0.235  & 0.511\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\nThe formation energies of selected Cr point defects in bcc Fe are presented in Table~\\ref{table:defect_energies}. In general, all four potential models are in fairly good agreement with the target DFT values, with percentage errors typically $<10$\\%. If we take the nine Cr point defects listed in Table~\\ref{table:defect_energies} and compute the symmetric mean absolute percentage error (SMAPE) of the predictions of each potential model~\\cite{[{The SMAPE is defined here as \n$100\\%\\sum_{j=1}^{N_E} \\abs{t_j-p_j}\/\\bigl(N_E \\abs{t_j}+N_E \\abs{p_j}\\bigr)$, where the sum is over the $N_E=9$ formation energies and $t_j$ and $p_j$ are their target values and potential-model predictions, respectively. See, e.g., }]Arm1985.book,*Flo1986.Omega14.93},\nwe obtain the following ranking (from best to worst): new Tersoff (SMAPE of 2.1\\%), old Tersoff (7.2\\%), CDEAM (13\\%), and 2BEAM (17\\%). On average, the new Tersoff potential is therefore in a better agreement with our \\emph{ab initio} point-defect energies than the other three potential models. We note in particular that the new Tersoff potential provides the best match for the negative DFT value of the Cr substitutional defect.\n\n\\softpekcom{Next, define the SRO parameter and present and analyze the SRO data (at three different temperatures?). In particular, compare the various model predictions with the experimental data at \\SI{700}{\\kelvin} (Fig.~\\ref{fig:sro_700K}).} \n\n\\begin{figure}[b]\n\\includegraphics[width=0.99\\columnwidth,keepaspectratio]{sro_beta_at_700K_vs_small_cr_concentration_20200818_cropped}\\\\\n\\vspace{4mm}\n\\includegraphics[width=0.99\\columnwidth,keepaspectratio]{sro_beta_at_700K_vs_cr_concentration_20200818_cropped}\n\\caption{\\label{fig:sro_700K}Short-range order parameter $\\beta$ [Eq.~\\eqref{eq:sro_beta}] at \\SI{700}{\\kelvin} in a bcc Fe--Cr alloy as a function of the chromium concentration $c_{\\mathrm{Cr}}$ for (a)~$0.02 \\leq c_{\\mathrm{Cr}} \\leq 0.16$ and (b)~$0.02 \\leq c_{\\mathrm{Cr}} \\leq 0.9$. The lines are guides to the eye. The experimental data points at $c_{\\mathrm{Cr}}=0.05$, $0.1$, and $0.15$ are from the diffuse--neutron-scattering measurements of Mirebeau, Hennion, and Parette~\\cite{Mir1984.PRL53.687}; fits A and B correspond to least-squares fits of the nuclear cross section with three and four order parameters, respectively.\n\\softpekcom{Could present these at other temperatures as well, but there is no experimental data to compare against.}\n} \n\\end{figure}\n\nBesides comparing formation energies, we may examine the degree of short-range ordering in a thermally equilibrated Fe--Cr alloy. To this end, the Cowley short-range order parameters $\\alpha$ can be defined as~\\cite{Cow1950.PR77.669}\n\\begin{equation}\n\\alpha_{\\mathrm{Cr}}^{(k)} = 1 - \n\\frac{Z^{(k)}_{\\mathrm{Fe}}}{\\left(Z^{(k)}_{\\mathrm{Fe}}+Z^{(k)}_{\\mathrm{Cr}}\\right)\\left(1-c_{\\mathrm{Cr}}\\right)},\n\\end{equation}\nwhere $Z^{(k)}_{\\mathrm{Fe}}$ and $Z^{(k)}_{\\mathrm{Cr}}$ are the average numbers of Fe and Cr atoms in the $k$th neighbor shell.\nWe further define the linear combination\n\\begin{equation}\\label{eq:sro_beta}\n\\beta=\\frac{8 \\alpha_{\\mathrm{Cr}}^{(1)} + 6\\alpha_{\\mathrm{Cr}}^{(2)}}{14}\n\\end{equation}\nrelevant to bcc lattices. If $\\beta < 0$, a Cr atom prefers to have Fe atoms as its nearest neighbors; if $\\beta > 0$, Cr prefers Cr neighbors; $\\beta=0$ corresponds to a random alloy. The configurations used to determine $\\beta$ are computed with a Monte Carlo method where possible moves \nconsist of atom displacements and exchanges of types (Fe or Cr) of pairs of atoms.\nThe displacements are performed with short sequences of MD simulations in the canonical ensemble, \nbecause this \nhas been found to be more efficient in moving the atoms than the conventional Metropolis algorithm~\\cite{Kur2015.PRB92.214113}. It should be noted that these Monte Carlo--MD calculations are pure equilibrium simulations with no kinetics involved.\n\nFigure~\\ref{fig:sro_700K} shows the order parameter $\\beta$ at \\SI{700}{\\kelvin} as a function of the Cr concentration \nfor the four potential models, along with three experimental data points determined by \ndiffuse--neutron-scattering measurements~\\cite{Mir1984.PRL53.687}.\nWhile none of the four potential models yields a particularly good fit to the experimental data, the CDEAM and 2BEAM potentials at least qualitatively reproduce the observed change of sign of $\\beta$ from negative to positive around $c_{\\mathrm{Cr}} = 0.1$  [Fig.~\\ref{fig:sro_700K}(a)].\nFor both Tersoff potentials, $\\beta$ remains negative up to high Cr concentrations, eventually turning positive at $c_{\\mathrm{Cr}}\\approx 0.37$ for the new Tersoff potential and at $c_{\\mathrm{Cr}}\\approx 0.76$ for the old Tersoff potential [Fig.~\\ref{fig:sro_700K}(b)].\nWe thus conclude that the new Tersoff potential, although largely failing to match the experimental data, still performs noticeably better than the old Tersoff potential in describing the short-range ordering in Fe--Cr alloys.\n\n\\softpekcom{As the final set of bulk results, present migration energy barriers in bulk bcc Fe (Fig.~\\ref{fig:energy_barriers} and Table~\\ref{table:migration_barriers}).}\n\nWe have also computed the migration energy barries related to the diffusion of a vacancy--Cr-substitutional pair in bulk bcc Fe using the nudged elastic band (NEB) method~\\cite{Mil1995.SurfaceScience324.305,Jon1998.chapter} implemented in LAMMPS. We consider both a process where a Cr subsitutional moves to a vacancy in the nearest-neighbor site and a process where the vacancy migrates from the nearest-neighbor to the second-nearest-neighbor site of the Cr atom (see Fig.~\\ref{fig:energy_barriers}). The obtained barrier energies for the four potential models are listed in Table~\\ref{table:migration_barriers} along with DFT results by Messina, Nastar, Garnier, Domain, and Olsson~\\cite{Mes2014.PRB90.104203}.\nBy far the best agreement with the DFT values is given by the 2BEAM potential (with the largest relative difference being $<6\\%$), followed by the CDEAM potential, which overestimates the Cr--vacancy migration energy by 53\\% but is otherwise close to the target values. The new Tersoff potential is in fairly good agreement with DFT for the Cr--vacancy migration energy $E_2^\\mathrm{mig}$ but noticeably overestimates both barries $E_{12}^\\mathrm{mig}$ and $E_{21}^\\mathrm{mig}$. The old Tersoff potential is also off by significant margins (26\\% for $E_2^\\mathrm{mig}$, $-32\\%$ for $E_{12}^\\mathrm{mig}$, and $16\\%$ for $E_{21}^\\mathrm{mig}$). \n\n\\begin{figure}[htb]\n\\includegraphics[width=0.65\\columnwidth,keepaspectratio]{migration_barriers_schematic}\n\\caption{\\label{fig:energy_barriers}Nomenclature used for the migration energy barriers. The Cr atom is shown in blue and the Fe atoms in red. $E_2^\\mathrm{mig}$ denotes the migration energy barrier for the solute--vacancy jump, and $E_{ij}^\\mathrm{mig}$ is for an iron atom moving from site $j$ to a vacant site $i$. Table~\\ref{table:migration_barriers} lists the values of these energies barriers within the different models under consideration. \n}\n\\end{figure}\n\n\\begin{table}[tbh]\n\\caption{\\label{table:migration_barriers}Migration energy barriers for Cr--vacancy diffusion and Fe self-diffusion in the neighboring site (see Fig.~\\ref{fig:energy_barriers}).}\n\\begin{ruledtabular}\n\\begin{tabular}{lddd}\nMethod & E_2^\\mathrm{mig} &  E_{12}^\\mathrm{mig}   & E_{21}^\\mathrm{mig}  \\\\\n\\cline{1-4}\nDFT~\\cite{Mes2014.PRB90.104203} & 0.575(45) & 0.69 & 0.64(1) \\\\\nCDEAM & 0.8772 & 0.6467 & 0.6160 \\\\\n2BEAM & 0.5725 & 0.6511 & 0.6260 \\\\\nOld Tersoff & 0.7267 & 0.4666 & 0.7413 \\\\\nNew Tersoff & 0.632 & 1.09 & 0.941 \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsection{Comparison of the potential models for surface-related properties}\\label{subsec:surface_results}\n\nWe now move to scenarios involving surfaces \nof Fe--Cr alloys and investigate whether we have achieved our goal of improving \nupon the performance of the existing potential models in \npredicting properties of such surfaces. \n\nLet us first consider the segregation of Cr atoms to the (100) surface of bcc Fe. The segregation energy of Cr from a given reference region A to another region B is defined as \nthe net change in energy when a Cr atom is transferred from A to B and an Fe atom from B to A,\n\\begin{equation}\nE^{\\mathrm{Cr}}_{\\mathrm{segr},\\mathrm{A}\\to\\mathrm{B}}=E_{\\mathrm{tot}}(\\textrm{Fe at A, Cr at B})-E_{\\mathrm{tot}}(\\textrm{Fe at B, Cr at A}),\n\\end{equation}\nwhere $E_{\\mathrm{tot}}$ is the total energy of the computational cell.  In our case, B will be one of the top surface layers and A will be a bulk site. The energies are calculated using a slab geometry with periodic boundary conditions in $y$ and $z$ directions and the slab thickness $L_x$ large enough that further increases in $L_x$ cause no changes in the segregation energies. The reference bulk position A is taken from the middle of the slab. For nonzero background Cr concentration, the segregation energy values are averaged over 10\\,000 random alloy configurations.\n\n\\begin{figure}[htb]\n\\includegraphics[width=0.891\\columnwidth,keepaspectratio]{segregation_energy_layer1_vs_cr_concentration_20200818_cropped}\n\\includegraphics[width=0.891\\columnwidth,keepaspectratio]{segregation_energy_layer2_vs_cr_concentration_20200818_cropped}\n\\includegraphics[width=0.891\\columnwidth,keepaspectratio]{segregation_energy_difference_vs_cr_concentration_20200818_cropped}\n\\caption{\\label{fig:segregation_energy}Segregation energy of a Cr atom in a bcc Fe--Cr alloy from the bulk to (a) the surface layer and (b) the first subsurface layer as a function of the chromium concentration $c_{\\mathrm{Cr}}$. (c) The difference between the surface- and first-subsurface-layer segregation energies. The asterisk is the GPAW \\emph{ab initio} result at $c_{\\mathrm{Cr}}=0$ and the squares are the \\emph{ab initio} results from Ref.~\\cite{Kur2015.PRB92.214113} obtained using the EMTO method. The lines are guides to the eye.}\n\\end{figure}\n\nThe Cr segregation energies to the topmost surface layer ($\\eseg{1}$) and the second topmost layer ($\\eseg{2}$) are shown in Figs.~\\ref{fig:segregation_energy}(a) and ~\\ref{fig:segregation_energy}(b), respectively, as a function of the Cr concentration $c_{\\mathrm{Cr}}$ for both the DFT and the four potential models. Due to computational limitations, the GPAW DFT calculations are limited to zero background Cr concentration. The DFT results for $c_{\\mathrm{Cr}} > 0$ are obtained \nwith a basis set of exact muffin-tin orbitals (EMTO)~\\cite{And1994.incollection,And2000.PRB62.R16219,Vit2007.book} in combination with the coherent potential approximation~\\cite{Sov1967.PR156.809,Vit2001.PRL87.156401}, \nwhich circumvents the need to average over a large number of alloy configurations as done for the potential models. \nNote that only the zero-concentration segregation energies of the new Tersoff potential have been explicitly fitted (with the target values being the GPAW ones); the data for $c_{\\mathrm{Cr}} > 0$ should be regarded as predictions of the model.\n\nThe old Tersoff potential from Ref.~\\cite{Hen2013.JPCM25.445401} is observed to be far away from the \\emph{ab initio} segregation energies for both layers.\nThe CDEAM and 2BEAM potentials give a fairly good fit to the EMTO results for $\\eseg{1}$ but are far off from both the GPAW and EMTO results for $\\eseg{2}$. \nImportantly, as can be observed from Fig.~\\ref{fig:segregation_energy}(c), all three pre-existing models (CDEAM, 2BEAM, and old Tersoff) yield $\\eseg{2}-\\eseg{1} < 0$ at all Cr concentrations shown and thus predict segregation of Cr to the second atomic layer instead of the topmost one. This is in stark contrast to the \\emph{ab initio} results, for which  $\\eseg{2}-\\eseg{1}$ is positive at all investigated concentrations. Fortunately, this shortcoming is fixed by our new Tersoff potential, for which $\\eseg{2}-\\eseg{1} > 0$ in the entire range $0 \\leq c_{\\mathrm{Cr}} \\leq 0.48$ and the fit to the GPAW value of  $\\eseg{2}-\\eseg{1}=\\SI{0.3696}{\\electronvolt}$ at $c_{\\mathrm{Cr}}=0$ is excellent, the relative error being $2.5\\%$.\n\nWe have also investigated the migration of Cr in the vicinity of an Fe(100) surface. \nIn Table~\\ref{table:surface_migration_barriers}, we list the migration energy barriers for the process where a surface Cr atom moves to a vacancy at a nearest-neighbor site in the second layer. Although we do not have a DFT value for this migration energy to compare against, it is worthwhile to note that the 2BEAM potential and the new Tersoff potential both yield \nan energy barrier close to zero \nwhereas the value given by the old Tersoff potential is much higher than the others.\n\n\\begin{table}[tbh]\n\\caption{\\label{table:surface_migration_barriers}Migration energy barriers for a surface Cr atom moving to a nearest-neighbor second-layer vacancy.}\n\\begin{ruledtabular}\n\\begin{tabular}{ld}\nPotential & \\multicolumn{1}{c}{Energy barrier (eV)}  \\\\\n\\cline{1-2}\nCDEAM & 0.2624 \\\\\n2BEAM & 0.0035 \\\\\nOld Tersoff & 1.1866 \\\\\nNew Tersoff & 0.0023 \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\softpekcom{Here we can also present the migration energy barriers as a function of the layer index (cf. Daniel's presentation). No DFT data to compare against, though.}\n\n\n\\section{Discussion}\\label{sec:discussion}\n\nIn summary, we have presented a new interatomic Fe--Cr potential that is suitable for molecular dynamics simulations of surface physics in Fe--Cr alloys. The potential was formulated in the Tersoff formalism, allowing it to be combined with previously developed Fe--C and Cr--C potentials~\\cite{Hen2013.JPCM25.445401} into a model of the stainless-steel system Fe--Cr--C. The new potential parameters were optimized by fitting to a structural database consisting of \\emph{ab initio} results not only for bulk alloys but also for the segregation of Cr atoms \nto the (100) surface of an Fe--Cr alloy. \nWe compared the performance of the new potential to that of three pre-existing Fe--Cr potentials with regard to the fitting database as well as to bulk- and surface-related quantities not involved in the fitting.\n\nFor the bulk properties, the new Tersoff potential was found to perform at the same overall level as the pre-existing EAM potentials considered, namely, the CDEAM and the 2BEAM potentials. On the one hand, the new Tersoff potential provided the closest match of all the tested potentials to our DFT-calculated mixing energies and point-defect energies. On the other hand, the 2BEAM potential was the best performer when it came to the short-range ordering in random Fe--Cr alloys \n(qualitatively reproducing the experimentally observed sign change of the order parameter at $c_{\\mathrm{Cr}} \\approx 0.1$)\nand to the vacancy--Cr diffusion in bcc Fe \n(yielding energy barriers within 6\\% of the DFT values). \nFor the tested bulk data, the new Tersoff potential performed significantly better than the old Tersoff potential from Ref.~\\cite{Hen2013.JPCM25.445401}, even though only the latter was fitted exclusively to bulk quantities.\n\nThe main objective we set at the beginning was\nfor the new potential model\nto outperform \nthe existing models \nby correctly predicting the surface-segregation behavior of Cr in Fe--Cr alloys.\n\nWe achieved this goal in the sense that the new Tersoff potential yields the same ordering of Cr segregation energies as the DFT calculations, namely, $\\eseg{1} < \\eseg{2}$. This is in contrast to the other potential models, for which $\\eseg{1} > \\eseg{2}$ \nand which thus predict Cr to segregate primarily to the second layer instead of the first.\nThe agreement between the new Tersoff potential and the DFT calculations is far from perfect, however. For example, the new Tersoff potential predicts \na sign change in the first-layer segregation energy\nat a significantly smaller Cr concentration ($\\sim 2.5\\%$) than do the DFT-based EMTO calculations ($\\sim 8\\%$)~\\cite{Kur2015.PRB92.214113,Rop2007.PRB76.220401}.\n\nIn light of the above, we conclude that the new Tersoff potential appears to be the best potential for scenarios where both surface and bulk properties of Fe--Cr alloys are of importance. In bulk systems without boundaries, however, the 2BEAM potential is perhaps the most optimal choice, especially if we take into account its lower computational cost compared to the Tersoff potentials. The old Tersoff potential performs worse than the new one in almost all of our tests, and hence \nit seems difficult to justify its future use.\n\n\\begin{acknowledgments}\nThis work was supported by the Academy of Finland (Grant No.~308632). The computational resources granted by the CSC -- IT Center for Science, Finland, and by the Finnish Grid and Cloud Infrastructure project (FGCI; urn:nbn:fi:research-infras-2016072533) are gratefully acknowledged, as are the facilities provided by the Turku University Centre for Materials and Surfaces (MatSurf).\n\\end{acknowledgments}\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sintro}\n\n\nProtoplanetary disks originate from dense cloud material consisting of\nsub-$\\mu$m sized, almost completely amorphous interstellar medium\n(ISM) dust grains \\citep{BE00,LD01,KE04,HG10}. The dust and gas in\nthese disks form the basic matter from which planets may form. At the\nsame time, mineralogical studies of primitive solar system bodies\nsuggest that a considerable fraction of the silicate grains in these\nobjects are of crystalline nature (\\citealt{WO07,PB10}, and references\ntherein). It is then naturally implied that the crystallinity fraction\nincreases, through thermal and chemical modification of these solids\nduring the general planet formation process, commonly referred to in\nthe literature as ``disk evolution''.\n\nAs time passes, the small dust responsible for the infrared (IR)\nexcess observed around young stars is subjected to different processes\nthat affect, and will eventually determine how this progression will\nend. Planets and planetary systems have been observed around hundreds\nof stars other than the Sun, showing that this result is rather common\n\\citep{US07}. IR observations have revealed a great number of {\\it\n  debris disks}, composed of large planetesimal rocks and smaller\nbodies, around a variety of stars spanning a large range in spectral\ntypes and ages \\citep{RI05,BR06,SU06,GA07,CA09}. A few debris disks\nare known to harbor planets (e.g., $\\beta$ Pictoris and Fomalhaut,\n\\citealt{LG10,KA08}), although it is still unclear whether this is\noften true \\citep{KO09}. The majority of main-sequence stars show no\nsigns of planets or debris within the current observational\nlimitations, however, indicating that the disks around such stars at\nthe time of their formation have dissipated completely, leaving no\ndust behind to tell the story. Which processes are important and\ndeterminant for the aftermath of disk evolution are still under\ndebate, and this topic is the subject of many theoretical and\nobservational studies over the last decade, stimulated in large by\nrecent IR and (sub-)millimeter facilities.\n\nSpecifically on the subject of the mineralogical composition, spectra\nfrom the ground and the {\\it Infrared Space Observatory} gave the\nfirst clues of a potential link between crystalline material in\nprotoplanetary disks and comets. A great similarity was noted between\nthe spectra of the disk around the Herbig star HD 100546 and that of\ncomet Hale-Bopp \\citep{CR97,MA98}. More recently, the InfraRed\nSpectrograph (IRS, 5 -- 38 $\\mu$m, \\citealt{HO04}) on-board the {\\it\n  Spitzer Space Telescope} allowed an unprecedented combination of\nhigh sensitivity and the ability to observe large numbers of disks,\ndown to the brown dwarf limit. The shape of the silicate features\nprobed by the IRS spectra at 10 and 20 $\\mu$m is affected by the\ncomposition, size and structure of its emitting dust. Amorphous\nsilicates show broad smooth mid-IR features, while the opacities of\ncrystalline grains show sharp features due to their large-scale\nlattice arrangement, such that even small fractions of crystalline\ngrains produce additional structure in the silicate features\n\\citep{MI05,BO08,JU09,OF10}. Because most protoplanetary disks are\noptically thick at optical and IR wavelengths, the silicate features\nobserved in the mid-IR are generally emitted by dust in the optically\nthin disk surface only. To probe the disk midplane, observations at\nlonger wavelengths are necessary. Additionally, the emission at 10 and\n20 $\\mu$m has been shown to arise from different grain populations,\nprobing different radii \\citep{KE06,OF09,OF10}. While the 10 $\\mu$m\nfeature probes a warmer dust population, at $\\leq$ 1 AU for T Tauri\nstars, the dust emitting at 20 $\\mu$m is colder, further out and\ndeeper into the disk \\citep{KE07}.\n\nTwo methods have been proposed to explain the formation of crystal\ngrains: thermal annealing of amorphous grains or vaporization followed\nby gas-phase condensation. Both methods require high temperatures\n(above $\\sim$1000 K, \\citealt{FA00,GA04}) which is inconsistent with\nouter disk temperatures. However, crystalline grains have been\nobserved in outer, as well as in inner disks \\citep{VB04}. Large-scale\nradial mixing has been invoked to explain the presence of crystals at\nlow temperatures in the outer disk \\citep{BM00,GA04,CI09}. A third\nproposed formation mechanism for crystal formation is that shock waves\ncould locally heat amorphous silicates and crystallize them\n\\citep{DC02,HD02}.\n\nFrom protoplanetary disks to comets, several authors have attempted to\ninfer the dust composition from IRS spectra and laboratory data on\namorphous and crystalline silicate dust, using a variety of analysis\ntechniques. Whether for individual objects\n\\citep{FO04,ME07,PI08,BY08}, for mixed disk samples\n\\citep{BO01,AP05,VB05,BO08,OF09,OF10,JU10}, or systematic studies of\nthe disk population of a given star-forming region\n\\citep{SI09,WA09,ST09}, it has been shown that a significant mass\nfraction of the dust in those disks must be in crystalline\nform. However, the many studies dealing with the mineralogical\ncomposition of dust to date focus on a specific region or object,\nfailing to investigate the hypothesis that the crystallinity fraction\nis a measure of the evolutionary stage of a region. That is, no study\nin the literature has yet investigated an increase of crystallinity\nfraction with cluster age.\n\nMineralogical studies of Solar System bodies show a range of\ncrystallinity fractions. Evidence from primitive chondrites shows that\nthe abundance of crystalline silicate material varies from nearly\nnothing up to 20 -- 30~\\% (e.g. Acfer 094 and ALH77307, \\citealt{PB10}\nand references therein). Oort cloud comets, with long periods and\nlarge distances from the Sun, have inferred crystallinity fractions up\nto 60 -- 80 \\% (e.g. Hale-Bopp, \\citealt{WO99,WO07}). Jupiter-family,\nor short period comets, have lower fractions, up to $\\sim$35 \\%\n(e.g. 9P\/Tempel 1, \\citealt{HA07}; 81P\/Wild 2, \\citealt{ZO06}). This\ndiscrepancy in fractions points to the existence of a radial\ndependence in crystallinity fraction in the protoplanetary disk around\nthe young Sun \\citep{HA05}. It is important to note that those values\nare model dependent, and the use of large amorphous grains (10 -- 100\n$\\mu$m) can lead to systematically lower crystalline fractions\n\\citep{HA02}. This is evident for Hale-Bopp, where \\citet{MI05a} find\na much lower fraction ($\\sim$7.5 \\%) than other authors, using a\ndistribution of amorphous grain sizes up to 100 $\\mu$m. What is clear\nis that even within the discrepancies, the crystallinity fractions\nderived for Solar System bodies are appreciably higher than those\nderived for the ISM dust ($< 2 \\%$, \\citealt{KE04}). Recent {\\it\n  Spitzer} data indicate further similarities between crystalline\nsilicate features seen in comets or asteroids with those seen in some\ndebris disks around solar mass stars \\citep{BE06,LI07,LI08}. One\nproposed explanation is that the observed spectral features in the\ndisk result from the catastrophic break-up of a single large body (a\n`super comet') which creates the small dust particles needed for\ndetection. At the even earlier protoplanetary disk stage, there is\nlimited observational evidence for radial gradients in crystallinity\nfrom mid-infrared interferometry data, with higher crystallinity\nfractions found closer to the young stars \\citep{VB04,SC08}. All of\nthis suggests that the crystallization occurs early in the disk\nevolution and is then incorporated into larger solid bodies.\n\nBesides dust composition, the evolution of grain sizes is an essential\nindicator of disk evolution. The initially sub-$\\mu$m size ISM grains\nmust grow astounding 14--15 orders of magnitude in diameter if they\nare to form planets. If grains were to grow orderly and steadily,\ntheoretical calculations predict disks to have fully dissipated their\nsmall grains within $\\sim$10$^5$ years \\citep{WE80,DD05}. The fact\nthat many disks a few Myr old are observed to have small grains\n\\citep{HE08} poses a serious problem for the paradigm that grain\ngrowth is a steady, monotonic process in disk evolution and planet\nformation. Additionally, small dust has been observed in the surface\nlayers of disks in clusters of different ages and environments for\nhundreds of systems. The implications, as discussed most recently by\n\\citet{OL10} and \\citet{OF10}, is that small grains must be\nreplenished by fragmentation of bigger grains, and that an equilibrium\nbetween grain growth and fragmentation is established. \\citet{OL10}\nhave shown that this equilibrium is maintained over a few million\nyears, as long as the disks are optically thick, and is independent of\nthe population or environment studied.\n\nIn this paper we present a comprehensive study of the mineralogical\ncomposition of disks around stars in young star-forming regions (where\nmost stars are still surrounded by optically thick disks) and older\nclusters (where the majority of disks has already dissipated).\nCorrelating the results on mean size and composition of dust grains\nper region, obtained in a homogeneous way using the same methodology,\nwith the properties of small bodies in our own Solar System can put\nconstraints on some of the processes responsible for disk evolution\nand planet formation. The Serpens Molecular Cloud, whose complete\nflux-limited YSO population has been observed by the IRS instrument\n\\citep{OL10}, is used as a prototype of a young star-forming region,\ntogether with Taurus, the best studied region to date. The sources\nthat have retained their protoplanetary disks in the $\\eta$\nChamaeleontis and Upper Scorpius clusters are used to probe the\nmineralogy in the older bin of disk evolution.\n\nSection~\\ref{sdata} describes the YSO samples in the 4 regions\nmentioned. The {\\it Spitzer} IRS observations and reduction are\nexplained. The spectral decomposition method B2C \\citep{OF10} is\nbriefly introduced in \\S~\\ref{s_spdecomp}, and its results for\nindividual and mean cluster grain sizes and composition are shown in\n\\S~\\ref{sres}. In \\S~\\ref{sdis} the results are discussed in the\ncontext of time evolution. There we demonstrate that no evolution is\nseen in either mean grain sizes or crystallinity fractions as clusters\nevolve from $\\sim$1 to 8 Myr. The implications for disk formation and\ndissipation, and planet formation are discussed. In \\S~\\ref{scon} we\npresent our conclusions.\n\n\n\\section{Spitzer IRS data}\n\\label{sdata}\n\n\nThe four regions presented here were chosen due to the availability of\ncomplete sets of IRS spectra of their IR-excess sources, while\nspanning a wide range of stellar characteristics, environment, mean\nages and disk fractions (the disk fraction of Serpens is still\nunknown, see Table \\ref{t_overview}).\n\nThe IRS spectra of a complete flux-limited sample of young stellar\nobjects (YSO) in the Serpens Molecular Cloud have been presented by\n\\citet{OL10}, based on program ID \\#30223 (PI: Pontoppidan). As\ndetailed there, the spectra were extracted from the basic calibration\ndata (BCD) using the reduction pipeline from the Spitzer Legacy\nProgram ``From Molecular Cores to Planet-Forming Disks'' (c2d,\n\\citealt{LH06}). A similarly large YSO sample in the Taurus\nstar-forming region has been presented by \\citet{FU06}. IRS spectra of\nall 18 members of the $\\eta$ Chamaeleontis cluster were first shown by\n\\citet{SI09}, while the spectra of 26 out of the 35 IR-excess sources\nin the Upper Scorpius OB association were shown by \\citet{DC09} (the\nremaining 9 objects were not known at the time the observations were\nproposed). For the latter 3 regions, the post-BCD data were downloaded\nfrom the SSC pipeline (version S18.4) and then extracted with the\nSpitzer IRS Custom Extraction software (SPICE, version 2.3) using the\nbatch generic template for point sources. As a test, the IRS spectra\nof the YSOs in Serpens were also reduced using SPICE to ensure that\nboth pipelines produce nearly identical results. On visual inspection,\nno discrepancies were found between the results from the two\npipelines, all objects showed the exact same features in both\nspectra. The similarity in outputs is such that the effects on the\nspectral decomposition results are within the cited error bars.\n\nSince the spectral decomposition method applied here aims to reproduce\nthe silicate emission from dust particles in circumstellar disks, the\nsample has been limited to spectra that show clear silicate features.\nThe few sources with PAH emission have been excluded from the\nsample. PAH sources amount to less than 8\\% in low-mass star-forming\nregions \\citep{GE06,OL10}. Furthermore, spectra with very low\nsignal-to-noise ratios (S\/N) are excluded from the analysed sample in\norder to guarantee the quality of the results. In addition, for\nobjects \\#114 and 137 in Serpens, and 04370+2559 and V955Tau in Taurus\nthe warm component fit contributes to most of the spectrum, leaving\nvery low fluxes to be fitted by the cold component. This produces\nlarge uncertainties in the cold component fit, and they are therefore\nnot further used in the analysis. The low S\/N objects rejected amount\nto less than 10~\\% of each of the Serpens and Taurus samples, so the\nstatistical results derived here should not be affected by this\nremoval. The final sample of 139 sources analysed is composed of 60\nobjects in Serpens, 66 in Taurus, 9 objects in Upper Scorpius, and 4\nin $\\eta$ Chamaeleontis. The statistical uncertainties of the spectra\nwere estimated as explained in \\citet{OF09}.\n\nThe great majority of the objects studied here are low-mass stars\n(spectral types K and M, see Table \\ref{t_comp}). The study of\nmineralogical evolution across stellar mass is not the focus of this\npaper. Such a study would require a separate paper, in which the same\ntechniques are used for low- and intermediate-mass stars. Thus, the\nstatistical results derived in the following sections concern T Tauri\nstars, and not necessarily apply to intermediate-mass Herbig Ae\/Be\nstars.\n\n\n\\section{Spectral Decomposition and the B2C method}\n\\label{s_spdecomp}\n\n\nIn order to reproduce the observed IRS spectra of these circumstellar\ndisks the B2C decomposition method, explained in detail and tested\nextensively in \\citet{OF10}, is applied. Two dust grain populations,\nor components, at different temperatures (warm and cold) are used in\nthe method, in addition to a continuum emission. The warm component\nreproduces the 10 $\\mu$m feature, while the cold component reproduces\nthe non-negligible residuals at longer wavelengths, over the full\nspectral range (see Figure \\ref{f_fit}). Each component, warm and\ncold, is the combination of five different dust species and three\ngrain sizes for amorphous silicates or two grain sizes for crystalline\nsilicates.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{f1.ps}\n\\end{center}\n\\caption{\\label{f_fit} Example of the B2C modeling for object \\#15 in\n  Serpens. The black line is the estimated continuum for this\n  source. The red line is the fit to the warm component and the blue\n  line is the fit to the cold component. The green line is the final\n  fit to the entire spectrum. The original spectrum is shown in black\n  with its uncertainties in light grey.}\n\\end{figure}\n\nThe three amorphous species are silicates of olivine stoichiometry\n(MgFeSiO$_4$), silicates of pyroxene stoichiometry (MgFeSiO$_6$), and\nsilica (SiO$_2$). The two crystalline species are both Mg-rich end\nmembers of the pyroxene and olivine groups, enstatite (MgSiO$_3$) and\nforsterite (Mg$_2$SiO$_4$). As further explained in \\citet{OF10}, the\ntheoretical opacities of the amorphous species are computed assuming\nhomogeneous spheres (Mie theory), while those for the crystalline\nspecies use the distribution of hollow spheres (DHS, \\citealt{MI05})\ntheory so that irregularly shaped particles can be simulated.\n\nIn addition, the three grain sizes used are 0.1, 1.5 and 6.0 $\\mu$m,\nrepresenting well the spectroscopic behaviour of very small,\nintermediate-sized and large grains. For the crystalline species,\nhowever, the code is limited to only 2 grain sizes (0.1 and 1.5\n$\\mu$m). This restriction is imposed because large crystalline grains\nare highly degenerate with large amorphous grains (as can be seen in\nFigure 1 of \\citealt{OF10}), and because the production of large 6.0\n$\\mu$m pure crystals is not expected via thermal annealing\n\\citep{GA04}.\n\nThe B2C method itself consists of three steps. First, the continuum is\nestimated and subtracted from the observed spectrum. The adopted\ncontinuum is built by using a power-law plus a black-body at\ntemperature $T_{\\rm cont}$. The power-law represents the mid-IR tail\nof emission from the star and inner disk rim. The black-body is\ndesigned to contribute at longer wavelengths, and is therefore\nconstrained to be less than 150 K. Each dust component is then fitted\nseparately to the continuum-subtracted spectrum.\n\nThe second step is to fit the warm component to reproduce the 10\n$\\mu$m silicate feature between $\\sim$7.5 and 13.5 $\\mu$m. This is\ndone by summing up the 13 mass absorption coefficients ($N_{\\rm\n  species}$ = 5, $N_{\\rm sizes}$ = 3 or 2, for amorphous and\ncrystalline species, respectively), multiplied by a black-body\n$B_{\\nu} (T_{\\rm w})$ at a given warm temperature $T_{\\rm w}$.\n\nThe third step is to fit the residuals, mostly at longer wavelengths,\nover the entire spectral range (5 -- 35 $\\mu$m). This is done in a\nsimilar manner, for a given cold temperature $T_{\\rm c}$. The final\nfit is a sum of the three fits described, as can be seen in Figure\n\\ref{f_fit}. The entire fitting process is based on a Bayesian\nanalysis, combined with a Monte Carlo Markov chain, in order to\nrandomly explore the space of free parameters. The resulting mean\nmass-average grain size is the sum of all sizes fitted, each size\nbeing weighted by their corresponding masses, as:\n\n\\begin{tiny}\n\\begin{eqnarray}\n  \\langle a_{\\rm warm\/cold} \\rangle = \n  \\left(\\sum_{j=1}^{\\mathrm{N_{\\rm sizes}}}  a_{j} \n    \\sum_{i=1}^{\\mathrm{N_{\\rm species}}}  \n    M_{{\\rm w\/c},i}^{j} \\right)\n  \\times \\left( \\sum_{j=1}^{\\mathrm{N_{\\rm sizes}}} \n    \\sum_{i=1}^{\\mathrm{N_{\\rm species}}}  \n    M_{{\\rm w\/c},i}^{j} \\right)^{-1}\n  \\label{eq:meana}\n\\end{eqnarray}\n\\end{tiny}\nwhere $a_1 = 0.1\\,\\mu$m (small grains), $a_2 = 1.5\\,\\mu$m\n(intermediate-sized grains) and $a_3 = 6\\,\\mu$m (large grains).\nFurther details and tests of the B2C procedure can be found in\n\\citet{OF10}. That paper also demonstrates that the procedure is\nrobust for statistical samples, and that the relative comparisons\nbetween samples, which are the focus of this paper, should not suffer\nfrom the assumptions that enter in the procedure. The robustness of\nthe procedure is evaluated by fitting synthetic spectra, and is\ndiscussed in detail in their Appendix A. The influence of the\ncontinuum estimate is also discussed, especially for the cold\ncomponent for both grain sizes and crystallinity fractions, and it is\nshown that prescriptions that do not use large 6 $\\mu$m grains (which\nare, to some degree, degenerate with the continuum) give fits that are\nnot so good.\n\nFor the amorphous grains, the B2C procedure uses the Mie scattering\ntheory to compute mass absorption coefficients. However, \\citet{MI07}\nfound that they could best reproduce the extinction profile toward the\ngalactic center using the DHS scattering theory, with a maximum\nfilling factor of 0.7. The most striking difference between Mie and\nDHS mass absorption coefficients is seen for the O--Si--O bending mode\naround 20 $\\mu$m. Here we investigate the influence of the use of DHS\ninstead of Mie for amorphous grain with an olivine or pyroxene\nstoichiometry. We conducted tests on a sub-sample of 30 objects (15 in\nSerpens and 15 in Taurus). The conclusion of such tests is that it has\na small influence on the quantities we discuss in this study. For the\nwarm component of the 30 objects, we find a change in the mean\ncrystallinity fraction of -1.6\\% (the mean crystallinity for this\nsub-sample using DHS is 9.3\\% versus 10.9\\% using Mie), which is in\nthe range of uncertainties claimed in this study. We also computed the\nmean slope of grain size distributions to gauge the effect of using\nDHS on grain sizes. On average, the grain size distribution indices\nare steeper by $\\sim$0.2 (with a mean slope of -3.01 for this\nsub-sample using DHS versus -2.80 using Mie). Therefore, our main\nconclusions are preserved for the warm component. Concerning the cold\ncomponent, the inferred crystallinity fraction using DHS is 22.5\\%\nversus 15.1\\% with Mie, a mean increase of 7.4\\%. For the mean slope\nof grain size distributions, a negligible decrease is found (-3.07\nusing DHS versus -3.01 for Mie). Again, the differences found are\nwithin our significant errors for the cold component and do not change\nany of our conclusions.\n\nIt is important to note that the S\/N generally degrades at longer\nwavelengths when compared to shorter wavelengths. The lower S\/N\nreflect on the cold component fits and will most likely result in\nlarger uncertainties. We evaluate that the fits to the cold component\nare reliable and add important information on the dust mineralogy\n(albeit with larger uncertainties) and thus those results are included\nin the following discussion.\n\n\n\\section{Results}\n\\label{sres}\n\n\nThe IRS spectra of the 139 YSOs with IR excess discussed in\n\\S~\\ref{sdata} were fitted with the B2C spectral decomposition\nprocedure. The relative abundances derived for all objects are shown\nin Appendix \\ref{sabun}. The S\/N drops considerably for the long\nwavelength module of some of the objects studied (including all\nobjects in Upper Scorpius and $\\eta$ Chamaeleontis). For this reason,\nthe cold component could not be satisfactorily fitted and no results\nfor this component are presented for these sources (see Appendix\n\\ref{sabun}).\n\nDue to the large number of objects, these results allow statistical\nstudies on both the mineralogy and size distribution of the grains\nthat compose the optically thin surface layers of disks in each\ncluster studied. The mean abundances of each species per region are\npresented in Table \\ref{t_mean_comp}, where it can be seen that the\nmajority of the dust studied is of amorphous form. In Table\n\\ref{t_mean} the mean mass-average grain sizes and crystallinity\nfractions per region are shown. Mean sizes are in the range 1 -- 3\n$\\mu$m, without significant difference between regions. These results\nare discussed in detail in the following sections.\n\nIt is important to note that the comparison of results derived here\nfor the different regions is valid because the same method, with exact\nsame species, is used for all sources. The comparison of samples\nanalyzed in distinct ways can lead to differences in results that do\nnot correspond to real differences in composition. Nevertheless, in\n\\S~\\ref{scomp} the results presented here are compared to literature\nresults for the same objects, when available, with generally good\nagreement.\n\n\n\\subsection{Grain Sizes}\n\\label{s_gsize}\n\n\nThe mean mass-averaged grain sizes for the warm ($\\langle a_{\\rm warm}\n\\rangle$) and cold ($\\langle a_{\\rm cold} \\rangle$) components are\nshown in Figure \\ref{f_size}, for Serpens and Taurus (for the objects\nin Upper Sco and $\\eta$ Cha no results for the cold component are not\navailable, see Appendix \\ref{sabun}). It is seen that the two clouds\noverlap greatly, and that the grain sizes derived from the different\ntemperature components do not seem to correlate. To quantify this\ncorrelation, a Kendall $\\tau$ correlation coefficient can be computed\ntogether with its associated probability $P$ (between 0 and 1). $\\tau$\n= 1(-1) defines a perfect correlation (anti-correlation), and $\\tau$ =\n0 means that the datasets are completely independent. A small $P$, on\nthe other hand, testifies to how tight the correlation is. For the\nwarm and cold mean mass-averaged grain sizes for both clouds, $\\tau$\nis found to be 0.14, with $P$ = 0.07. This lack of correlation\nindicates that different processes are likely responsible for\nregulating the size distribution at different radii \\citep{OF10}.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{f2.ps}\n\\end{center}\n\\caption{\\label{f_size} Mass-averaged mean grain sizes for the warm\n  ($\\langle a_{\\rm warm} \\rangle$) and cold ($\\langle a_{\\rm cold}\n  \\rangle$) components. Black dots are the objects in Serpens, and red\n  squares are the objects in Taurus.}\n\\end{figure}\n\nAlthough the average grain size in the warm component is bigger than\nthat in the cold component within a given star-forming region, as\nshown in Table \\ref{t_mean}, this difference is mostly not\nsignificant. However, Figures \\ref{f_size} and \\ref{f_sizedist}\nclearly show a difference between the range of grain sizes spanned in\nboth components, with $\\langle a_{\\rm cold} \\rangle$ never reaching\nnear the biggest grain size modeled (6.0 $\\mu$m) for any object. A\npossible explanation for larger grains at smaller radii, suggested by\n\\citet{ST09}, is that grains coagulate faster in the inner disk where\ndynamical timescales are shorter.  However as discussed by\n\\citet{OL10} and in \\S~\\ref{sover}, the mean dust size at the disk\nsurface is not regulated by grain growth alone, but also by\nfragmentation and vertical mix. This means that faster coagulation at\nsmaller radii cannot be uniquely responsible for bigger grains in the\ninner disk. Future modeling should try to understand this difference\nin mean grain sizes observed.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{f3.ps}\n\\end{center}\n\\caption{\\label{f_sizedist} Distribution of mass-averaged mean grain\n  sizes for the warm ($\\langle a_{\\rm warm} \\rangle$, left panel) and\n  cold ($\\langle a_{\\rm cold} \\rangle$, right panel) components. Due\n  to the low number statistics, the objects in Upper Sco and $\\eta$\n  Cha have been merged together as an older cluster.}\n\\end{figure}\n\nFurthermore, Serpens and Taurus occupy an indistinguishable locus in\nFigure \\ref{f_size}, explicitly seen in Figure \\ref{f_sizedist}. A two\nsample Kolmogorov-Smirnov test (KS-test) was performed and the results\nshow that the null hypothesis that the two distributions come from the\nsame parent population cannot be rejected to any significance\n(14\\%). The older regions, although lacking statistical significance,\nshow a distribution of mass-average grain sizes in the same range\nprobed by the young star-forming regions (Figure\n\\ref{f_sizedist}). This supports the evidence that the size\ndistribution of the dust in the surface layers of disks is\nstatistically the same independent of the population studied\n\\citep{OL10}. \n\nThe results here confirm those from \\citet{OF10} that the mean\ndifferential grain size distributions slope for the three grain sizes\nconsidered are shallower than the reference MRN differential size\ndistribution ($\\alpha$ = -3.5). The mean grain size distributions\nslopes ($\\alpha$) for each region can be found in Table \\ref{t_mean}.\n\n\n\\subsection{Disk Geometry}\n\\label{s_dgeo}\n\n\nThe amount of IR-excess in a disk is directly related to its geometry\n\\citep{KE87,GM01,DU01}. Specifically using the IRS spectra, disk\ngeometry can be inferred from the flux ratio between 30 and 13 $\\mu$m\n($F_{30}\/F_{13}$, \\citealt{JB07,OL10,ME10}). A flared geometry ($1.5\n\\lesssim F_{30}\/F_{13} \\lesssim 5$), with considerable IR excess and\nsmall dust, allows the uppermost dust layers to intercept stellar\nlight at both the inner and outer disk. For flat disks ($F_{30}\/F_{13}\n\\lesssim 1.5$) with little IR excess, only the inner disk can easily\nintercept the stellar radiation as the outer disk is\nshadowed. Moreover, cold or transitional disks are interesting objects\nthat present inner dust gaps or holes, producing a region with little\nor no near-IR excess ($5 \\lesssim F_{30}\/F_{13} \\lesssim 15$). It is\ninteresting to explore the effect of disk geometry on both the mean\nmass-average grain sizes and crystallinity fractions of the disks\nstudied.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{f4.ps}\n\\end{center}\n\\caption{\\label{f_f30} Top: Flaring index $F_{30}\/F_{13}$, used as a\n  proxy for disk geometry, versus warm (left) and cold (right)\n  mass-averaged mean grain sizes. Bottom: $F_{30}\/F_{13}$ versus warm\n  (left) and cold (right) crystallinity fractions. The YSOs in Serpens\n  (black dots), Taurus (red squares), Upper Sco (blue stars), and\n  $\\eta$ Cha (green triangles) are compared. }\n\\end{figure}\n\nFigure \\ref{f_f30} shows $F_{30}\/F_{13}$ as a proxy for disk geometry\ncompared with the mean mass-averaged grain sizes and crystallinity\nfractions for both components and all regions studied here. No\npreferential grain size (correlation coefficient $\\tau$ = -0.14, $P$ =\n0.02, and $\\tau$ = 0.07, $P$ = 0.33 for warm and cold components,\nrespectively) nor crystallinity fraction ($\\tau$ = 0.09, $P$ = 0.10\nfor the warm, and $\\tau$ = -0.19, $P$ = 0.01 for the cold component)\nis apparent for any given disk geometry. Similar scatter plots result\nfor the mean mass-average grains sizes for only amorphous ($\\tau$ =\n-0.12, $P$ = 0.08 for the warm, and $\\tau$ = 0.13, $P$ = 0.11 for the\ncold component), or only crystalline grains ($\\tau$ = 0.08, $P$ = 0.17\nfor the warm, and $\\tau$ = -0.13, $P$ = 0.10 for the cold\ncomponent). Furthermore, no clear separation is seen between the\ndifferent regions studied. The statistically relevant samples in\nSerpens and Taurus define a locus where the majority of the objects is\nlocated in each plot, which is followed by the lower number statistics\nfor older regions. Figure \\ref{f_f30} therefore shows not only that\ngrain size and crystallinity fraction are not a function of disk\ngeometry, but also that younger and older regions show similar\ndistributions of those two parameters.\n\n\n\\subsection{Crystallinity Fraction}\n\\label{s_cryst}\n\n\nThe crystallinity fractions derived from the warm and cold components\n($C_{\\rm Warm}$ and $C_{\\rm Cold}$, respectively) for Serpens and\nTaurus are show in Figure \\ref{f_cryst}. No strong trend of warm and\ncold crystallinity fractions increasing together is seen ($\\tau$ =\n0.10, with $P$ = 0.10 for the entire sample). This fact implies that,\nif an unique process is responsible for the crystallization of dust at\nall radii, this process is not occurring at the same rate in the\ninnermost regions as further out in the disk. This is opposite to the\nconclusion of \\citet{WA09}, who derive a correlation between inner and\nouter disk crystallinity from the simultaneous presence of the 11.3\nand 33 $\\mu$m features. The opacities of the crystalline species are\nmore complex than those two features alone, making the analysis here\nmore complete than that of \\citet{WA09}. Our finding that the fraction\nof crystalline material in disk surfaces varies with radius can\nconstrain some of the mechanisms for formation and distribution of\ncrystals.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{f5.ps}\n\\end{center}\n\\caption{\\label{f_cryst} Crystalline fraction of the warm and cold\n  components in Serpens (black dots) and Taurus (red squares).}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{f6.ps}\n\\end{center}\n\\caption{\\label{f_warmdist} Distribution of crystalline fractions for\n  Serpens (top), Taurus (middle), and Upper Sco and $\\eta$ Cha\n  combined (bottom). Similar distributions and the same range of\n  fractions are seen for all clusters.}\n\\end{figure}\n\nA wider spread in crystallinity fraction is observed for the cold\ncomponent than for the warm component (Figure \\ref{f_warmdist}), which\nis reflected in the mean crystallinity fractions for each sample\n(Table \\ref{t_mean}). This discrepancy could be real, or an artifact\ndue to the signal-to-noise ratio (S\/N) being frequently lower at\nlonger wavelengths (cold component) than that at shorter wavelengths\n(warm component), introducing a larger scatter. The difference in\nSerpens is more significant ($\\langle C_{\\rm warm} \\rangle \\simeq$\n11.0\\% and $\\langle C_{\\rm cold} \\rangle \\simeq$ 17.5\\%). The left\npanel of Figure \\ref{f_cum} shows the cumulative fractions as\nfunctions of crystallinity fractions. Despite small differences\nbetween the warm (red line) and cold (blue line) components, the two\ncumulative fractions have similar behavior. If this difference is\ntrue, there is a small fraction of T Tauri disks with a higher cold\n(outer) than warm (inner) crystallinity fraction. This finding\ncontrasts with that derived by \\citet{VB04} for the disks around 3\nHerbig stars. Their spatially resolved observations infer higher\ncrystallinity fractions in the inner than in the outer disks, albeit\nbased on only 10 $\\mu$m data. A larger sample of objects with good S\/N\nincluding both 10 and 20 $\\mu$m data is needed to better constrain\nthis point. In addition, Figure \\ref{f_warmdist} shows that younger\nand older clusters have similar distributions of crystallinity\nfractions.\n\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{f7.ps}\n\\end{center}\n\\caption{\\label{f_cum} Left: Cumulative fractions of the crystallinity\n  fractions, for Serpens (dashed line) and Taurus (dotted\n  line). Right: Cumulative fraction of the ration between the\n  forsterite and enstatite fractions, for Serpens (dashed line) and\n  Taurus (dotted line). The warm component is shown in red while the\n  cold component is blue.}\n\\end{figure}\n\nAs discussed by many authors, both the grain size and the degree of\ncrystallinity affect the silicate features, therefore it is\ninteresting to search for trends between these two parameters. In\nFigure \\ref{f_warm}, the mass-average grain sizes are compared to the\ncrystallinity fraction for both warm (left panel) and cold (right\npanel) components. No obvious trends are seen in either component,\nneither any separation between regions. This result supports the\ndiscussion of \\citet{OF10} that whatever processes govern the mean\ngrain size and the crystallinity in disks, they are independent from\neach other.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{f8.ps}\n\\end{center}\n\\caption{\\label{f_warm} Mass-averaged mean grain sizes versus the\n  crystalline fraction for Serpens (black dots), Taurus (red squares),\n  Upper Sco (blue stars), and $\\eta$ Cha (green triangles).}\n\\end{figure}\n\n\n\\subsubsection{Enstatite vs. Forsterite}\n\\label{scomp2}\n\n\nThe disk models of \\citet{GA04} consider chemical equilibrium of a\nmixture of solid and gas at high temperatures, allowing radial mixing\nof material. These models predict a predominance of forsterite in the\ninnermost regions of the disk, while enstatite dominates at lower\ntemperatures (being converted from forsterite). From the observational\npoint of view, data on disks around T Tauri \\citep{BO08} and Herbig\nAe\/Be stars \\citep{JU10} have shown the opposite trend: enstatite is\nmore concentrated in the inner disk, while forsterite dominates the\ncolder, outer disk region. \\citet{BO08} interpret this result as a\nradial dependence of the species formation mechanisms, or a\nnon-equilibrium of the conditions under which the species formed,\ncontrary to the models assumptions.\n\nFor the regions presented in this study, it can be seen in Table\n\\ref{t_mean_comp} for mean cluster values and in Table \\ref{t_comp}\nfor individual objects that the results derived from this study\ngenerally follow those of \\citet{BO08}, with more enstatite in the\nwarm component and, to a lesser extent, more forsterite in the cold\ncomponent. The right panel of Figure \\ref{f_cum} illustrates this for\nthe cumulative fraction of the forsterite over enstatite ratios for\nindividuals disks. However, this trend is not very significant given\nthe uncertainties.\n\n\n\\subsection{The Silicate Strength-Shape Relation}\n\\label{ssil}\n\n\nA correlation between the shape and the strength of the 10 $\\mu$m\nsilicate feature from disks has been discussed extensively in the\nliterature \\citep{VB03,KE06,OF09,PA09,OL10,OF10}. Synthetic 10 $\\mu$m\nfeatures generated for different grain sizes and compositions have\nbeen shown to fit well with observations, yielding grain size as the\nimportant parameter responsible for such a relationship. The degree of\ncrystallinity of the dust also plays a role on the shape of this\nfeature. However, as clearly shown for EX Lup \\citep{AB09}, and\nsupported by models \\citep{MI08,OF09}, an increase in crystallinity\nfraction does not change the strength of the feature, even though its\nshape does change. Crystallinity is then understood as responsible for\nthe scatter in the strength-shape relationship, and not the\nrelationship itself. As a result, the strength and shape of the 10\n$\\mu$m silicate feature yield the typical size of the grains in the\nupper layers of the disk at a few AU from the star \\citep{KE07}. The\ntop panel of Figure \\ref{f_s10} shows the results for Serpens, Taurus,\nUpper Sco and $\\eta$ Cha. The bottom panel presents the median values\nper region, indicating the 15 -- 85 percentile ranges of the\ndistributions. Overlaid are the models of \\citet{OF09} for different\ngrain sizes (0.1 -- 6.0 $\\mu$m) generated for amorphous silicates of\nolivine and pyroxene stoichiometry, and a 50:50 mixture. The\ndifference in mean ages does not correspond to a significant\ndifference in mean grain sizes between the different regions.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth]{f9.ps}\n\\end{center}\n\\caption{\\label{f_s10} Top: The ratio of normalized fluxes at 11.3 to\n  9.8 $\\mu$m ($S_{11.3}\/S_{9.8}$) is plotted against the peak at 10\n  $\\mu$m ($S^{10\\mu{\\rm m}}_{{\\rm peak}}$) for Serpens (black dots),\n  Taurus (red squares), Upper Scorpius (blue stars), $\\eta$\n  Chamaeleontis (green triangles). Bottom: Squares show the median\n  values and crosses indicate the 15 -- 85 percentile ranges of the\n  distributions (top panel). Colored curves are derived from\n  theoretical opacities for different mixtures by \\citet{OF09}. The\n  open circles correspond to different grain sizes, from left to right\n  6.25, 5.2, 4.3, 3.25, 2.7, 2.0, 1.5, 1.25, 1.0 and 0.1 $\\mu$m. }\n\\end{figure}\n\nWith the mean grain sizes derived from the spectral decomposition, it\nis possible to further explore the validity of using the strength of\nthe 10 $\\mu$m silicate feature to trace the sizes of grains in the\nsurfaces of disks. The left panel of Figure \\ref{f_s10_2} shows the\ncorrelation between $\\langle a_{\\rm warm} \\rangle$ and $S^{10\\mu{\\rm\n    m}}_{{\\rm peak}}$ for all 4 samples. The Kendall $\\tau$\ncoefficient of -0.29, $P$ = 0.01 supports the effectiveness of\n$S^{10\\mu{\\rm m}}_{{\\rm peak}}$ as a proxy for grain sizes, with\nsmaller values of $S^{10\\mu{\\rm m}}_{{\\rm peak}}$ implying larger\ngrain sizes.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{f10.ps}\n\\end{center}\n\\caption{\\label{f_s10_2} Left panel: Strength of the 10 $\\mu$m silicate\n  feature ($S^{10\\mu{\\rm m}}_{{\\rm peak}}$) versus the mass-averaged\n  mean grain size for the warm component. Right panel: Strength of the\n  10 $\\mu$m silicate feature versus crystalline fraction for the warm\n  component. The best fit relationships are shown for reference. }\n\\end{figure}\n\nOn the other hand, it is also possible to test how the degree of\ncrystallinity can influence the strength of the 10 $\\mu$m silicate\nfeature. The lack of correlation between $C_{\\rm warm}$ and\n$S^{10\\mu{\\rm m}}_{{\\rm peak}}$ ($\\tau$ = -0.07, $P$ = 0.14), shown in\nthe right panel of Figure \\ref{f_s10_2} for all samples, supports that\nthe degree of crystallinity is not the dominant parameter setting the\nstrength of the 10 $\\mu$m silicate feature. These results argue\nagainst the results of \\citet{ST09} that find a high crystallinity\nfraction and small grains fitting low strengths of the 10 $\\mu$m\nsilicate feature. Although it may be possible to fit a few spectra\nwith a certain prescription, a good model should be able to explain\nthe robust relationship between the strength and shape of the 10\n$\\mu$m silicate feature observed for large numbers of disks. Despite\nthe many processes able to change the shape or the strength of this\nfeature, only grain size has so far demonstrated capability to explain\nthe observed trend. Our conclusion is that $S^{10\\mu{\\rm m}}_{{\\rm\n    peak}}$ and dust sizes are appropriately correlated.\n\n\n\\subsection{Comparison with other studies}\n\\label{scomp}\n\n\nDust composition results are available in the literature for the disks\nin Taurus and $\\eta$ Cha (see Table \\ref{t_lit} for an overview).\n\\citet{SI09} present their analysis in $\\eta$ Cha considering the same\n5 dust species and three grain sizes (enstatite in their model is the\nonly species for which only the 2 smaller grain sizes are considered),\nbut for a distribution of temperatures derived using the Two Layer\nTemperature Distribution (TLTD, \\citealt{JU09}) decomposition\nprocedure. For the same 4 objects, their mean amorphous fraction is\n80.1 $\\pm$ 9.3 \\%. This result is consistent with the 82.8 $\\pm$ 12.9\n\\% mean amorphous fraction found here. The mean crystalline fractions\nderived are 18.4 $\\pm$ 10.7 \\% with TLTD and 17.1 $\\pm$ 12.8 \\%\nderived here.\n\n\\citet{ST09} present their decomposition procedure for 65 YSOs in\nTaurus. This method also takes into consideration a warm and a cold\ntemperature, and makes use of two amorphous species (olivine and\npyroxene) with two grain sizes (small and large), and 3 crystalline\nspecies (enstatite, forsterite and crystalline silica) of a single\nsize. Their mean warm amorphous fraction is 82.9 $\\pm$ 19.3 \\% and\nwarm crystalline fraction is $17.1 \\pm 19.3$ \\%, while here the\nderived fractions are 89.0 $\\pm$ 6.6 \\% and 10.9 $\\pm$ 6.6 \\% for the\nwarm amorphous and crystalline fractions, respectively. For the cold\ncomponent, \\citet{ST09} derive a mean cold amorphous fraction of 77.3\n$\\pm$ 19.9 \\% and cold crystalline fraction of 22.6 $\\pm$ 19.9 \\%,\nwhile here the values are 85.9 $\\pm$ 10.6 \\% and 13.9 $\\pm$ 10.5 \\%,\nrespectively. The consistently lower amorphous (higher crystalline)\nfractions found by \\citet{ST09} could be a result of their choice to\nuse silica in crystalline rather than amorphous form (as used\nhere).\n\n\n\\section{Discussion}\n\\label{sdis}\n\n\\subsection{Dust Characteristics}\n\\label{sover} \n\nSection \\ref{sres} has shown that the disk populations in the four\nregions presented here, young and older, have very similar\ndistributions in the two main dust parameters: grain size and\ncomposition. The large number of objects in the two young regions\nstudied occupy a region in parameter space of either grain size or\ncrystallinity fraction that is also populated by the small number of\nolder disks. The grain sizes derived for the cold component never\nreach the biggest grain size modeled (6 $\\mu$m), different from the\nwarm component results that span the entire range in sizes. The\ncrystallinity fraction does not seem to be correlated with mean grain\nsize, warm or cold. Whatever processes are responsible for the\ncrystallization of the initially amorphous grains, they should not\nonly be independent from the processes that govern the grain size\ndistribution, but they should also be able to work on bigger amorphous\ngrains. Alternatively, the crystalline lattice should be able to keep\nitself regular during the coagulation of small crystalline dust to\ncreate big crystalline grains.\nThe correlation between the strength of the 10 $\\mu$m feature and the\nmean grain size in disk surfaces, combined with the lack of\ncorrelation between crystallinity fraction and $S^{10\\mu{\\rm m}}_{{\\rm\n    peak}}$, supports the wide usage of $S^{10\\mu{\\rm m}}_{{\\rm\n    peak}}$ as a proxy for dust size in literature\n\\citep{VB03,KE06,PA09}.\n\n\\citet{BO08} found a strong correlation between disk geometry and the\nstrength of the 10 $\\mu$m silicate feature for a very small sample of\nT Tauri stars (7 disks), which points to flatter disks having\nshallower 10 $\\mu$m features (i.e., big grains in the disk\nsurface). Using results from similar decomposition procedures,\n\\citet{OF10} and \\citet{JU10} confirm this trend for larger samples of\nT Tauri (58 disks) and Herbig Ae\/Be stars (45 disks),\nrespectively. Those trends are much weaker than that found by\n\\citet{BO08}, showing a larger spread. For the current even larger\nsample (139 disks), no significant trend is seen, indicating that the\nearlier small sample trends may have been affected by a few\noutliers. This result is similar to that found by \\citet{OL10} for a\nlarge YSO sample ($\\sim$~200 objects) using the strength of the 10\n$\\mu$m silicate feature as a proxy for grain size (Figure 14 in that\npaper). As discussed by Oliveira et al., the sedimentation models of\n\\citet{DD08} expect a strong correlation of larger grains in flatter\ndisks that is not seen. This means that sedimentation alone cannot be\nresponsible for the distribution of mean grain sizes in the upper\nlayers of protoplanetary disks around T Tauri stars. Furthermore, the\nlack of correlation between crystallinity fraction and disk geometry\nis not in support of the results of \\citet{WA09} and \\citet{ST09}, who\nfind a link between increasing crystallinity fraction and dust\nsedimentation.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{f11.ps}\n\\end{center}\n\\caption{\\label{f_grain} Left: Mean mass-average grain sizes vs. disk\n  fraction. Serpens is not included because its disk fraction is not\n  yet known. Filled circles represent mean warm grain sizes, and open\n  triangles represent mean cold grain sizes. Error bars for the mean\n  mass-average grain sizes are estimated using a Monte Carlo approach,\n  sampling the errors of the individual objects. Right: Mean grain\n  sizes vs. mean cluster age. Filled circles represent results for the\n  warm component, while open triangles represent the cold\n  component. The black points are YSOs in Serpens, red in Taurus, blue\n  in Upper Sco and green in $\\eta$ Cha. }\n\\end{figure}\n\nAs discussed in \\citet{OL10} for Serpens and Taurus, and confirmed by\nthe addition of considerably older samples, there is no clear\ndifference in the mean grain sizes in the disk surfaces with mean\ncluster age, which can be seen in Figure \\ref{f_grain}. This evidence\nsupports the discussion in that paper that the dust population\nobserved in the disk surface cannot be a result of a progressive,\nmonotonic change of state from small amorphous grains, to large, more\ncrystalline grains, or `grain growth and processing'. The fact that\nthe distribution of grain sizes in the upper layers of disks does not\nchange with cluster age implies that an equilibrium of the processes\nof dust growth and fragmentation must exist, which also supports the\nexistence of small grains in disks that are millions of years old whereas\ndust growth is a rapid process \\citep{WE80,DD05}. That small dust is\nstill seen in disks in older regions like Upper Sco and $\\eta$ Cha\nargues that this equilibrium of processes is maintained for millions\nof years, as long as the disks are optically thick, but independent of\nthem having a flared or flatter geometry.\n\n\n\\subsection{Evolution of Crystallinity with Time?}\n\\label{sevol}\n\n\nLiterature studies of disk fractions of different YSO clusters with\ndifferent mean ages show a trend of decreasing disk fraction,\ni.e. disks dissipating with time, over some few millions of years\n\\citep{HA01,HE08}. This decrease is clearly confirmed by the lower\nfraction of disks still present in the older regions studied here\n(Upper Sco and $\\eta$ Cha). According to current planet formation\ntheories, if giant planets are to be formed from gas rich disks, the\noptically thin, gas-poor disks in those older regions should already\nharbor (proto-)planets. Considering the evidence from small bodies in\nour own Solar System that suggest considerably higher crystallinity\nfractions than ISM dust (see \\citealt{WO07} and \\citealt{PB10} for\nreviews of latest results), a crystallinity increase must occur.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{f12.ps}\n\\end{center}\n\\caption{\\label{f_comp2} Left: Crystallinity fraction vs. disk\n  fraction. Serpens is not included because its disk fraction is not\n  yet known. Filled circles represent mean warm crystallinity, and\n  open triangles represent mean cold crystallinity. Uncertainty for\n  crystallinity fractions are estimated using a Monte Carlo approach,\n  sampling the errors of the individual objects. Right: Crystallinity\n  fraction vs. mean cluster age. Filled circles represent mean warm\n  crystallinity, and open triangles represent mean cold\n  crystallinity. The black points are YSOs in Serpens, red in Taurus,\n  blue in Upper Sco and green in $\\eta$ Cha. }\n\\end{figure}\n\nIn Figure \\ref{f_comp2}, the mean crystallinity fraction per region is\nplotted against two evolutionary parameters: disk fraction (left) and\nmean age (right). Within the spread in individual fractions it is seen\nthat, just as for grain sizes, there is no strong evidence of an\nincrease of crystallinity fraction with either evolutionary parameter.\nThis implies that there is no evolution in grain sizes or\ncrystallinity fraction for the dust in the surface of disks over\ncluster ages in the range 1 -- 8 Myr, as probed by the observations\npresented here. Essentially, there is no change in these two\nparameters until the disks disperse. Starting from the assumption that\ninitially the dust in protoplanetary disks is of ISM origin\n(sub-$\\mu$m in size and almost completely amorphous), it appears that\na modest level of crystallinity is established in the disk surface\nearly in the evolution ($\\leq$ 1 Myr) and then reaches some sort of\nsteady state, irrespective of what is taking place in the disk\nmidplane. Thus, the dust in the upper layers of disks does not seem to\nbe a good tracer of the evolution that is taking place in the disk\ninterior, where dust is growing further for the formation of\nplanetesimals and planets, at many times higher crystallinity\nfractions, to be consistent with evidence from Solar system bodies.\n\nIf this is the case, within 1 Myr this surface dust must be\ncrystallized to the observed fraction ($\\sim$10 -- 20~\\%). This result\nputs constraints on the formation of circumstellar disks. One\npossibility is that this crystallization of the dust in disks mostly\noccurs during the embedded phase. In this early stage of star\nformation, where large quantities of material are still accreting\ntowards the protostar, a fraction of the infalling material comes very\nclose to the protostar and is heated to temperatures $>$800 K before\nit moves outwards in the disk. Alternatively, accretion shocks or\nepisodic heating events could be responsible for thermally annealing\nthe dust in the disk surface.\n\nThe 2-D models of \\citet{VD10} treat the radial evolution of crystals\nin time. According to these models, 100~\\% of the dust in the inner\ndisk ($\\leq$ 1 AU) is crystallized within 1 Myr. With time, the inner\ndisk crystalline fraction drops as the disk spreads, and crystalline\nmaterial is transported to outer parts of the disk. These models can\nhelp explain the rapid crystallization required to account for our\nresults. However, the models do show a decrease in inner disk ($\\leq$\n1 AU) crystallinity fraction with time, which is not supported by our\nresults. Since these models do not discriminate on vertical structure,\nbut rather present crystallinity fractions that are integrated over\nall heights at a given radius, this decrease in crystallinity fraction\nis not necessarily connected to the surface of the disk. Thus the\ndecrease in crystallinity fraction with time found in the models of\n\\citet{VD10} could be explained as a decrease in crystallinity\nfraction just in the disk midplane where the bulk of the mass resides,\nbut not in the surface layers, as our data indicate. That would imply\nthat radial mixing of these crystals is more efficient than vertical\nmixing, which is responsible for the crystallinity fraction decrease\nin the disk midplane.\n\nAccording to the models of \\citet{CI07} for outward transport of high\ntemperature materials, variations in radial transport dynamics with\nheight produce vertical gradients in the crystalline fractions, such\nthat the upper layers of the disk will have lower crystallinity\nfractions than the midplane population. If that is the case, the\nobservations discussed here, which probe the disk surface only, lead\nto lower limits on the real crystalline fraction of disk midplanes. In\nthis scenario, planets (and comets) forming in the disk midplane would\nhave higher crystalline abundances than those derived here for the\ndisk surfaces, which are compatible with what has been observed in our\nSolar System. However, this model does not make predictions for the\ntime evolution of the systems. Combining the vertical and radial\nmixing processes with evolutionary models such as those of\n\\citet{VD10} are needed to investigate whether older and younger disks\ncould still show the same distribution of crystallinity fractions in\nthe upper layers of disks, as observed here.\n\n\n\\section{Conclusions}\n\\label{scon}\n\n\nThis paper presents the spectral decomposition of Spitzer\/IRS spectra\nusing the B2C decomposition model of \\citet{OF10}. Mineralogical\ncompositions and size distributions of dust grains in the surface\nlayers of protoplanetary disks are derived for 139 YSOs belonging to\nfour young star clusters using the same method.\n\nSerpens and Taurus are used as prototypes of young regions, where most\nstars are still surrounded by disks, while Upper Sco and $\\eta$ Cha\nrepresent the older bin of disk evolution, where a large fraction of\nthe disks have already dissipated but some massive protoplanetary\ndisks are left. The large number of objects analyzed allows\nstatistical results that point to the main processes that affect the\ngrain size distribution and composition of dust in protoplanetary\ndisks. Furthermore, the usage of the same analysis method for regions\nof different mean ages allow a study of evolution of the dust\nparameters with time.\n\nOur large sample does not show a preferential grain size or\ncrystallinity fraction with disk geometry, contrary to earlier\nanalyses based on smaller samples. Also, younger and older regions\nhave very similar distributions. The difference between mean\nmass-averaged grain sizes for the warm and cold components of a given\nstar-forming region is small, however a considerable difference is\nseen between the ranges of grain sizes spanned in both components. The\ncold mass-averaged grain sizes never reach the biggest size modelled\n(6 $\\mu$m) while the warm mass-averaged grain sizes span the entire\nrange of sizes modeled. The crystallinity fractions derived for inner\n(warm) and outer (cold) disks are typically 10 -- 20\\%{}, and not\ncorrelated. The cold crystallinity fraction shows a larger spread than\nthe warm. No strong difference is seen between the overall mean warm\nand cold crystallinity fraction. Within the crystalline dust\npopulation, more enstatite is found in the warm component and more\nforsterite in the cold component. The differences are not very\nsignificant, however.\n\nThe results of the spectral decomposition support the usage of the\nstrength of the 10 $\\mu$m silicate feature ($S^{10\\mu{\\rm m}}_{{\\rm\n    peak}}$) as a proxy of the mean grain size of dust in the disk\nsurface. This is supported by the correlation between $S^{10\\mu{\\rm\n    m}}_{{\\rm peak}}$ and mean grain size and lack of correlation with\nthe mean crystallinity.\n\nMean cluster ages and disk fractions are used as indicators of the\nevolutionary stage of the different populations. Our results show that\nthe different regions have similar distributions of mean grain sizes\nand crystallinity fractions regardless of the spread in mean ages of 1\n-- 8 Myr. Thus, despite the fact that the majority of disks dissipate\nwithin a few Myr, the surface dust properties do not depend on age for\nthose disks that have not yet dissipated in the 1 -- 8 Myr range. This\npoints to a rapid change in the composition and crystallinity of the\ndust in the early stages ($\\leq$ 1 Myr) that is maintained essentially\nuntil the disks dissipate.\n\n\n\\acknowledgements Astrochemistry at Leiden is supported by a Spinoza\ngrant from the Netherlands Organization for Scientific Research (NWO)\nand by the Netherlands Research School for Astronomy (NOVA)\ngrants. This work is based on observations made with the Spitzer Space\nTelescope, which is operated by the Jet Propulsion Laboratory,\nCalifornia Institute of Technology under a contract with NASA. Support\nfor this work was provided by NASA through an award issued by\nJPL\/Caltech.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nEver since L. Gross proved that a logarithmic Sobolev inequality is equivalent to hypercontractivity of the associated semigroup (see \\cite{G}), these inequalities have been the subject of much research and interest.  They have proved extremely useful as a tool in the control of the rate of convergence to equilibrium of spin systems, and were extensively studied (see for example \\cite{B-H1},\\cite{G-Z}, \\cite{Led}, \\cite{O-R}, \\cite{S-Z1},\\cite{Yo2}, \\cite{Ze}).  Up until recently, however, most of the attention has been focused on the case of elliptic generators, for which there are some very powerful methods for proving such inequalities.  Our aim here is to show that a certain class of infinite dimensional measures corresponding to non-elliptic H\\\"ormander type generators satisfy logarithmic Sobolev inequalities.\n\nOne method that exists for proving coercive inequalities such as the logarithmic Sobolev inequality and the spectral gap inequality, as well as gradient bounds (which are closely related) involves showing that the so-called $CD(\\rho, \\infty)$ condition holds (see \\cite{A-B-C-F-G-M-R-S}, \\cite{Bakry}).  Indeed, let $L$ be the generator of a Markov semigroup $P_t$, and define the operators\n\n\\begin{align*}\n\\Gamma(f,f) &= \\frac{1}{2}\\left(L(f^2) - 2fLf\\right)\\\\\n\\Gamma_2(f,f) &= \\frac{1}{2}\\left[L\\Gamma(f,f) - 2\\Gamma(f, Lf)\\right].\n\\end{align*}\nWe say that the $CD(\\rho, \\infty)$ holds when there exists $\\rho\\in\\mathbb{R}$ such that\n\\[\n\\Gamma_2(f,f) \\geq \\rho \\Gamma(f,f).\n\\]\nWhen $L$ is elliptic such a condition holds in many situations.  In the case when $M$ is a complete connected Riemannian manifold, and $\\nabla$ and $\\Delta$ are the standard Riemannian gradient and Laplace-Beltrami operators, taking $L=\\Delta$, the condition reads\n\\[\n|\\nabla\\nabla f|^2 + \\mathrm{Ric}(\\nabla f, \\nabla f) \\geq \\rho|\\nabla f|^2.\n\\]\nThis holds for some $\\rho\\in\\mathbb{R}$ when $M$ is compact, or for $\\rho =0$ when $M=\\mathbb{R}^n$ with the usual metric, since $\\mathrm{Ricci}=0$.\n\nHowever, in this paper we will consider non-elliptic H\\\"ormander generators. For such generators these methods do not work, since the $CD(\\rho, \\infty)$ condition does not hold.  Indeed, the Ricci tensor of our generators can be thought of as being $-\\infty$ almost everywhere.  \n\nWe consider an $N$-dimensional lattice and impose interactions between points in the lattice described by an unbounded quadratic potential.  In the standard case where the underlying space is Euclidean, the $CD(\\rho, \\infty)$ condition allows us to prove that the finite dimensional measures on the lattice, which depend on the boundary conditions, satisfy logarithmic Sobolev inequalities uniformly on the boundary conditions.  It is then possible to pass to the infinite dimensional measure.  We aim for a comparable result in a more complicated sub-Riemannian setting, using different methods.\n\nIn \\cite{L-Z} a similar situation is studied, in that the authors consider a system of H\\\"ormander generators in infinite dimensions and prove logarithmic Sobolev inequalities as well as some ergodicity results.  The main difference between the present set up and their situation is that we consider a non-compact underlying space, namely the Heisenberg group, in which the techniques of \\cite{L-Z} cannot be applied.\n\n\n\n\n\\section{Logarithmic Sobolev inequalities on the Heisenberg group}\n\\label{Logarithmic Sobolev inequalities on the Heisenberg group}\nWe consider the Heisenberg group, $\\mathbb{H}$, which can be described as $\\mathbb{R}^3$ with the following group operation:\n$$x\\cdot\\tilde{x} = (x_1, x_2, x_3)\\cdot(\\tilde{x}_1, \\tilde{x}_2, \\tilde{x}_3) = (x_1 + \\tilde{x}_1, x_2 + \\tilde{x}_2, x_3 + \\tilde{x}_3 + \\frac{1}{2}(x_1\\tilde{x}_2 - x_2\\tilde{x}_1)).$$\n$\\mathbb{H}$ is a Lie group, and its Lie algebra $\\mathfrak{h}$ can be identified with the space of left invariant vector fields on $\\mathbb{H}$ in the standard way.  By direct computation we see that this space is spanned by \n\\begin{eqnarray}\nX_1 &=& \\partial_{x_1} - \\frac{1}{2}x_2\\partial_{x_3} \\nonumber \\\\\nX_2 &=& \\partial_{x_2} + \\frac{1}{2}x_1\\partial_{x_3} \\nonumber \\\\\nX_3 &=& \\partial_{x_3} = [X_1, X_2]. \\nonumber\n\\end{eqnarray}\nFrom this it is clear that $X_1, X_2$ satisfy the H\\\"ormander condition (i.e. $X_1, X_2$ and their commutator $[X_1, X_2]$ span the tangent space at every point of $\\mathbb{H}$).  It is also easy to check that the left invariant Haar measure (which is also the right invariant measure since the group is nilpotent) is the Lebesgue measure $dx$ on $\\mathbb{R}^3$. \n\n$\\mathbb{H}$ is naturally equipped with a 1-parameter family of automorphisms $\\{\\delta_\\lambda\\}_{\\lambda>0}$ defined by\n\\[\n\\delta_\\lambda(x) := \\left(\\lambda x_1, \\lambda x_2, \\lambda^2 x_3\\right).\n\\]\n$\\{\\delta_\\lambda\\}_{\\lambda>0}$ is called a family of \\textit{dilations}.  Thus $\\mathbb{H}$ is an example of a homogeneous Carnot group (see \\cite{B-L-U} for an extensive study of such groups).\n \nOn $C^\\infty_0(\\mathbb{H})$, define the \\textit{sub-gradient} to be the operator given by\n$$\\nabla := (X_1,X_2)$$ \nand the \\textit{sub-Laplacian} to be the second order operator given by \n$$\\Delta := X_1^2 + X_2^2.$$\n$\\nabla$ can be treated as a closed operator from $L^2(\\mathbb{H}, dx)$ to $L^2(\\mathbb{H}; \\mathbb{R}^2, dx)$.  Similarly, since $\\Delta$ is densely defined and symmetric in $L^2(\\mathbb{H}, dx)$, we may treat $\\Delta$ as a closed self-adjoint operator on $L^2(\\mathbb{H}, dx)$ by taking the Friedrich extension.  \n\nWe introduce the logarithmic Sobolev inequality on $\\mathbb{H}$ in the following way.\n\n\\begin{definition}\nLet $q\\in(1,2]$, and let $\\mu$ be a probability measure on $\\mathbb{H}$.  $\\mu$ is said to satisfy a \\textit{$q$-logarithmic Sobolev inequality} $(LS_q)$ on $\\mathbb{H}$ if there exists a constant $c>0$ such that for all smooth functions $f: \\mathbb{H}\\to \\mathbb{R}$\n\\begin{equation}\n\\label{LSq}\n\\mu\\left(|f|^q\\log\\frac{|f|^q}{\\mu|f|^q}\\right) \\leq c\\mu\\left(|\\nabla f|^q\\right)\n\\end{equation}\nwhere $\\nabla$ is the sub-gradient on $\\mathbb{H}$.\n\\end{definition}\n\n\\begin{rem}\nThe $(LS_q)$ was introduced in \\cite{B-L} and further studied in \\cite{B-Z}, as a variation of the more standard $(LS_2)$ inequality.  Here it is noted that for $q<2$, the $(LS_q)$ inequality serves as a certain sharpening of $(LS_2)$, at least when the underlying space is finite dimensional.\n\\end{rem}\n\n\\begin{rem}\n\\label{tensorisation}\nWe recall four important standard properties of $(LS_q)$ inequalities that will be used below (see \\cite{B-Z} and \\cite{G-Z}):\n\\begin{itemize}\n\\item[\\rm{(i)}] {\\rm$(LS_q)$ is stable under tensorisation:} Suppose $\\mu_1$ and $\\mu_2$ satisfy $(LS_q)$ inequalities with constants $c_1$ and $c_2$ respectively.  Then $\\mu_1\\otimes\\mu_2$ satisfies an $(LS_q)$ inequality with constant $\\max\\{c_1, c_2\\}$.\n\\item[\\rm{(ii)}] {\\rm$(LS_q)$ is stable under bounded perturbations:} Suppose $d\\mu = \\frac{e^{-U}}{Z}dx$ satisfies an $(LS_q)$, and that $W$ is bounded.  Then $\\tilde\\mu(dx) = \\frac{e^{-U - W}}{\\tilde Z}dx$ satisfies an $(LS_q)$ inequality.\n\\item[\\rm{(iii)}] {\\rm$(LS_q) \\Rightarrow (SG_q)$:} Suppose $\\mu$ satisfies an $(LS_q)$ inequality with constant $c$.  Then $\\mu$ satisfies a $q$-spectral gap inequality (we say $\\mu$ satisfies an $(SG_q)$ inequality) with constant $\\frac{4c}{\\log2}$ i.e. \n\\[\n\\mu\\left|f - \\mu f\\right|^q \\leq \\frac{4c}{\\log2} \\mu\\left(|\\nabla f|^q\\right)\n\\]\nfor all smooth $f$.\n\\item[\\rm{(iv)}] When the underlying space is finite dimensional, $(LS_q) \\Rightarrow (LS_{q'})$ and $(SG_q) \\Rightarrow (SG_{q'})$ for $q<q'$.\n\\end{itemize}\n\\end{rem} \n\nThe operator $\\Delta$ is non-elliptic, but by H\\\"ormander's theorem it is hypoelliptic, so that the associated heat semigroup $P_t$ has a smooth convolution kernel with respect to the Haar measure.  Note that we can calculate the $\\Gamma$ and $\\Gamma_2$ functions for this generator explicitly, and we can easily see that there does not exist a constant $\\rho\\in\\mathbb{R}$ such that $\\Gamma_2\\geq\\rho\\Gamma$.  However, despite this, in \\cite{B-B-B-C} and \\cite{Li} it was recently shown that there exists a constant $C$ such that\n\\[\n|\\nabla P_t f|(x) \\leq CP_t(|\\nabla f|)(x), \\qquad \\forall x\\in \\mathbb{H},\n\\]\nwhich is a surprising result.  It follows directly from this gradient bound that the heat kernel measure on $\\mathbb{H}$ satisfies an $(LS_2)$ inequality on $\\mathbb{H}$.\n\n\\begin{rem}\nSince we have the sub-gradient on the right-hand side, \\eqref{LSq} is a logarithmic Sobolev inequality corresponding to a H\\\"ormander type generator.  Indeed,  if $\\mu(dx) = \\frac{e^{-U}}{Z}dx$ then it is clear that $\\mathcal{L} = \\Delta - \\nabla U.\\nabla$ is a Dirichlet operator satisfying\n\\[\n\\mu\\left(f\\mathcal{L}f\\right) = - \\mu\\left(|\\nabla f|^2\\right),\n\\]\nwhere $\\Delta$ is the sub-Laplacian, and $\\nabla$ the sub-gradient.\n\\end{rem}\n\nIndependently, and by very different methods, in \\cite{H-Z} the authors were able to show that a related class of measures on $\\mathbb{H}$ satisfy $(LS_q)$ inequalities (see Theorem \\ref{hz} below).  To describe these we first need to introduce the natural distance function on $\\mathbb{H}$, which is the so-called \\textit{Carnot-Carath\\'eodory distance}.  This distance is more natural than the usual Euclidean one, since it takes into account the extra structure that the Heisenberg group posseses.  \n\nWe define the Carnot-Carath\\'eodory distance between two points in $\\mathbb{H}$ by considering only \\textit{admissible} curves between them.  A Lipschitz curve $\\gamma:[0,1]\\to\\mathbb{H}$ is said to be \\textit{admissible} if $\\gamma'(s) = a_1(s)X_1(\\gamma(s)) + a_2(s)X_2(\\gamma(s))$ almost everywhere with measurable coefficients $a_1, a_2$ i.e. if $\\gamma'(s) \\in sp\\{X_1(\\gamma(s)),X_2(\\gamma(s))\\}$ a.e.  Then the length of $\\gamma$ is given by\n$$l(\\gamma) = \\int_0^1\\left(a_1^2(s) + a_2^2(s)\\right)^{1\/2}ds$$\nand we define the Carnot-Carath\\'eodory distance between two points $x,y\\in\\mathbb{H}$ to be \n$$d(x,y) := \\inf\\{l(\\gamma):\\gamma\\ \\textrm{is an admissible path joining}\\ x\\ \\textrm{to}\\ y\\}.$$\nWrite $d(x)=d(x,e)$, where $e$ is the identity.\n\n\n\\begin{rem}\nThis distance function is well defined as a result of Chow's theorem, which states that every two points in $\\mathbb{H}$ can be joined by an admissible curve (see for example \\cite{B-L-U},\\cite{Grom}).\n\\end{rem}\n \nIt is well know that $d$ is a homogeneous norm on $\\mathbb{H}$ i.e. $d(\\delta_\\lambda(x)) = \\lambda d(x)$ for all $\\lambda>0$, where $\\delta_\\lambda$ is the dilation as defined above (see for example \\cite{B-L-U}).\n \nGeodesics are smooth, and are helices in $\\mathbb{R}^3$.  They have an explicit parameterisation.  For details see \\cite{B-B-B-C}, \\cite{Beals}, \\cite{Bel}, \\cite{Monti}.\nWe also have that $x=(x_1, x_2, x_3)\\mapsto d(x)$ is smooth for $(x_1, x_2) \\neq 0$, but not at points $(0,0, x_3)$, so that the unit ball has singularities on the $x_3$-axis (one can think of it as being 'apple' shaped).\n\nIn our analysis, we will frequently use the following two results.  The first is the well-known fact that the Carnot-Carath\\'eodory distance satisfies the eikonal equation (see for example \\cite{Monti}):\n\n\\begin{proposition}\n\\label{eik}\nLet $\\nabla$ be the sub-gradient on $\\mathbb{H}$.  Then $|\\nabla d(x)|=1$ for all $x=(x_1, x_2, x_3)\\in\\mathbb{H}$ such that $(x_1, x_2)\\neq0$.\n\\end{proposition}\n\nWe must be careful in dealing with the notion of $\\Delta d$, since it will have singularities on the $x_3$-axis.  However, the following (proved in \\cite{H-Z}) provides some control of these singularities.\n\n\\begin{proposition}\n\\label{sub-laplacian lem}\nLet $\\Delta$ be the sub-Laplacian on $\\mathbb{H}$. There exists a constant $K$ such that $\\Delta d \\leq \\frac{K}{d}$ in the sense of distributions.\n\\end{proposition}\n\n\\begin{proof}\nFor the sake of completeness, we recall part of the proof given in \\cite{H-Z}.  It suffices to show that $\\Delta d \\leq K$ on $\\{d(x) =1\\}$.  Indeed, using dilations and homogeneity, we have that \n\\[\n\\Delta d(x)=\\lambda\\Delta d(\\delta_\\lambda(x))\n\\]\nfor all $x\\neq0, \\lambda>0$, so that for any $x\\in\\mathbb{H}\\backslash\\{0\\}$\n\\begin{equation}\n\\label{laplacian estimate}\n\\Delta d(x) \\leq \\frac{1}{d(x)}\\sup_{\\{d(y) = 1\\}}\\Delta d(y).\n\\end{equation}\nSince everything is smooth away from the $x_3$ axis, in order to prove that  $\\Delta d \\leq K$ on $\\{d(x) =1\\}$, it suffices to look at what happens in a small neighbourhood of $(0,0, z)$, where $z$ is such that $d\\left((0,0,z)\\right)=1$.\nTo do this, let\n\\[\nA_\\eta := \\left\\{(r,s) \\in \\mathbb{R}^2: s>0, r>-\\eta s\\right\\},\n\\]\n and for $x = (x_1, x_2, x_3)\\in\\mathbb{H}$ write $\\|x\\| := \\left(x_1^2 + x_2^2\\right)^{1\/2}$.  Then it is shown that there exists $\\eta>0$ and a smooth function $\\psi(r,s)$ defined on $A_\\eta$ such that for $x=(x_1, x_2, x_3)\\in \\mathbb{H}$,\n\\[\nd(x) = \\psi(\\|x\\|, |x_3|),\n\\]\nand moreover that $\\partial_r\\psi<0$ when $r=0$.  One can then compute that\n\\begin{equation}\n\\label{sub-laplacian}\n\\Delta d(x) = \\frac{1}{\\|x\\|}\\partial_r\\psi(\\|x\\|, |x_3|) + \\partial_r^2\\varphi(\\|x\\|, |x_3|) + \\frac{\\|x\\|^2}{4}\\partial_{s}\\varphi(\\|x\\|, |x_3|).\n\\end{equation}\nFrom \\eqref{sub-laplacian} it follows that $\\Delta d$ is bounded from above in a small neighbourhood of $(0,0,z)$, since although the first term is unbounded, it is negative.\n\\end{proof}\n\nThe following result is also found in \\cite{H-Z}.\n\n\\begin{theorem}\n\\label{hz}\nLet $\\mu_p$ be the probability measure on $\\mathbb{H}$ given by \n\\[\n\\mu_p (dx) = \\frac{e^{-\\beta d^p(x)}}{\\int_\\mathbb{H} e^{-\\beta d^p(x)}dx}dx\n\\]\nwhere $p\\geq2$, $\\beta >0$, $dx$ is the Lebesgue measure on $\\mathbb{R}^3$ and $d(x)$ is the Carnot-Carath\\'edory distance.  Then $\\mu_p$ satisfies an $(LS_q)$ inequality, where $\\frac{1}{p}+\\frac{1}{q} = 1$.\n\\end{theorem}\n\nIn the remainder of this paper we use the methods contained in \\cite{H-Z}, together with an iterative procedure based on ideas contained in \\cite{G-Z}, \\cite{Led}, \\cite{Ze2} and \\cite{Ze} to prove an $(LS_q)$ inequality for a class of infinite dimensional measures on $(\\mathbb{H})^{\\mathbb{Z}^N}$. \n\n\n\n\n\n\n\n\n\n\n\\section{Infinite dimensional setting and main result}\n\\label{Infinite dimensional setting}\n\\paragraph{\\textit{The Lattice:}}\nLet $\\mathbb{Z}^N$ be the $N$-dimensional square lattice, for some fixed $N\\in\\mathbb{N}$.  We equip $\\mathbb{Z}^N$ with the $l^1$ lattice metric $dist(\\cdot,\\cdot)$, defined by\n\\[\ndist(i,j) := \\sum_{l=1}^N|i_l - j_l|\n\\]\nfor $i=(i_1,\\dots, i_N), j=(j_1, \\dots, j_N) \\in \\mathbb{Z}^N$.  For $i,j\\in\\mathbb{Z}^N$ we will also write \n\\[\ni\\sim j \\qquad \\Leftrightarrow \\qquad dist(i,j) = 1\n\\]\ni.e. $i\\sim j$ when $i$ and $j$ are nearest neighbours in the lattice.\n\n\nFor $\\Lambda \\subset \\mathbb{Z}^N$, we will write $|\\Lambda|$ for the cardinality of $\\Lambda$, and $\\Lambda \\subset \\subset \\mathbb{Z}^N$ when $|\\Lambda|<\\infty$.\n  \n\\paragraph{\\textit{The Configuration Space:}}\nLet $\\Omega = (\\mathbb{H})^{\\mathbb{Z}^N}$ be the \\textit{configuration space}.  We introduce the following notation.  Given $\\Lambda \\subset \\mathbb{Z}^N$ and $\\omega = (\\omega_i)_{i\\in\\mathbb{Z}^N}\\in\\Omega$, let $\\omega_\\Lambda := (\\omega_i)_{i\\in\\Lambda} \\in\\mathbb{H}^\\Lambda$ (so that $\\omega\\mapsto \\omega_\\Lambda$ is the natural projection of $\\Omega$ onto $\\mathbb{H}^\\Lambda$).  \n\n\nLet $f\\colon \\Omega \\to \\mathbb{R}$.  Then for $i\\in\\mathbb{Z}^N$ and $\\omega \\in \\Omega$ define $f_i(\\cdot|\\omega)\\colon\\mathbb{H} \\to \\mathbb{R}$ by\n$$f_i(x|\\omega) := f(x\\bullet_i\\omega)$$\nwhere the configuration $x\\bullet_i\\omega\\in \\Omega$ is defined by declaring its $i$th coordinate to be equal to $x\\in\\mathbb{H}$ and all the other coordinates coinciding with those of $\\omega\\in\\Omega$.  Let $C^{(n)}(\\Omega)$, $n\\in\\mathbb{N}$ denote the set of all functions $f$ for which we have $f_i(\\cdot|\\omega) \\in C^{(n)}(\\mathbb{H})$ for all $i\\in\\mathbb{Z}^d$ .  For $i\\in\\mathbb{Z}^N, k\\in\\{1,2\\}$ and $f\\in C^{(1)}(\\Omega)$, define\n$$X_{i,k}f(\\omega) := X_kf_i(x|\\omega)\\vert_{x = \\omega_i},$$\nwhere $X_1, X_2$ are the left invariant vector fields on $\\mathbb{H}$ defined in section \\ref{Logarithmic Sobolev inequalities on the Heisenberg group}.\n\n\nDefine similarly $\\nabla_if(\\omega) := \\nabla f_i(x|\\omega)\\vert_{x = \\omega_i}$ and $\\Delta_if(\\omega) := \\Delta f_i(x|\\omega)\\vert_{x = \\omega_i}$ for suitable $f$, where $\\nabla$ and $\\Delta$ are the sub-gradient and the sub-Laplacian on $\\mathbb{H}$ respectively.  For $\\Lambda \\subset \\mathbb{Z}^N$, set $\\nabla_\\Lambda f = (\\nabla_if)_{i\\in\\Lambda}$ and\n$$|\\nabla_\\Lambda f|^q := \\sum_{i\\in\\Lambda}|\\nabla_if|^q.$$\nWe will write $\\nabla_{\\mathbb{Z^d}} = \\nabla$, since it will not cause any confusion.\n\nFinally, a function $f$ on $\\Omega$ is said to be \\textit{localised} in a set $\\Lambda\\subset\\mathbb{Z}^N$ if $f$ is only a function of those coordinates in $\\Lambda$.\n\n\\paragraph{\\textit{Local Specification and Gibbs Measure:}}\nLet $\\Phi = (\\phi_{\\{i,j\\}})_{\\{i,j\\}\\subset\\mathbb{Z}^N, i\\sim j}$ be a family of $C^2$ functions such that $\\phi_{\\{i,j\\}}$ is localised in $\\{i,j\\}$.  Assume that there exists an $M\\in(0,\\infty)$ such that $\\|\\phi_{\\{i,j\\}}\\|_\\infty \\leq M$ and $\\|\\nabla_i\\nabla_j\\phi_{\\{i,j\\}}\\|_\\infty\\leq M$ for all $i,j\\in\\mathbb{Z}^N$ such that $i\\sim j$.  We say $\\Phi$ is a bounded potential of range 1.  For $\\omega\\in\\Omega$, define\n$$H_\\Lambda^\\omega(x_\\Lambda) = \\sum_{\\substack{\\{i,j\\}\\cap\\Lambda\\neq\\emptyset \\\\ i\\sim j}}\\phi_{\\{i,j\\}}(x_i, x_j),$$\nfor $x_\\Lambda = (x_i)_{i\\in\\Lambda} \\in \\mathbb{H}^\\Lambda$, where the summation is taken over couples of nearest neighbours $i\\sim j$ in the lattice with at least one point in $\\Lambda$, and where $x_i = \\omega_i$ for $i\\not\\in \\Lambda$.\n\nNow let $(\\mathbb{E}^{\\omega}_\\Lambda)_{\\Lambda\\subset\\subset\\mathbb{Z}^N,\\omega\\in\\Omega}$ be the local specification defined by\n\\begin{equation}\n\\label{local spec}\n\\mathbb{E}^{\\omega}_\\Lambda(dx_\\Lambda)=\\frac{e^{-U^{\\omega}_\\Lambda(x_\\Lambda)}}{\\int e^{-U^{\\omega}_\\Lambda(x_\\Lambda)}dx_\\Lambda} dx_{\\Lambda}\\equiv\\frac{e^{-U^{\\omega}_\\Lambda(x_\\Lambda)}}{Z^\\omega_\\Lambda} dx_{\\Lambda}\n\\end{equation}\nwhere $dx_\\Lambda$ is the Lebesgue product measure on $\\mathbb{H}^\\Lambda$ and \n\\begin{equation}\n\\label{hamiltonian}\nU^{\\omega}_\\Lambda (x_\\Lambda) = \\alpha\\sum_{i\\in\\Lambda}d^p(x_i) + \\varepsilon\\sum_{\\substack{ \\{i,j\\}\\cap\\Lambda \\neq \\emptyset \\\\ i\\sim j}}(d(x_i) + \\rho d(x_j))^{2} + \\theta H_\\Lambda^\\omega(x_\\Lambda),\n\\end{equation}\nfor $\\alpha>0$, $\\varepsilon, \\rho, \\theta \\in\\mathbb{R}$, and $p\\geq2$, where as above  $x_i = \\omega_i$ for $i\\not\\in \\Lambda$.\n\n\\begin{rem}\n\\label{indices}\nIn the case when $p=2$, we must have that $\\varepsilon> -\\frac{\\alpha}{2N}$ to ensure that $\\int e^{-U_\\Lambda}dx_\\Lambda <\\infty$.\n\\end{rem}\n\nWe define an infinite volume \\textit{Gibbs measure} $\\nu$ on $\\Omega$ to be a solution of the (DLR) equation:\n\\[\n\\nu \\mathbb{E}^{\\cdot}_\\Lambda f = \\nu f\n\\]\nfor all bounded measurable functions $f$ on $\\Omega$.  $\\nu$ is a measure on $\\Omega$ which has $\\mathbb{E}^{\\omega}_\\Lambda$ as its finite volume conditional measures.\n\n~\n\nThe main result of this paper is the following:\n\n\\begin{theorem}\n\\label{main}\nLet $\\nu$ be a Gibbs measure corresponding to the local specification defined by \\eqref{local spec} and \\eqref{hamiltonian}.  Let $q$ be dual to $p$ i.e. $\\frac{1}{p}+\\frac{1}{q} =1$ and suppose $\\varepsilon\\rho>0$, with $\\varepsilon>-\\frac{\\alpha}{2N}$ if $p=2$. Then there exists $\\varepsilon_0, \\theta_0>0$ such that for $|\\varepsilon|<\\varepsilon_0$ and $|\\theta|<\\theta_0$, $\\nu$ is unique and satisfies an $(LS_q)$ inequality i.e. there exists a constant $C$ such that\n\\[\n\\nu\\left(|f|^q\\log\\frac{|f|^q}{\\nu |f|^q}\\right) \\leq C\\nu\\left(\\sum_i|\\nabla_if|^q\\right)\n\\]\nfor all $f$ for which the right-hand side is well defined.\n\\end{theorem}\n\nWe briefly mention some consequences of this result.  The first follows directly from Remark \\ref{tensorisation} part (iii).\n\n\\begin{corollary}\nLet $\\nu$ be as in Theorem \\ref{main}.  Then $\\nu$ satisfies the $q$-spectral gap inequality.  Indeed\n\\[\n\\nu\\left|f-\\nu f\\right|^q \\leq \\frac{4C}{\\log2}\\nu\\left(\\sum_i|\\nabla_if|^q\\right)\n\\]\nwhere $C$ is as in Theorem \\ref{main}.\n\\end{corollary}\n\nThe proofs of the next two can be found in \\cite{B-Z}.\n\n\\begin{corollary}\nLet $\\nu$ be as in Theorem \\ref{main} and suppose $f:\\Omega \\to \\mathbb{R}$ is such that $\\||\\nabla f|^q\\|_\\infty  < 1$. Then\n\\[\n\\nu\\left( e^{\\lambda f}\\right) \\leq \\exp\\left\\{\\lambda \\nu(f) +   \\frac{C}{q^q(q-1)}\\lambda^q\\right\\}\n\\]\nfor all $\\lambda>0$ where $C$ is as in Theorem \\ref{main}.  Moreover, by applying Chebyshev's inequality, and optimising over $\\lambda$, we arrive at the following `decay of tails' estimate\n\\[\n\\nu\\left\\{\\left|f - \\int fd\\nu\\right| \\geq h\\right\\} \\leq 2\\exp\\left\\{-\\frac{(q-1)^p}{C^{p-1}}h^p\\right\\}\n\\]\nfor all $h>0$, where $\\frac{1}{p} + \\frac{1}{q} =1$.\n\\end{corollary}\n\n\\begin{corollary}\nSuppose that our configuration space is actually finite dimensional, so that we replace $\\mathbb{Z}^N$ by some finite graph $G$, and $\\Omega = (\\mathbb{H})^G$.  Then Theorem \\ref{main} still holds, and implies that if $\\mathcal{L}$ is a Dirichlet operator satisfying\n\\[\n\\nu\\left(f\\mathcal{L} f\\right) = -\\nu\\left(|\\nabla f|^2\\right),\n\\]\nthen the associated semigroup $P_t = e^{t\\mathcal {L}}$ is ultracontractive.\n\\end{corollary}\n\n\\begin{rem}\nIn the above we are only considering interactions of range $1$, but we can easily extend our results to deal with the case where the interaction is of finite range $R$.\n\\end{rem}\n\n\\section{Results for the single site measure}\n\\label{results for single site measure}\nThe aim of this section is to show that the single site measures\n\\[\n\\mathbb{E}^{\\omega}_{\\{i\\}}(dx_i) =: \\mathbb{E}^{\\omega}_i(dx_i) = \\frac{e^{-U^{\\omega}_i(x_i)}}{Z^{\\omega}_i}dx_i, \\qquad i\\in\\mathbb{Z}^N,\n\\]\nsatisfy an $(LS_q)$ inequality uniformly on the boundary conditions $\\omega\\in\\Omega$.  We will often drop the $\\omega$ in the notation for convenience.  The work is strongly motivated by the methods of Hebisch and Zegarlinski described in \\cite{H-Z}.\n\\begin{theorem}\n\\label{uniform LSq}\nLet $\\frac{1}{q} +\\frac{1}{p}=1$, and $\\varepsilon\\rho>0$ with $\\varepsilon>-\\frac{\\alpha}{2N}$ if $p=2$.  Then there exists a constant $c$, independent of the boundary conditions $\\omega \\in \\Omega$ such that\n\\[\\mathbb{E}^{\\omega}_i\\left(|f|^q\\log\\frac{|f|^q}{\\mathbb{E}_i^\\omega|f|^q}\\right) \\leq c \\mathbb{E}^{\\omega}_i(|\\nabla_if|^q)\\]\nfor all smooth $f:\\Omega \\to \\mathbb{R}$.\n\\end{theorem}\n\nIt will be convenient to work with alternative measures to the ones defined above.  Indeed, if we can prove uniform $(LS_q)$ inequalities for the single site measures when $\\theta=0$ (so that we no longer have the the bounded interaction term in \\eqref{hamiltonian}), then by Remark \\ref{tensorisation} (ii), which states that $(LS_q)$ inequalities are stable under bounded perturbations, Theorem \\ref{uniform LSq} will hold.  Moreover, it is clear that\n\\[\n\\frac{e^{-\\alpha d^p(x_i) - \\varepsilon\\sum_{j:j\\sim i}(d(x_i) + \\rho d(\\omega_j))^{2}}}{\\int e^{-\\alpha d^p(x_i) - \\varepsilon\\sum_{j:j\\sim i}(d(x_i) + \\rho d(\\omega_j))^{2}}dx_i} = \\frac{e^{-\\tilde{U}^\\omega_i}}{\\int e^{-\\tilde{U}^\\omega_i}dx_i},\n\\]\nwhere \n$$\n\\tilde{U}_i(x_i) = \\alpha d^p(x_i) + 2N\\varepsilon d^{2}(x_i) + 2\\varepsilon\\rho d(x_i)\\sum_{j:j\\sim i}d(\\omega_j).\n$$\nIt is therefore sufficient to work with the measures defined by $\\tilde{\\mathbb{E}}_i^\\omega(dx_i) = (\\tilde{Z}_i^\\omega)^{-1}e^{-\\tilde{U}_i^\\omega}$, instead of $\\mathbb{E}_i^\\omega$.\n\nThe proof of the theorem will be in three steps.  We first prove the following inequality, designated a `$U$-bound' in \\cite{H-Z}.\n\n\n\\begin{lemma}\n\\label{lem q U-Bound}\nLet $\\frac{1}{p}+\\frac{1}{q} =1$ and suppose $\\varepsilon\\rho>0$, with $\\varepsilon>-\\frac{\\alpha}{2N}$ if $p=2$.  Then the following inequality holds:\n\\[\n\\tilde{\\mathbb{E}}_i^\\omega\\left(|f|^q\\left(d^{p}  + d \\sum_{j:j\\sim i}d(\\omega_j)\\right)\\right) \\leq A\\tilde{\\mathbb{E}}_i^\\omega|\\nabla_if|^q + B\\tilde\\mathbb{E}_i^\\omega |f|^q,\n\\]\nfor all smooth $f:\\Omega\\to\\mathbb{R}$, and some constants $A, B\\in(0,\\infty)$ independent of $\\omega$. \n\\end{lemma}\n\n\n\n\n\\begin{proof}\nWithout loss of generality assume $f\\geq0$.  By the Liebniz rule, we have\n\\begin{equation}\n\\label{leibniz}\n(\\nabla_i f)e^{-\\tilde{U}_i} = \\nabla_i(fe^{-\\tilde{U}_i}) + f\\nabla_i \\tilde{U}_ie^{-\\tilde{U}_i}. \n\\end{equation}\n\nTaking the inner product of both sides of this equation with $d(x_i)\\nabla_id(x_i)$ and integrating yields\n\\begin{align*}\n\\int_{\\mathbb{H}}fd\\nabla_id.\\nabla_i\\tilde{U}_ie^{-\\tilde{U}_i}dx_i &\\leq \\int_{\\mathbb{H}}d|\\nabla_id||\\nabla_i f|e^{-\\tilde{U}_i}dx_i - \\int_{\\mathbb{H}}d\\nabla_id.\\nabla_i\\left(fe^{-\\tilde{U}_i}\\right)dx_i \\\\\n&= \\int_{\\mathbb{H}}d|\\nabla_i f|e^{-\\tilde{U}_i}dx_i + \\int_{\\mathbb{H}}f\\nabla_i\\cdot(d\\nabla_id)e^{-\\tilde{U}_i}dx_i,\n\\end{align*}\nwhere we have used integration by parts and Proposition \\ref{eik}.  Now by Proposition \\ref{sub-laplacian lem}, we have\n\\[\n\\nabla_i\\cdot(d\\nabla_id) = |\\nabla_id|^2 + d\\Delta_id \\leq 1+K\n\\]\nin terms of distributions.  Therefore we have\n\\begin{align*}\n\\int_{\\mathbb{H}}fd\\nabla_id.\\nabla_i\\tilde{U}_ie^{-\\tilde{U}_i}dx_i &\\leq  \\int_{\\mathbb{H}}d|\\nabla_i f|e^{-\\tilde{U}_i}dx_i + (1+K)\\int_{\\mathbb{H}}fe^{-\\tilde{U}_i}dx_i.\n\\end{align*}\n\nReplacing $f$ by $f^q$ in this inequality, and using Young's inequality, we arrive at\n\\begin{align}\n\\label{U-bound one}\n\\int_{\\mathbb{H}}f^qd\\nabla_id.\\nabla_i\\tilde{U}_ie^{-\\tilde{U}_i}dx_i &\\leq \\frac{1}{\\tau}\\int_{\\mathbb{H}}|\\nabla_i f|^qe^{-\\tilde{U}_i}dx_i + \\frac{q}{p}\\tau^{p-1}\\int_{\\mathbb{H}}f^qd^pe^{-\\tilde{U}_i}dx_i \\nonumber \\\\\n& \\qquad+ (1+K)\\int_{\\mathbb{H}}f^qe^{-\\tilde{U}_i}dx_i,\n\\end{align}\nfor all $\\tau>0$.\n\nWe now calculate that\n\\begin{align*}\n\\nabla_i d(x_i).\\nabla_i\\tilde{U}_i(x_i) &= p\\alpha d^{p-1}(x_i) + 4N\\varepsilon d(x_i) + 2\\varepsilon\\rho \\sum_{j:j\\sim i}d(\\omega_j),\n\\end{align*}\nalmost everywhere, again using Proposition \\ref{eik}.\n\nFor $\\varepsilon\\rho>0$, we therefore have that there exist constants $a_1, b_1 \\in (0, \\infty)$ such that\n\\begin{equation}\n\\label{lower estimate1}\nd(x_i)\\nabla_i d(x_i).\\nabla_i\\tilde{U}_i(x_i) \\geq a_1\\left(d^{p}(x_i)  + d(x_i) \\sum_{j:j\\sim i}d(\\omega_j)\\right) - b_1.\n\\end{equation}\nThis is clear if $\\varepsilon>0$.  If $\\varepsilon<0$ and $p>2$ then we use the fact that for any $\\delta\\in(0,1)$ there exists a constant $C(\\delta)$ such that $d \\leq \\delta d^{p} + C(\\delta)$.  If $p=2$, recall from Remark \\ref{indices} that we must assume $\\varepsilon>-\\frac{\\alpha}{2N}$, and the assertion follows.\n\n\nUsing the estimate \\eqref{lower estimate1} in \\eqref{U-bound one} and taking $\\tau$ small enough, we see that there exist constants $A, B\\in(0, \\infty)$ independent of $\\omega$  such that\n\\begin{align*}\n&\\int f^q\\left(d^{p}(x_i)  +  d(x_i) \\sum_{j:j\\sim i}d(\\omega_j)\\right)e^{-\\tilde{U}_i}dx_i \\\\\n& \\qquad \\leq A\\int |\\nabla_if|^qe^{-\\tilde{U}_i}dx_i + B\\int f^qe^{-\\tilde{U}_i}dx_i,\n\\end{align*}\nwhich proves the lemma.\n\\end{proof}\n\n\nThe second step is to use this to prove that $\\tilde\\mathbb{E}_{i}^{\\omega}$ satisfies a $q$-spectral gap inequality uniformly on the boundary conditions $\\omega$.\n\n\n\\begin{lemma}\n\\label{qSG}\nLet $\\frac{1}{p} +\\frac{1}{q}=1$ and suppose $\\varepsilon\\rho>0$, with $\\varepsilon>-\\frac{\\alpha}{2N}$ if $p=2$.  Then $\\tilde{\\mathbb{E}}_i^\\omega$ satisfies the q-spectral gap inequality uniformly on the boundary conditions i.e. there exists a constant $c_0\\in(0,\\infty)$ independent of $\\omega$ such that\n\\[\\tilde\\mathbb{E}^{\\omega}_i|f-\\tilde\\mathbb{E}^{\\omega}_if|^q \\leq c_0 \\tilde\\mathbb{E}^{\\omega}_i|\\nabla_i f|^q\\]\nfor all smooth $f:\\Omega\\to\\mathbb{R}$.\n\\end{lemma}\n\n\n\n\n\\begin{proof}\nFirst note that\n\\begin{equation}\n\\label{note}\n\\tilde{\\mathbb{E}}_i|f-\\tilde{\\mathbb{E}}_if|^q \\leq 2^{q} \\tilde{\\mathbb{E}}_i|f-m|^q\n\\end{equation}\nfor any $m\\in\\mathbb{R}$.\n\nLet $\\mathcal{W}_i^\\omega(x_i) := d^{p}(x_i)  +  d(x_i) \\sum_{j:j\\sim i}d(\\omega_j)$.   Then, for any $L\\in(0,\\infty)$, we have\n\\begin{align*}\n\\tilde{\\mathbb{E}}_i|f-m|^q & = \\tilde{\\mathbb{E}}_i|f - m|^q\\mathbf{1}_{\\{\\mathcal{W}_i \\leq L\\}} + \\tilde{\\mathbb{E}}_i|f - m|^q\\mathbf{1}_{\\{\\mathcal{W}_i \\geq L\\}}\n\\end{align*}\nwhere $\\mathbf{1}_{\\{\\mathcal{W}_i \\leq L\\}}$ is the indicator function of the set $A^\\omega(L) := \\{x_i \\in \\mathbb{H}: \\mathcal{W}^\\omega_i(x_i) \\leq L\\}$.  Let\n\\[\nI_1:= \\tilde{\\mathbb{E}}_i|f - m|^q\\mathbf{1}_{\\{\\mathcal{W}_i  \\leq L\\}}, \\qquad I_2 :=\\tilde{\\mathbb{E}}_i|f - m|^q\\mathbf{1}_{\\{\\mathcal{W}_i\\geq L\\}}.\n\\]\nWe estimate each of these terms separately.  We can treat $f$ as a function of $x_i$ only by fixing all the others.  Take \n\\[\nm = m(f) := \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)} f(x_i)dx_i\n\\]\nwhere $|A^\\omega(L)| = \\int_{A^\\omega(L)}dx_i$ is the Lebesgue measure of $A^\\omega(L)$.  Then we have that\n\\begin{align}\n\\label{I_1}\nI_1 & = \\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^q\\frac{e^{-\\tilde U^\\omega_i(x_i)}}{\\tilde Z_i}dx_i\\nonumber\\\\\n&\\leq \\frac{e^{2N|\\varepsilon| L^\\frac{2}{p}}}{\\tilde Z_i} \\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^qdx_i\n\\end{align}\nsince on $A^\\omega(L)$ we have that $\\tilde U^\\omega_i \\geq - 2N|\\varepsilon|L^{\\frac{2}{p}}$.  Now, using the invariance of the Lebesgue measure with respect to the group translation\n\\begin{align}\n\\label{interp1}\n&\\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^qdx_i\\nonumber\\\\\n&\\qquad \\leq \\frac{1}{|A^\\omega(L)|^q} \\int_{A^\\omega(L)}\\left(\\int_\\mathbb{H}|f(x_i) - f(x_iy_i)|\\mathbf{1}_{A^\\omega(L)}(x_iy_i)dy_i\\right)^qdx_i\\nonumber\\\\\n&\\qquad \\leq \\frac{1}{|A^\\omega(L)|}\\int_\\mathbb{H}\\int_\\mathbb{H}|f(x_i) - f(x_iy_i)|^q \\mathbf{1}_{A^\\omega(L)}(x_iy_i)\\mathbf{1}_{A^\\omega(L)}(x_i)dy_idx_i\n\\end{align}\nusing H\\\"older's inequality.  Let $\\gamma : [0, t] \\to \\mathbb{H}$ be a geodesic in $\\mathbb{H}$ from $0$ to $y_i$ such that $|\\dot\\gamma(s)|\\leq1$.  Then\n\\begin{align*}\n|f(x_i) - f(x_iy_i)|^q &= \\left| \\int_0^t \\frac{d}{ds}f(x_i\\gamma(s))ds\\right|^q \\\\\n&=  \\left| \\int_0^t \\nabla_i f(x_i\\gamma(s)).\\dot\\gamma(s)ds\\right|^q\\\\\n&\\leq t^\\frac{q}{p}\\int_0^t|\\nabla_i f(x_i \\gamma(s))|^qds\n\\end{align*}\nagain by H\\\"older's inequality, where $\\frac{1}{p} +\\frac{1}{q}=1$.  Here $t = d(y_i)$.  Using this estimate in \\eqref{interp1} we see that\n\\begin{align}\n\\label{interp2}\n&\\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^qdx_i\\nonumber\\\\\n&\\qquad \\leq \\frac{1}{|A^\\omega(L)|}\\int_\\mathbb{H}\\int_\\mathbb{H} d^\\frac{q}{p}(y_i)\\int_0^t|\\nabla_i f(x_i \\gamma(s))|^qds \\mathbf{1}_{A^\\omega(L)}(x_iy_i)\\mathbf{1}_{A^\\omega(L)}(x_i)dy_idx_i.\n\\end{align}\nNote that when $x_iy_i\\in A^\\omega(L)$ and $x_i\\in A^\\omega(L)$ then we have $d(x_iy_i)\\leq L^{\\frac{1}{p}}$ and $d(x_i)\\leq L^\\frac{1}{p}$, so that\n\\[\nd(y_i) = d(x_i^{-1}x_iy_i) \\leq d(x_i) + d(x_iy_i) \\leq 2L^\\frac{1}{p}.\n\\]\nTherefore, continuing \\eqref{interp2}, \n\\begin{align}\n\\label{interp3}\n&\\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^qdx_i\\nonumber\\\\\n&\\qquad \\leq\\frac{(2L^\\frac{1}{p})^\\frac{q}{p}}{|A^\\omega(L)|}\\int_\\mathbb{H}\\int_\\mathbb{H}\\int_0^t|\\nabla_i f(x_i \\gamma(s))|^q \\mathbf{1}_{A^\\omega(L)}(x_iy_i)\\mathbf{1}_{A^\\omega(L)}(x_i)dsdy_idx_i.\n\\end{align}\n\nNext we note that for $x_iy_i\\in A^\\omega(L)$ and $x_i\\in A^\\omega(L)$ we have\n\\[\nd^p(y_i) + d(y_i) \\sum_{j:j\\sim i}d(\\omega_j) \\leq 2^pL\n\\]\nand \n\\begin{align*}\nd^p(x_i\\gamma(s)) + d(x_i\\gamma(s)) \\sum_{j:j\\sim i}d(\\omega_j) &\\leq 2^{p-1}\\left(d^p(x_i) + d(x_i) \\sum_{j:j\\sim i}d(\\omega_j)\\right) \\\\\n&\\quad + 2^{p-1}\\left(d^p(\\gamma(s)) + d(\\gamma(s)) \\sum_{j:j\\sim i}d(\\omega_j)\\right)\\\\\n&\\leq 2^{p-1}L + 2^{p-1}\\left(d^p(y_i) + d(y_i) \\sum_{j:j\\sim i}d(\\omega_j)\\right)\\\\\n&\\leq 2^{p-1}L(1+2^p) =: R.\n\\end{align*}\nThus we can continue \\eqref{interp3} by writing\n\\begin{align}\n\\label{interp4}\n&\\int_{A^\\omega(L)}\\left| f(x_i) - \\frac{1}{|A^\\omega(L)|}\\int_{A^\\omega(L)}f(y_i)dy_i\\right|^qdx_i\\nonumber\\\\\n&\\quad \\leq\\frac{(2L^\\frac{1}{p})^\\frac{q}{p}}{|A^\\omega(L)|}\\int_\\mathbb{H}\\int_\\mathbb{H}\\int_0^t|\\nabla_i f(x_i \\gamma(s))|^qds \\mathbf{1}_{A^\\omega(R)}(x_i\\gamma(s))\\mathbf{1}_{A^\\omega(2^pL)}(y_i)dy_idx_i\\nonumber\\\\\n& \\quad \\leq\\frac{(2L^\\frac{1}{p})^\\frac{q}{p}}{|A^\\omega(L)|}\\int_\\mathbb{H} d(y_i) \\left(\\int_\\mathbb{H}|\\nabla_i f(x_i)|^q\\mathbf{1}_{A^\\omega(R)}(x_i)dx_i\\right)\\mathbf{1}_{A^\\omega(2^pL)}(y_i)dy_i\\nonumber\\\\\n& \\quad \\leq\\frac{(2L^\\frac{1}{p})^{\\frac{q}{p}+1}}{|A^\\omega(L)|}\\int_\\mathbb{H} \\left(\\int_\\mathbb{H}|\\nabla_i f(x_i)|^q\\mathbf{1}_{A^\\omega(R)}(x_i)dx_i\\right)\\mathbf{1}_{A^\\omega(2^pL)}(y_i)dy_i\\nonumber\\\\\n& \\quad = 2^qL^\\frac{q}{p}\\frac{|A^\\omega(2^pL)|}{|A^\\omega(L)|}\\int_{A^\\omega(R)}|\\nabla_i f(x_i)|^qdx_i\\nonumber\\\\\n& \\quad \\leq  2^qL^\\frac{q}{p}e^R\\frac{|A^\\omega(2^pL)|}{|A^\\omega(L)|}\\int |\\nabla_i f(x_i)|^qe^{-\\tilde U_i^\\omega}dx_i\n\\end{align}\nwhere in the last line we have used the fact that on the set $A^\\omega(R)$ we have $e^{-\\tilde U_i^\\omega} \\geq e^{-R}$.  We finally note that $|A^\\omega(2^pL)|\/|A^\\omega(L)|$ can be bounded above by a constant $C_1$ independent of $\\omega$.  This is because $|A^\\omega(2^pL)|\/|A^\\omega(L)|\\geq 1$ and\n\\[\n\\frac{|A^\\omega(2^pL)|}{|A^\\omega(L)|} \\to 1 \\quad {\\rm as} \\quad \\sum_{j:j\\sim i}d(\\omega_j) \\to \\infty.\n\\]\nThen, using \\eqref{interp4} in \\eqref{I_1} yeilds\n\\[\nI_1 \\leq C_2 \\tilde\\mathbb{E}_i|\\nabla_i f|^q\n\\]\nwhere $C_2 = 2^qL^\\frac{q}{p}e^{2N|\\varepsilon|L^\\frac{2}{p} + R}C_1$ is independent of $\\omega$.\n\nFor the second term, we have that\n\\begin{align*}\nI_2 &\\leq \\frac{1}{L}\\tilde\\mathbb{E}_i\\left(|f -m|^q\\mathcal{W}_i\\right) \\\\\n& \\leq \\frac{A}{L}\\tilde\\mathbb{E}_i\\left|\\nabla f \\right|^q + \\frac{B}{L}\\tilde\\mathbb{E}_i\\left|f-m\\right|^q\n\\end{align*}\nwhere we have used Lemma \\ref{lem q U-Bound}.  Putting the estimates for $I_1$ and $I_2$ together, we see that\n\\begin{align*}\n\\tilde\\mathbb{E}_i\\left|f-m\\right|^q & \\leq  \\left(C_2 + \\frac{A}{L}\\right) \\tilde\\mathbb{E}_i|\\nabla_i f|^q + \\frac{B}{L}\\tilde\\mathbb{E}_i\\left|f-m\\right|^q \\\\\n\\Rightarrow \\tilde\\mathbb{E}_i\\left|f-m\\right|^q & \\leq \\frac{C_2 + \\frac{A}{L}}{1-  \\frac{B}{L}}\\tilde\\mathbb{E}_i|\\nabla_i f|^q\n\\end{align*}\nfor $L>B$, where all constants are independent of $\\omega$.  We can finally use this in \\eqref{note} to get the result. \n\n\\end{proof}\n\nWe can now prove Theorem \\ref{uniform LSq}:\n\\begin{proof}[of Theorem  \\ref{uniform LSq}]\n\nOur starting point is the classical Sobolev inequality on the Heisenberg group for the Lebesgue measure (\\cite{Var}):  there exists a $t>0$ such that\n\\begin{equation}\n\\label{CS}\n\\left(\\int|f|^{1+t}dx_i\\right)^{\\frac{1}{1+t}} \\leq a \\int|\\nabla_if|dx_i + b\\int|f|dx_i,\n\\end{equation}\nfor some constants $a,b\\in(0,\\infty)$. Without loss of generality, we may assume that $f\\geq0$.  Suppose also, to begin with, that $\\tilde\\mathbb{E}_i(f)=1$.  Now, if we set\n\\[\ng\\equiv \\frac{fe^{-\\tilde{U}_i}}{\\tilde{Z}_i}\n\\]\nthen\n\\begin{equation}\n\\label{LS1}\n\\tilde\\mathbb{E}_i(f\\log f) = \\int_\\mathbb{H} g\\log g dx_i + \\tilde\\mathbb{E}_i(f\\tilde{U}_i) + \\log \\tilde{Z}_i.\n\\end{equation}\nNow by Jensen's inequality\n\\begin{align*}\n\\int g\\log gdx_i & = \\frac{1}{t}\\int g\\log g^tdx_i  \\\\\n& = \\frac{1}{t}\\int g\\log \\left(d^{1+t}g^t\\right)dx_i - \\frac{1+t}{t}\\int g\\log ddx_i \\\\\n&\\leq \\frac{1+t}{t}\\log\\left(\\int (dg)^{1+t}dx_i\\right)^{\\frac{1}{1+t}} +\\frac{1+t}{t}\\int g ddx_i \\\\\n& \\leq\\frac{1+t}{t}\\left(\\int (dg)^{1+t}dx_i\\right)^{\\frac{1}{1+t}} +\\frac{1+t}{t}\\tilde\\mathbb{E}_i(f d) \\\\\n&\\leq \\frac{a(1 + t)}{t}\\int |\\nabla_i (dg)|dx_i + \\frac{1 +t}{t}(b+1)\\tilde\\mathbb{E}_i(f d)\\\\\n&\\leq  \\frac{a(1 + t)}{t}\\int d|\\nabla_i g|dx_i + \\frac{1 +t}{t}(b+1)\\tilde\\mathbb{E}_i(f d) + \\frac{a(1 + t)}{t},\n\\end{align*}\nwhere we have used the classical Sobolev inequality \\eqref{CS}, the fact that we have assumed $\\tilde\\mathbb{E}_i(f)=1$, and the elementary inequality $\\log x\\leq x$.  Hence by \\eqref{LS1}\n\\begin{align}\n\\label{LS2}\n\\tilde\\mathbb{E}_i(f\\log f) &\\leq \\frac{a(1 + t)}{t}\\int d\\left|\\nabla_i \\left(\\frac{fe^{-\\tilde{U}_i}}{\\tilde{Z}_i}\\right)\\right|dx_i + \\tilde\\mathbb{E}_i(f\\tilde{U}_i) + \\frac{1 +t}{t}(b+1)\\tilde\\mathbb{E}_i(f d)\\nonumber\\\\\n&\\quad +  \\frac{a(1 + t)}{t} + \\log \\tilde{Z}_i \\nonumber\\\\\n& \\leq \\frac{a(1 + t)}{t}\\tilde\\mathbb{E}_i(d |\\nabla_i f|) + \\frac{a(1 + t)}{t}\\tilde\\mathbb{E}_i(f d|\\nabla_ i \\tilde U_i|) + \\tilde\\mathbb{E}_i(f\\tilde{U}_i)  \\nonumber\\\\\n& \\qquad + \\frac{1 +t}{t}(b+1)\\tilde\\mathbb{E}_i(fd^p) + 1 +  \\frac{a(1 + t)}{t} + \\log \\tilde{Z}_i.\n\\end{align}\n\nNow, since $\\varepsilon\\rho>0$ we have that $\\tilde Z_i^\\omega \\leq C_3$ for some constant $C_3\\in(0, \\infty)$ independent of $\\omega$.\n\nMoreover, we can directly calculate that\n\\begin{align}\n\\label{est1}\nd(x_i)|\\nabla_i \\tilde{U}^\\omega_i|(x_i) &\\leq \\alpha pd^p(x_i) + 4N|\\varepsilon|d^2(x_i) + 2\\varepsilon\\rho d(x_i)\\sum_{j:j\\sim i}d(\\omega_j)\\nonumber\\\\\n&\\leq  \\left(\\alpha p + 4N|\\varepsilon|\\right)d^p(x_i) + 2\\varepsilon\\rho d(x_i)\\sum_{j:j\\sim i}d(\\omega_j) + 4N|\\varepsilon|\\nonumber\\\\\n& \\leq a_3\\mathcal{W}_i^\\omega(x_i) + b_3\n\\end{align}\nalmost everywhere, where $\\mathcal{W}_i^\\omega(x_i) = d^{p}(x_i) + d(x_i)\\sum_{j:j\\sim i}d(\\omega_j)$ as in Lemma \\ref{qSG}, and $a_3 = \\max\\{\\alpha p + 4N|\\varepsilon|, 2\\varepsilon\\rho\\}$ and $b_3=4N|\\varepsilon|$ are constants independent of $\\omega$.\nSimilarly there exist constants $a_4, b_4\\in[0, \\infty)$ independent of $\\omega$ such that\n\\begin{equation}\n\\label{est2}\n\\tilde{U}^\\omega_i(x_i) \\leq a_4\\mathcal{W}_i^\\omega(x_i) + b_4.\n\\end{equation}\n\nWe can substitute estimates \\eqref{est1} and \\eqref{est2} into \\eqref{LS2}.  This yields\n\\begin{align}\n\\label{LS2.5}\n\\tilde\\mathbb{E}_i(f\\log f) &\\leq c_1\\tilde\\mathbb{E}_i(d |\\nabla_i f|) + c_2\\tilde\\mathbb{E}_i(f \\mathcal{W}_i) + c_3\n\\end{align}\nwhere \n\\begin{align*}\nc_1= &\\frac{a(1 + t)}{t}, \\qquad c_2 =  \\frac{aa_3(1 + t)}{t} + a_4 + \\frac{1+t}{t}(b+1)\\\\\n&c_3 =  1 +  \\frac{a(1 + t)}{t} + \\log C_3 + b_3 + b_4\n\\end{align*}\nare all independent of $\\omega$.\n\n\nNow, by fiirst replacing $f$ by $\\frac{f}{\\tilde\\mathbb{E}_i f}$ and then $f$ by $f^q$ in \\eqref{LS2.5}, after an application of Young's inequality we see that\n\\begin{align}\n\\label{LS3}\n\\tilde\\mathbb{E}_i\\left(f^q\\log \\frac{f^q}{\\tilde\\mathbb{E}_i f^q}\\right) & \\leq \\frac{c_1}{q}\\tilde\\mathbb{E}_i(|\\nabla_i f|^q) + \\left(c_2 + \\frac{c_1}{p}\\right)\\tilde\\mathbb{E}_i\\left(f^q\\mathcal{W}_i\\right) + c_3\\tilde\\mathbb{E}_i(f^q).\n\\end{align}\n\nWe recognise that the second term in \\eqref{LS3} can be bounded using Lemma \\ref{lem q U-Bound}.  Indeed, using this estimate\n\n\\begin{equation}\n\\label{GLS}\n\\tilde\\mathbb{E}_i\\left(f^q\\log \\frac{f^q}{\\tilde\\mathbb{E}_i f^q}\\right)  \\leq \\tilde c_1\\tilde\\mathbb{E}_i(|\\nabla_i f|^q) + \\tilde c_2\\tilde\\mathbb{E}_i(f^q),\n\\end{equation}\nwhere $\\tilde c_1 = \\frac{c_1}{q} + A\\left(c_2 + \\frac{c_1}{p}\\right)$ and $\\tilde c_2 = c_3 + B\\left(c_2 + \\frac{c_1}{p}\\right)$. Thus $\\tilde\\mathbb{E}^\\omega_i$ satisfies the generalised $(LS_q)$ inequality uniformly on the boundary conditions.\n\nFinally, we have the $q$-Rothaus inequality (see \\cite{B-Z}, \\cite{Ro}), which states that\n\\[\n\\tilde\\mathbb{E}_i\\left(f^q\\log\\frac{f^q}{\\tilde\\mathbb{E}_if^q}\\right) \\leq \\tilde\\mathbb{E}_i\\left(\\left|f-\\tilde\\mathbb{E}_if\\right|^q\\log\\frac{\\left|f-\\tilde\\mathbb{E}_if\\right|^q}{\\tilde\\mathbb{E}_i\\left|f-\\tilde\\mathbb{E}_if\\right|^q}\\right) + 2^{q+1}\\tilde\\mathbb{E}_i\\left|f-\\tilde\\mathbb{E}_if\\right|^q.\n\\]\nUsing this together with \\eqref{GLS} and the $q$-spectral gap inequality proved in Lemma \\ref{qSG} we thus arrive at a constant $c$ independent of $\\omega$ such that\n\\[\n\\tilde\\mathbb{E}_i\\left(f^q\\log\\frac{f^q}{\\tilde\\mathbb{E}_if^q}\\right) \\leq c\\tilde\\mathbb{E}_i(|\\nabla_i f|^q),\n\\]\nwhich proves Theorem \\ref{uniform LSq}.\n\\end{proof}\n\n\\section{The logarithmic Sobolev inequality for Gibbs measures}\n \n  \nIn this section we show how to pass from the uniform $(LS_q)$ inequality for the single site measures $\\mathbb{E}^{\\omega}_i$, to the $(LS_q)$ inequality for the corresponding Gibbs measure $\\nu$ on the entire configuration space $\\Omega = (\\mathbb{H})^{\\mathbb{Z}^N}$.  In the more standard Euclidean model, this problem has been extensively studied in the case $q=2$ , for example in \\cite{B-H1}, \\cite{G-Z}, \\cite{Led}, \\cite{Marton} and more recently in \\cite{O-R}, as well as in many of the afore mentioned papers.  The case $q<2$ was looked at in \\cite{B-Z}.  The following argument is strongly related to these methods, though it is based on the work contained in \\cite{Ze2} and \\cite{Ze}.\n\nWe work in greater generality than is required for Theorem \\ref{main}, though the results of section \\ref{results for single site measure} show that in the specific case where the local specification is defined by \\eqref{local spec} and \\eqref{hamiltonian}, the hypotheses \\textbf{(H0)} and \\textbf{(H1)} below are satisfied.  Then Theorem \\ref{main} follows as an immediate corollary of Theorem \\ref{thm1}.\n\nConsider a local specification $\\{\\mathbb{E}^{\\omega}_\\Lambda\\}_{\\Lambda\\subset\\subset\\mathbb{Z}^N, \\omega\\in\\Omega}$ defined by \n\\begin{equation}\n\\label{gen local spec}\n\\mathbb{E}^{\\omega}_\\Lambda(dx_{\\Lambda})=\\frac{e^{-\\sum_{i\\in\\Lambda}\\phi(x_i)-\\sum_{\\{i,j\\}\\cap\\Lambda\\neq\\emptyset, i\\sim j}J_{ij}V(x_i,x_j)}dx_\\Lambda} {Z^{\\omega}_\\Lambda}\n\\end{equation}\nwhere $Z^{\\omega}_\\Lambda$ is the normalisation factor and the summation is taken over couples of nearest neighbours $i\\sim j$ in the lattice with at least one point in $\\Lambda$ and where $x_i = \\omega_i$ for $i\\not\\in \\Lambda$, as before.  We suppose that $|J_{ij}| \\in [0, J_0]$ for some $J_0>0$.\n\n~\n\nWe will work with the following hypotheses:\n \n \\begin{itemize}\n \\item[\\textbf{(H0):}] The one dimensional single site measures $\\mathbb{E}^{\\omega}_i$ satisfy $(LS_q)$ with a constant $c$ which is independent of the boundary conditions $\\omega$.  \n \n~\n \n\\item[\\textbf{(H1):}] The interaction $V$ is such that\n$$\n\\left\\Vert \\nabla_i \\nabla_j V(x_i,x_j) \\right\\Vert_{\\infty}<\\infty.\n$$\n\\end{itemize}\n \n\\begin{rem}\nIn the situation where {\\rm \\textbf{(H1)}} is not satisfied, i.e. when the interaction potential grows faster than quadratically, a number of results have been obtained in \\cite{Ioannis1} and \\cite{Ioannis2} under some additional assumptions.\n\\end{rem}\n \n\n \n \n \n \n \n \n \n\\begin{theorem}\n\\label{thm1}\nSuppose the local specification $\\{\\mathbb{E}^{\\omega}_\\Lambda\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^N,\\omega \\in\n\\Omega}$ defined by \\eqref{gen local spec} satisifies {\\rm\\textbf{(H0)}} and {\\rm \\textbf{(H1)}}.  Then, for sufficiently small $J_0$, the corresponding infinite dimensional Gibbs measure $\\nu$ is unique and satisfies the $(LS_q)$ inequality \n$$\\nu \\left(|f|^q \\log\\frac{|f|^q}{\\nu |f|^q}\\right)\\leq C \\nu (\\left| \\nabla f\\right|^q)$$                               \nfor some positive constant $C$. \n\\end{theorem}\n\nFor notational sake we only prove this result for the case $N=2$, but our methods are easily generalised.  Before proving Theorem \\ref{thm1} we will present some useful lemmata.  \n \n\\subsection{Lemmata:}\n\nDefine the following sets\n\\begin{align*}\n&\\Gamma_0=(0,0)\\cup\\{j \\in \\mathbb{Z}^2 : dist(j,(0,0))=2m \\text{\\; for some \\;}m\\in\\mathbb{N}\\}, \\\\  \n&\\Gamma_1=\\mathbb{Z}^2\\smallsetminus\\Gamma_0 .\n\\end{align*}\nwhere $dist(\\cdot, \\cdot)$ is as in section \\ref{Infinite dimensional setting}.  Note that $dist(i,j)>1$ for all $i,j \\in\\Gamma_k,k=0,1$ and $\\Gamma_0\\cap\\Gamma_1=\\emptyset$.  Moreover $\\mathbb{Z}^2=\\Gamma_0\\cup\\Gamma_1$.  As above, for the sake of notation, we will write $\\mathbb{E}_{\\Gamma_k}=\\mathbb{E}_{\\Gamma_k}^{\\omega}$ for $k=0,1$. We will also define \n$$ \\mathcal{P}:=\\mathbb{E}_{\\Gamma_1}\\mathbb{E}_{\\Gamma_{0}}.$$\n \n\\begin{lemma} \n\\label{lem1}\nIf the local specification $\\{\\mathbb{E}^{\\omega}_\\Lambda\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^N,\\omega \\in\n\\Omega}$ satisfies {\\rm \\textbf{(H0)}} and {\\rm \\textbf{(H1)}}, then, for sufficiently small $J_0$, there exist constants $\\tilde D>0$ and $\\tilde\\eta\\in(0,1)$ such that\n\\begin{equation}\n\\label{first lem}\n\\nu\\left\\vert \\nabla_{\\Gamma_k}(\\mathbb{E}_{\\Gamma_{l}}f)\n\\right\\vert^q \\leq \\tilde D\\nu\\left\\vert \\nabla_{\\Gamma_k} f\n\\right\\vert^q+\\tilde\\eta\\nu\\left\\vert \\nabla_{\\Gamma_l} f\n\\right\\vert^q\\end{equation}\nfor $k,l\\in\\{0,1\\}$ such that $k\\neq l$.\n\\end{lemma}\n\n\n\n\n \\begin{proof}   \n \nFor convenience, suppose $k=1$ and $l=0$. The case $k=0,l=1$ follows similarly. We can write \n\\begin{align}\n\\label{lem1-1}\n\\mathcal{I} &:=\\nu\\left\\vert \\nabla_{\\Gamma_1 }\n(\\mathbb{E}_{\\Gamma_0}f) \\right\\vert ^q = \\nu\\sum_{i\\epsilon \\Gamma_1}\\left\\vert \\nabla_{i }\n(\\mathbb{E}_{\\Gamma_0}f) \\right\\vert ^q \\nonumber \\\\\n& \\leq \\nu\\sum_{i\\epsilon \\Gamma_1}\n\\left\\vert \\nabla_{i }\n(\\mathbb{E}_{\\{\\sim i\\}}f) \\right\\vert ^q \\nonumber \\\\\n&\\leq2^{q-1}\\nu\\sum_{i\\epsilon \\Gamma_1}\n\\left\\vert \\mathbb{E}_{\\{\\sim i\\}}\\nabla_{i } f\n\\right\\vert\n^q+\n2^{q-1}J_{0}^{q}\\nu\\sum_{i\\epsilon \\Gamma_1}\n\\left\\vert\\mathbb{E}_{\\{\\sim i\\}}(f\\mathcal{U}_i) \\right\\vert ^q\n\\end{align}\nwhere above we have denoted $\\{ \\sim i\\}=\\{j:j \\sim i\\}$, $W_i=\\sum_{j\\in\\{\\sim i\\}}\\nabla_iV(x_{i}, x_{j})$ and \n$$\\mathcal{U}_i=W_i-\\mathbb{E}_{\\{ \\sim i\\}}W_i.$$\nThen\n\\begin{align}\n\\label{lem1-2}\n\\mathcal{I} &\\leq2^{q-1}\\nu\\sum_{i\\epsilon \\Gamma_1}\n\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert \\nabla_{i } f \\right\\vert^q + 2^{q-1}J_{0}^q\\nu\\sum_{i\\epsilon\\Gamma_1}\\left\\vert\\mathbb{E}_{\\{\\sim i\\}}(f-\\mathbb{E}_{\\{\\sim i\\}}f)\\mathcal{U}_i \\right\\vert ^q \\nonumber \\\\ \n&\\leq 2^{q-1}\\nu\\sum_{i\\epsilon \\Gamma_1}\n\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert \\nabla_{i } f\n\\right\\vert^q+2^{q-1}J_{0}^q\\nu\\left(\\sum_{i\\epsilon \\Gamma_1}\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q\\left(\\mathbb{E}_{\\{\\sim i\\}}\\left|\\mathcal{U}_i\\right|^p\\right)^{q\/p}\\right)\n\\end{align}\nusing H\\\"older's inequality and the fact that $\\mathbb{E}_{\\{\\sim i\\}}\\mathcal{U}_i =0$.  Since interactions occur only between nearest neighbours in the lattice, we have that no interactions occur between points of the set $\\{\\sim i\\}$.  Hence the measure $\\mathbb{E}_{\\{\\sim i\\}}^{\\omega}$ is the product measure of the single site measures i.e.  $\\mathbb{E}_{\\{\\sim i\\}}^{\\omega}=\\otimes_{j \\in\\{\\sim i\\}}\\mathbb{E}^{\\omega}_j$.\nMoreover, by \\textbf{(H0)}, all measures $\\mathbb{E}^{\\omega}_j, j \\in\\{\\sim i\\}$ satisfy the $(LS_q)$ inequality with a constant $c$ uniformly on the\nboundary conditions.  Therefore, since the $(LS_q)$ inequality is stable under tensorisation (see Remark \\ref{tensorisation} (i)), we have that the product measure $\\mathbb{E}_{\\{\\sim i\\}}^{\\omega}$ also satisfies the $(LS_q)$ inequality with the same constant $c$.  By Remark \\ref{tensorisation} (iii), it follows that $\\mathbb{E}_{\\{\\sim i\\}}^{\\omega}$ also satisfies the $q$-spectral gap inequality with constant $c_0 = \\frac{4c}{\\log2}$.\n\nHence we have\n\\begin{equation}\n\\label{lem1-3}\n\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q\\leq c_0\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert\n\\nabla_{\\{\\sim i\\}} f\\right\\vert^q.\n\\end{equation}\nMoreover, by Remark \\ref{tensorisation} (iv), since $q<p$ and $\\mathbb{E}_{\\{\\sim i\\}}$ is a measure on a finite dimensional space, we have there exists a constant $\\tilde c_0$ such that\n\\begin{align}\n\\label{lem1-4}\n\\mathbb{E}_{\\{\\sim i\\}}\\left|\\mathcal{U}_i\\right|^p=& \\mathbb{E}_{\\{\\sim i\\}}\\left|W_i-\\mathbb{E}_{\\{\\sim\ni\\}}W_i\\right|^p\\leq \\tilde{c}_0\\sum_{_{ j \\in\\{\\sim i\\}}}\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert \\nabla_{j} W_i\\right\\vert\n^p \\nonumber\\\\\n&\\leq \\tilde{c}_0\\sum_{_{ j \\in\\{\\sim i\\}}}\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert \\nabla_{j} \\nabla_iV(x_{i}, x_{j})\\right\\vert\n^p\\leq 4\\tilde{c}_0M^p\\end{align}\nwhere $M=\\|\\nabla_i\\nabla_j V(x_i, \\omega_j)\\|_\\infty < \\infty$ by \\textbf{(H1)}. \n\nIf we combine \\eqref{lem1-2}, \\eqref{lem1-3} and \\eqref{lem1-4} we obtain\n \\begin{align*}\n\\nu\\left\\vert \\nabla_{\\Gamma_1 }\n(\\mathbb{E}_{\\Gamma_0}f) \\right\\vert ^q &\\leq 2^{q-1}\\nu\\left(\\sum_{i\\epsilon \\Gamma_1}\n\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert \\nabla_{i } f\n\\right\\vert^q\\right)  \\\\\n&\\qquad + 2^{q-1}c\\left(4\\tilde{c}_0\\right)^{q\/p}M^qJ^q_0\\nu\\left(\\sum_{i\\epsilon \\Gamma_1}\\mathbb{E}_{\\{\\sim i\\}}\\left\\vert\n\\nabla_{\\{\\sim i\\}} f\\right\\vert^q\\right)\\\\\n&\\leq  2^{q-1}\\nu\\left(\\sum_{i\\epsilon \\Gamma_1}\n\\left\\vert \\nabla_{i } f\n\\right\\vert^q\\right)  \\\\\n&\\qquad + 2^{q+1}c\\left(4\\tilde{c}_0\\right)^{q\/p}M^qJ^q_0\\nu\\left(\\sum_{i\\epsilon \\Gamma_0}\\left\\vert\n\\nabla_{i} f\\right\\vert^q\\right).\n\\end{align*}\nTherefore, choosing $J_0$ sufficiently small so that  $ 2^{q+1}c\\left(4\\tilde{c}_0\\right)^{q\/p}M^qJ^q_0<1$, we see that\n$$\\nu\n\\left\\vert \\nabla_{\\Gamma_1 }\n(\\mathbb{E}^{\\Gamma_0}f) \\right\\vert ^q\\leq \\tilde D\\nu\\left\\vert \\nabla_{ \\Gamma_1 } f\n\\right\\vert^q+\\tilde\\eta\\nu\\left\\vert\\nabla_{ \\Gamma_0} f\\right\\vert^q$$\n with $\\tilde{D} =2^{q-1}$ and $\\tilde\\eta= 2^{q+1}c\\left(4\\tilde{c}_0\\right)^{q\/p}M^qJ^q_0<1$, as required.\n\n \\end{proof}\n \n \\begin{lemma}\n \\label{lem2}\nSuppose the local specification  $\\{\\mathbb{E}_{\\Lambda}^{\\omega}\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^N, \\omega \\in\n\\Omega}$ satisfies {\\rm \\textbf{(H0)}} and {\\rm \\textbf{(H1)}}, and let $W_i=\\sum_{j\\in\\{\\sim i\\}}\\nabla_iV(x_{i}, x_{j})$ be as in the previous lemma.  Then there exists a constant $\\kappa$, independent of the boundary conditions, such that\n\\begin{align*}\n\\mathbb{E}_{\\{\\sim i\\}}&\\left(|f|^q;  W_{i}\\right) \\leq \\left(\\mathbb{E}_{\\{\\sim i\\}}f^q\\right)^{\\frac{1}{p}}\\left( \\kappa\\mathbb{E}_{\\{\\sim i \\}}\\left|\\nabla_{\\{ \\sim i\\}} f\\right|^q\\right)^\\frac{1}{q},\n\\end{align*}\nwhere $\\frac{1}{p}+\\frac{1}{q}=1$ and $\\mathbb{E}_{\\{\\sim i \\}}(g;h):= \\mathbb{E}_{\\{\\sim i \\}}(gh) - \\mathbb{E}_{\\{\\sim i \\}}(g)\\mathbb{E}_{\\{\\sim i \\}}(h)$ for any functions $g,h$.\n\\end{lemma}\n \\begin{proof} \n \n Without loss of generality, we may suppose that $f\\geq0$.  Let $\\hat\\mathbb{E}_{\\{\\sim i\\}}$ be an isomorphic copy of $\\mathbb{E}_{\\{\\sim i\\}}$. Then we have \n\\begin{align*}\n\\mathbb{E}_{\\{\\sim i\\}}(f^q;W_{i}) &=\\frac{1}{2}\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(\\left(f^q- \\hat f^{q}\\right)(W_{i}-\\hat W_i  )\\right) \\\\\n& = \\frac{1}{2}\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left[\\left(\\int_0^1\\frac{d}{ds}F_s^qds\\right)\\left(W_i - \\hat W_i\\right)\\right]\n\\end{align*}\nwhere $F_s = sf + (1-s)\\hat f$ for $s\\in[0,1]$.  Then\n\\begin{align}\n\\label{covlem1}\n\\mathbb{E}_{\\{\\sim i\\}}(f^q;W_{i}) &= \\frac{q}{2} \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left[\\left(\\int_0^1F_s^{q-1}ds\\right)\\left(f-\\hat f\\right)\\left(W_i - \\hat W_i\\right)\\right] \\nonumber\\\\\n&\\leq \\frac{q}{2}\\left\\{ \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(\\int_0^1F_s^{q-1}ds\\right)^p\\right\\}^\\frac{1}{p} \\nonumber\\\\\n& \\quad \\times \\left\\{ \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\hat f\\right|^q\\left|W_i - \\hat W_i\\right|^q\\right\\}^{\\frac{1}{q}}.\n\\end{align}\n\nNow by Jensen's inequality and convexity of the function $y\\mapsto y^q$ we have\n\\begin{align}\n\\label{covlem1a}\n\\left\\{ \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(\\int_0^1F_s^{q-1}ds\\right)^p\\right\\}^\\frac{1}{p} &\\leq \\left\\{ \\int_0^1\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}F_s^{q}ds\\right\\}^\\frac{1}{p}\\nonumber\\\\\n&\\leq  \\left\\{ \\int_0^1\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(sf^{q} + (1-s)\\hat f^q\\right)ds\\right\\}^\\frac{1}{p}\\nonumber\\\\\n&= \\left(\\mathbb{E}_{\\{\\sim i\\}} f^q\\right)^{\\frac{1}{p}}.\n\\end{align}\n\nMoreover,\n\\begin{align}\n\\label{covlem2}\n&\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\hat f\\right|^q\\left|W_i - \\hat W_i\\right|^q \\nonumber \\\\\n& \\qquad \\leq  2^q\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q\\left|W_i - \\hat W_i\\right|^q.\n \\end{align}\n\nWe have the following relative entropy inequality (see eg \\cite{A-B-C-F-G-M-R-S}, \\cite{D-S}): if $\\mu$ is a probability measure then\n\\[\n\\mu(uv) \\leq \\frac{1}{\\tau}\\mu(u)\\log\\mu(e^{\\tau v}) + \\frac{1}{\\tau}\\mu\\left(u\\log\\frac{u}{\\mu (u)}\\right), \\qquad \\forall \\tau>0.\n\\]\nApplying this to the right hand side of \\eqref{covlem2} with $\\mu = \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}$ we see that $\\forall \\tau>0$\n\\begin{align}\n\\label{covlem3}\n&\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\hat f\\right|^q\\left|W_i - \\hat W_i\\right|^q \\nonumber\\\\\n&\\qquad \\qquad \\leq \\frac{2^q}{\\tau}\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q\\log \\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(e^{\\tau|W_i - \\hat W_i|^q}\\right) \\nonumber \\\\\n& \\quad\\quad\\quad + \\frac{2^q}{\\tau} \\mathbb{E}_{\\{\\sim i\\}}\\left(\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q\\log\\frac{\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q}{\\mathbb{E}_{\\{\\sim i\\}} \\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q}\\right).\n\\end{align}\n\nNow, by the Herbst argument, see for example \\cite{Helffer} or \\cite{Ledconc} , and using both \\textbf{(H0)} and \\textbf{(H1)}, we have that for some $\\tau>0$ there exists a constant $\\Theta>0$ independent of $\\omega$ such that\n\\[\n\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left(e^{\\tau|W_i - \\hat W_i|^q}\\right) \\leq \\Theta.\n\\]\nWe can also use \\textbf{(H0)} to bound the second term of \\eqref{covlem3}.  This gives\n\\begin{align}\n\\label{covlem4}\n\\mathbb{E}_{\\{\\sim i\\}}\\otimes\\hat\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\hat f\\right|^q\\left|W_i - \\hat W_i\\right|^q \\nonumber &\\leq \\frac{2^q\\log\\Theta}{\\tau}\\mathbb{E}_{\\{\\sim i\\}}\\left|f-\\mathbb{E}_{\\{\\sim i\\}}f\\right|^q \\nonumber \\\\\n& \\quad\\quad + \\frac{2^qc}{\\tau}\\mathbb{E}_{\\{\\sim i\\}}\\left|\\nabla_{\\{\\sim i\\}}f\\right|^q\\nonumber\\\\\n&\\leq \\frac{2^q}{\\tau}\\left(c_0\\log\\Theta + c\\right) \\mathbb{E}_{\\{\\sim i\\}}\\left|\\nabla_{\\{\\sim i\\}}f\\right|^q\n\\end{align}\nwhere $c_0=\\frac{4c}{\\log2}$ as above, by Remark \\ref{tensorisation} (iii).\n\nPutting estimates \\eqref{covlem1a} and \\eqref{covlem4} into \\eqref{covlem1} we see that \n\\begin{align*}\n\\mathbb{E}_{\\{\\sim i\\}}(f^q;W_{i}) &\\leq \\left(\\mathbb{E}_{\\{\\sim i\\}} f^q\\right)^{\\frac{1}{p}}\\left(\\frac{q^q}{\\tau}(c_0\\log\\Theta +c)\\mathbb{E}_{\\{\\sim i\\}}\\left|\\nabla_{\\{\\sim i\\}}f\\right|^q\\right)^{\\frac{1}{q}}\n\\end{align*}\nwhich gives the desired result.\n\\end{proof}\n \n \n \n \n\\begin{lemma}\n\\label{lem3}\nSuppose the local specification  $\\{\\mathbb{E}_{\\Lambda}^{\\omega}\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^N,\\omega \\in\n\\Omega}$ satisfies {\\rm \\textbf{(H0)}} and {\\rm \\textbf{(H1)}}.  Then, for sufficiently small $J_0$, there exist constants $D >0$ and $\\eta\\in(0,1)$ such that\n\\begin{equation}\n\\label{lem3ineq}\n\\nu\\left\\vert \\nabla_{\\Gamma_k}(\\mathbb{E}_{\\Gamma_{l}}|f|^q)^\\frac{1}{q}\n\\right\\vert^q \\leq D\\nu\\left\\vert \\nabla_{\\Gamma_k} f\n\\right\\vert^q+\\eta\\nu\\left\\vert \\nabla_{\\Gamma_l} f\n\\right\\vert^q\\end{equation} \nfor $k,l\\in\\{0,1\\}, k\\neq l$. \n\\end{lemma}\n\n\\begin{proof} \nAgain we may suppose $f\\geq0$.  For $k=1,l=0$ (the other case is similar),  we can write \n\\begin{align}\n\\label{lem3-1}\n\\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_0}  f^q)^{\\frac{1}{q}} \\right\\vert\n^q &\\leq \\nu\\sum_{i\\epsilon\\Gamma_1}\\left\\vert \\nabla_i (\\mathbb{E}_{\\{\\sim i\\}}f^q)^{\\frac{1}{q}}\\right\\vert ^q \\nonumber \\\\ \n&= \\nu\\sum_{i\\epsilon\\Gamma_1} \\frac{1}{q^q}(\\mathbb{E}_{\\{\\sim i\\}}f^q)^{-\\frac{q}{p}}  \\left\\vert\\nabla_i( \\mathbb{E}_{\\{\\sim i\\}}  f^q)\\right\\vert ^q.\n\\end{align}  \n\n\nWe will compute the terms in the sum on the right hand side of \\eqref{lem3-1}.\nFor $i \\in \\Gamma_1$, we have\n\\begin{align*}\n\\nabla_i (\\mathbb{E}_{\\{\\sim i\\}} f^q) &=  q(\\mathbb{E}_{\\{\\sim i\\}} f^{q-1} \\nabla_i f )-\\sum_{ j\\in\\{\\sim i\\}}J_{i,j}\\mathbb{E}_{\\{\\sim i\\}}\\left(f^q; \\nabla_iV(x_i, x_j)\\right) \\\\\n\\Rightarrow \\left|\\nabla_i (\\mathbb{E}_{\\{\\sim i\\}} f^q)\\right| \n&\\leq q\\left( \\mathbb{E}_{\\{\\sim i\\}} f^q\\right)^{1\/p} \\left( \\mathbb{E}_{\\{\\sim i\\}}|\\nabla_if|^q\\right)^{1\/q} + J_0\\left|\\mathbb{E}_{\\{\\sim i\\}}\\left(f^q; W_i\\right)\\right|,\n\\end{align*}\nso that\n\\begin{align*}\n \\left|\\nabla_i (\\mathbb{E}_{\\{\\sim i\\}} f^q)\\right|^q &\\leq 2^{q-1}q^q\\left( \\mathbb{E}_{\\{\\sim i\\}} f^q\\right)^{\\frac{q}{p}} \\left( \\mathbb{E}_{\\{\\sim i\\}}|\\nabla_if|^q\\right)\\\\\n&\\qquad + 2^{q-1}J^q_0\\left|\\mathbb{E}_{\\{\\sim i\\}}\\left(f^q; W_i\\right)\\right|^q,\n\\end{align*}\nwhere $W_i = \\sum_{j\\in\\{\\sim i\\}} \\nabla_iV(x_i, x_j)$ as above.  \n\n\nWe can use Lemma \\ref{lem2} to bound the correlation in the second term.  Indeed, this gives\n\\begin{align*}\n\\left|\\nabla_i (\\mathbb{E}_{\\{\\sim i\\}} f^q)\\right|^q &\\leq  \\left(\\mathbb{E}_{\\{\\sim i\\}} f^q\\right)^{\\frac{q}{p}} \\left(2^{q-1}q^q\\mathbb{E}_{\\{\\sim i\\}}|\\nabla_if|^q + 2^{q-1}\\kappa J^q_0\\mathbb{E}_{\\{\\sim i\\}}\\left|\\nabla_{\\{\\sim i\\}} f\\right|^q\\right).\n\\end{align*}\n\nUsing this in \\eqref{lem3-1} yields\n\\begin{align*}\n\\nu \\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_0}  f^q)^{\\frac{1}{q}} \\right\\vert\n^q &\\leq \\nu\\sum_{i\\in\\Gamma_1}\\left(2^{q-1}\\mathbb{E}_{\\{\\sim i\\}}|\\nabla_if|^q + \\frac{2^{q-1}}{q^q}\\kappa J^q_0\\mathbb{E}_{\\{\\sim i\\}}\\left|\\nabla_{\\{\\sim i\\}} f\\right|^q\\right) \\\\\n& = 2^{q-1} \\nu\\left|\\nabla_{\\Gamma_1} f\\right|^q + \\frac{2^{q-1}}{q^q}\\kappa J_0^q\\nu\\sum_{i\\in\\Gamma_1}\\left|\\nabla_{\\{\\sim i\\}} f\\right|^q \\\\\n& = 2^{q-1} \\nu\\left|\\nabla_{\\Gamma_1} f\\right|^q + \\frac{2^{q+1}}{q^q}\\kappa J_0^q\\nu\\left|\\nabla_{\\Gamma_0} f\\right|^q.\n\\end{align*}\nFinally, taking $J^q_0<\\frac{q^q}{2^{q+1}\\kappa}$ we see that\n\\[\n\\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_0}  f^q)^{\\frac{1}{q}} \\right\\vert\n^q \\leq D\\nu \\left|\\nabla_{\\Gamma_1} f\\right|^q + \\eta\\nu\\left|\\nabla_{\\Gamma_0} f\\right|^q,\n\\]\nwhere $D=2^{q-1}$ and $\\eta= \\frac{2^{q+1}}{q^q}\\kappa J_0^q <1$, as required.\n\\end{proof}    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \\begin{lemma}\n \\label{lem4}\n Suppose the local specification  $\\{\\mathbb{E}_{\\Lambda}^{\\omega}\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^N,\\omega \\in\\Omega}$ satisfies {\\rm \\textbf{(H0)}} and {\\rm \\textbf{(H1)}}.  Then $\\mathcal{P}^nf$ converges\n $\\nu$-almost everywhere to $\\nu f$,  where we recall that $\\mathcal{P}=\\mathbb{E}_{\\Gamma_1}\\mathbb{E}_{\\Gamma_{0}}$.  In particular, $\\nu$ is unique.\n \\end{lemma}\n \n \n \\begin{proof}\n \n\\noindent  \nWe will follow \\cite{G-Z}. We have \n \\begin{align}\\nonumber\\nu\\left|f- \\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}} f\\right|^q&\\leq 2^{q-1}\\nu\\mathbb{E}_{\\Gamma_{0}}\\left|f- \\mathbb{E}_{\\Gamma_{0}} f\\right|^q+2^{q-1}\\nu\\mathbb{E}_{\\Gamma_{1}}\\left|\\mathbb{E}_{\\Gamma_{0}}f- \\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}} f\\right|^q\\\\ &\n \\leq2^{q-1}c_0\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q+2^{q-1}c_0\\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_{0}}f\n)\\right\\vert^q,\n\\end{align}           \nsince by \\textbf{(H0)} and Remark \\ref{tensorisation} both the measures $\\mathbb{E}_{\\Gamma_{0}}$ and $\\mathbb{E}_{\\Gamma_{1}}$ satisfy the $(SG_q)$ inequality with constant $c_0 = \\frac{4c}{\\log2}$ independant of the boundary conditions. If we use Lemma \\ref{lem1} we get \n\\[\n\\nu\\left|f- \\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}} f\\right|^q \\leq 2^{q-1}c_0\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q+2^{q-1}c_0(\\tilde D\\nu\\left\\vert \\nabla_{\\Gamma_1} f\n\\right\\vert^q+\\tilde\\eta\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q)\n\\]\nFrom the last inequality we obtain that for any   $n\\in \\mathbb{N}$,  \n\\begin{align*}\n\\nu\\left|\\mathcal{P}^nf- \\mathcal{P}^{n+1} f\\right|^q &\\leq 2^{q-1}c_0\\nu\\left\\vert \\nabla_{\\Gamma_0} \\mathcal{P}^nf\n\\right\\vert^q+2^{q-1}c_0\\tilde\\eta\\nu\\left\\vert \\nabla_{\\Gamma_0} \\mathcal{P}^nf\n\\right\\vert^q \\\\\n&= 2^{q-1}c_0(1+\\tilde\\eta)\\nu\\left\\vert \\nabla_{\\Gamma_0} \\mathcal{P}^nf\\right\\vert^2,\n\\end{align*}\nusing the fact that $\\mathcal{P}^n$ does not depend on coordinates in $\\Gamma_1$ by definition, so that $\\nabla_{\\Gamma_1}\\mathcal{P}^n = 0$.  By repeated applications of Lemma \\ref{lem1} we see that,\n\n\\begin{align*}\n\\nu\\left|\\mathcal{P}^nf- \\mathcal{P}^{n+1} f\\right|^q &\\leq 2^{q-1}c_0(1+\\tilde\\eta)\\tilde\\eta^{2n-1}\\nu\\left|\\nabla_{\\Gamma_1}\\mathbb{E}_{\\Gamma_0}f\\right|^q \\\\\n&\\leq 2^{q-1}c_0(1+\\tilde\\eta)\\tilde\\eta^{2n-1}\\left(\\tilde D\\nu\\left|\\nabla_{\\Gamma_1}f\\right|^q + \\tilde\\eta\\nu\\left|\\nabla_{\\Gamma_0}f\\right|^q\\right).\n\\end{align*}\n\nSince $\\tilde\\eta<1$, this clearly tends to zero as $n\\to\\infty$, so that the sequence $\\{\\mathcal{P}^n\\}$ is Cauchy in $L^q(\\nu)$.  Moreover, by the Borel-Cantelli lemma, the sequence\n $$\\{ \\mathcal{P}^n f-\\nu \\mathcal{P}^n f\\}_{n \\in \\mathbb{N}}$$  \n converges $\\nu-a.s$. The limit of $\\mathcal{P}^n f-\\nu \\mathcal{P}^n f=\\mathcal{P}^n f-\\nu  f$    is therefore constant and hence identical to zero. \n \\end{proof}      \n \n \n \n \n \n \n \n \n \n \\subsection{Proof of Theorem \\ref{thm1} }\n \n \n \n \n \n \n \n\\begin{proof}\nRecall that we want to extend the $(LS_q)$ inequality from the single-site measures $\\mathbb{E}_{i}^{\\omega}$ to the Gibbs measure corresponding to the local specification $\\{\\mathbb{E}_{\\Lambda}^{\\omega}\\}_{\\Lambda\\subset\\subset \\mathbb{Z}^2,\\omega \\in\n\\Omega}$ on the entire lattice (since we are taking $N=2$ for convenience).  As mentioned, to do so, we will follow the iterative method developed by B. Zegarlinski in  \\cite{Ze2} and \\cite{Ze}. \n\nAgain without loss of generality, suppose $f\\geq0$.  We can write \n \\begin{align}\n \\label{thm1-1}\n \\nonumber \\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right)=&\\nu\\mathbb{E}_{\\Gamma_0} \\left(f^q \\log\\frac{f^q}{\\mathbb{E}_{\\Gamma_0} f^q}\\right)+\\nu\\mathbb{E}_{\\Gamma_{1}} \\left(\\mathbb{E}_{\\Gamma_0}f^q \\log\\frac{\\mathbb{E}_{\\Gamma_0}f^q}{\\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_0} f^q}\\right) \\\\ \n&\\qquad + \\nu \\left(\\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}}f^q \\log\\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}}f^q\\right)- \\nu\\left(f^q \\log \\nu f^q\\right).\n\\end{align}\nAs mentioned above, by \\textbf{(H0)} and since the measures $\\mathbb{E}_{\\Gamma_0}$\nand $\\mathbb{E}_{\\Gamma_1}$ are in fact product measures, we know that they both satisfy $(LS_q)$ with constant $c$ independent of the boundary conditions.  Using this fact in \\eqref{thm1-1} yields\n\n\\begin{align}\n\\label{thm1-2}\n\\nonumber  \\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right) \\leq c\\nu(\\mathbb{E}_{\\Gamma_0}&\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q)+c\\nu \\mathbb{E}_{\\Gamma_1}\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_0} f^q)^{\\frac{1}{q}}\n\\right\\vert^q\\\\\n& \\qquad + \\nu \\left(\\mathcal{P}f^q \\log\\mathcal{P}f^q\\right)-\\nu \\left(f^q \\log \\nu f^q\\right).\n\\end{align}\nFor the third term of \\eqref{thm1-2} we can similarly write \n\\begin{align}\n\\nonumber\\nu \\left(\\mathcal{P}f^q \\log \\mathcal{P} f^q\\right)&= \\nu \\mathbb{E}_{\\Gamma_{0}}\\left(\\mathcal{P}f^q \\log \\frac{\\mathcal{P} f^q}{\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P} f^q}\\right)+\\nu \\mathbb{E}_{\\Gamma_{1}}\\left(\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P}f^q \\log \\frac{\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P} f^q}{\\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P} f^q}\\right)\\\\\n& \\qquad + \\nonumber\n\\nu\\left(\\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P}f^q \\log \\mathbb{E}_{\\Gamma_{1}}\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P}f^q \\right).\n\\end{align}\nIf we use again the $(LS_q)$ inequality  for the measures $\\mathbb{E}_{\\Gamma_{k}},k=0,1$\nwe get\n \\begin{equation}\n \\label{thm1-3}\n \\nu\\left(\\mathcal{P}f^q \\log \\mathcal{P} f^q\\right)\\leq c \\nu\\left\\vert \\nabla_{\\Gamma_0}(\\mathcal{P}f^q)^\\frac{1}{q}\n\\right\\vert^q+c \\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_{0}} \\mathcal{P}f^q)^\\frac{1}{q}\n\\right\\vert^q+\\nu \\left(\\mathcal{P}^2f^q \\log \\mathcal{P}^2f^q\\right).\n\\end{equation}\nWorking similarly for the last term $\\nu \\left(\\mathcal{P}^2f^q \\log \\mathcal{P}^2f^q\\right)$  of \\eqref{thm1-3} and inductively for any term $\\nu (\\mathcal{P}^kf^q \\log \\mathcal{P}^kf^q)$, then after $n$ steps \\eqref{thm1-2} and \\eqref{thm1-3} will give\n\\begin{align}\n\\label{thm1-4}\n\\nonumber\\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right) &\\leq c \\sum_{k=0}^{n-1} \\nu \\left\\vert \\nabla_{\\Gamma_0}(\\mathcal{P}^kf^q)^\\frac{1}{q}\\right\\vert^q+c \\sum_{k=0}^{n-1} \\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_{0}} \\mathcal{P}^kf^q)^\\frac{1}{q}\\right\\vert^q \\\\\n& \\qquad + \\nu \\left(\\mathcal{P}^n f^q \\log\\mathcal{P}^n f^q\\right)-\\nu \\left(f^q \\log \\nu f^q\\right).\n \\end{align}\n \nIn order to deal with the first and second term on the right-hand side of \\eqref{thm1-4} we will use Lemma \\ref{lem3}.    If we apply inductively relationship \\eqref{lem3ineq}, for any $k\\in\\mathbb{N}$ we obtain\n\\begin{equation}\n\\label{thm1-5}\n\\nu\\left\\vert \\nabla_{\\Gamma_0}(\\mathcal{P}^kf^q)^\\frac{1}{q}\n\\right\\vert^q \\leq \\eta^{2k-1}D\\nu\\left\\vert \\nabla_{\\Gamma_1} f\n\\right\\vert^q+\\eta^{2k}\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q\n\\end{equation}\nand\n\\begin{equation}\n\\label{thm1-6}\n\\nu\\left\\vert \\nabla_{\\Gamma_1}(\\mathbb{E}_{\\Gamma_{0}}\\mathcal{P}^kf^q)^\\frac{1}{q}\n\\right\\vert^q \\leq \\eta^{2k}D\\nu\\left\\vert \\nabla_{\\Gamma_1} f\n\\right\\vert^q+\\eta^{2k+1}\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q.\n\\end{equation}\n \n \n \n \n \nUsing \\eqref{thm1-5} and \\eqref{thm1-6} in \\eqref{thm1-4} we see that\n\\begin{align}\n\\label{thm1-7}\n\\nonumber\\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right)& \\leq cD\\left(\\eta^{-1}  + 1\\right)\\left(\\sum_{k=0}^{n-1}\\eta^{2k}\\right)\\nu\\left\\vert \\nabla_{\\Gamma_1} f\\right\\vert^q \\\\\n& \\qquad + c\\left(1+\\eta\\right)\\left(\\sum_{k=0}^{n-1}\\eta^{2k}\\right)\\nu\\left\\vert \\nabla_{\\Gamma_0} f\\right\\vert^q \\nonumber \\\\ \n& \\qquad + \\nu \\left(\\mathcal{P}^n f^q \\log \\mathcal{P}^n f^q\\right)-\\nu (f^q \\log \\nu f^q).\n\\end{align}\n \n \n \n \n \n \nBy Lemma \\ref{lem4} we have that $\\lim_{n\\to \\infty}\\mathcal{P}^nf^q=\\nu f^q$,  $\\nu-a.s$.   Therefore, taking the limit as $n\\to\\infty$ in \\eqref{thm1-7} yields\n\\[\n\\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right) \\leq cD\\left(\\frac{1}{\\eta}+1\\right)K\\nu\\left\\vert \\nabla_{\\Gamma_1} f\n\\right\\vert^q+c(1 + \\eta)K\\nu\\left\\vert \\nabla_{\\Gamma_0} f\n\\right\\vert^q\n\\]\n where $K=\\sum_{k=0}^{\\infty}\\eta^{2k} = \\frac{1}{1-\\eta^2}$ for  $\\eta<1$.  Hence\n \\[\n\\nu \\left(f^q \\log\\frac{f^q}{\\nu f^q}\\right) \\leq C\\nu|\\nabla f|^q\n\\]\nfor $C = \\max\\left\\{ cD\\left(\\frac{1}{\\eta}+1\\right)K, c(1 + \\eta)K\\right\\}$, as required.\n\\end{proof}  \n\n\n\n\n\\bibliographystyle{siam}     \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nAround half of the abundances of the heaviest isotopes in the periodic table, including gold and europium, are produced through the rapid neutron-capture process \\citep[$r$-process,][]{Burbidge57,Cameron57}. Since the first discussion of the $r$-process in the 1950s, there has been debate over which astrophysical sites produce $r$-process material. \nRecently, the detection of an optical transient associated with the neutron star merger GW170817 \\citep{LIGOGW170817a,coulter17} provided strong evidence for $r$-process production in neutron star mergers \\citep[e.g.,][]{Drout17, Pian17}. Neutron star mergers thus appear to be a source of $r$-process elements, but it is unclear if they are the dominant source in the early universe. One concern stems from observations of $r$-process abundances in metal-poor ([Fe\/H] $< -2.5$) stars in the Galactic halo.\n\nIt is unclear whether the delay time to form and coalesce a binary neutron star system is too long to provide $r$-process material to near-pristine gas before the formation of metal-poor stars \\citep[e.g.,][]{Argast04,Skuladottir19,Cescutti15,Wehmeyer15,Haynes19,Kobayashi20}. Possible solutions include processes like inhomogeneous metal mixing or inefficient star formation mitigating the delay time \\citep[e.g.,][]{Ishimaru15,Shen15,vandeVoort15,RamirezRuiz15,Ji16b,Dvorkin20} or common envelope producing a large number of rapidly merging neutron star binaries \\citep[e.g.,][]{Beniamini16,Safarzadeh19,Zevin19,Andrews20}, but concerns have not been eradicated.\n\nNatal kicks received from the supernova explosions that give birth to neutron stars may have also made it unlikely for small, early galaxies to retain neutron star binaries \\citep{Bramante16,Beniamini16,Bonetti19}. For example, the highly $r$-process-enriched metal-poor stars in the ultra-faint dwarf galaxy Reticulum II could potentially be explained by a neutron star merger \\citep{Ji16b}, but the natal kick would have to have been very small ($v < v_{esc} \\sim 10-20 $ km s\\textsuperscript{-1}) and\/or the merger time extremely short to avoid kicking the binary out of the tiny galaxy \\citep{Tarumi20,Safarzadeh19b,Bramante16}. This is in contrast to larger estimates of $20-140$ km s$^{-1}$ based on the offset distribution of short-duration $\\gamma$-ray bursts from their host galaxies \\citep{Fong2013}, and $5-5450$ km s$^{-1}$ from galactic double neutron star systems \\citep{Wong2010}.\n\nIn light of these concerns --- coupled with the inference that the ejecta from GW170817 was dominated by an accretion disk wind, rather than dynamical tidal tails \\citep[e.g.][]{siegel2019b} --- \\citet{siegel19} revived the idea that collapsars (the supernova- and $\\gamma$-ray-burst-triggering collapse of rapidly rotating massive stars) may be an important source of $r$-process material (see also, e.g., \\citealt{MacFadyen99,McLaughlin2005}). In particular, the accretion disks formed in collapsars can have similar conditions to the $r$-process producing disk of GW170817. \\citet{siegel19} found that for accretion rates $\\gtrsim$ 10$^{-3}$ M$_\\odot$ s$^{-1}$, these disks produce neutron-rich outflows that synthesize heavy $r$-process nuclei. They also found that collapsars can yield sufficient $r$-process material to explain over 80\\% of the $r$-process content of the Universe. \nAlthough the electron fraction in collapsar disk winds is still debated \\citep{Surman06,Miller19}, this is currently one of the most promising ways for core-collapse supernovae (CCSN) to make $r$-process elements (other than magnetorotationally driven CCSN, e.g., \\citealt{Nishimura15}, but see \\citealt{Mosta18}).\n\n\nThe scatter in the abundances of metal-poor stars is a useful probe of different $r$-process origins.\nIn this paper, we investigate collapsars as a source of the $r$-process in the early universe by investigating whether they can self-consistently reproduce the scatter of europium (Eu, Z = 63) in the most metal-poor stars. Our results also hold for any prompt $r$-process site with a power law distribution of effective $r$-process yields. As our representation was directly inspired by collapsar properties for the purposes of studying $r$-process collapsars, though, we call the $r$-process site in our model ``collapsars\" and discuss alternative interpretations in more detail in Section \\ref{subsec:NSMvscollapsar}.\n\nOur model assumes the $r$-process material in metal-poor stars was formed exclusively in collapsars with stochastic $r$-process yields. Previous stochastic models primarily assume the $r$-process is produced in fixed amounts, but comes from multiple different sources and\/or mixes into different environments \\citep[e.g.,][]{Tsujimoto+Shigeyama14,Cescutti15,Wehmeyer15,Shen15}. \nIn contrast, our model assumes the $r$-process source has an intrinsically stochastic production: each collapsar synthesizes a different amount of $r$-process material.\n\nIn Section \\ref{sec:model}, we outline our stochastic collapsar enrichment model in which we assume each collapsar contributes an $r$-process yield that is independently drawn from a power law distribution, inspired by models of collapsar jet fits to $\\gamma$-ray burst data. Our model is constrained using stellar abundance data described in Section \\ref{sec:data}, and the parameter constraints are described in Section \\ref{sec:results}. The implications of these results are discussed in Section \\ref{sec:disc}, where we put our results in context with collapsar jet property distributions, different types of core-collapse supernovae, and different estimates for the amounts of $r$-process material which may be produced by collapsars. Our conclusions are summarized in Section \\ref{sec:conc}.\n\n\n\\section{Collapsar $r$-Process Yield Model} \\label{sec:model}\n\n\\begin{figure*}[t!]\n\\gridline{\n\t\\fig{schematic.pdf}{0.45\\textwidth}{(a) Schematic of our model. Core-collapse supernovae explode into the total gas, yielding iron (Fe), and some fraction of these are collapsars which also yield stochastic amounts of europium (Eu). This produces stars with different [Eu\/Fe] and [Fe\/H] abundances.}\n\t\\fig{stellar_iqr.pdf}{0.4\\textwidth}{(b) The $r$-process abundance scatter in the RPA data. The data is shown as dots (Eu detections) and triangles (Eu upper limits). The grey boxes show the estimated $\\text{IQR}_{\\text{Eu}}$ (the abundance scatter) in several metallicity bins, which decreases with increasing metallicity.}}\n\\caption{Schematic of our theoretical model and scatter plot of the stellar data our models attempts to reproduce. Our model attempts to reproduce the observed Eu scatter at low metallicity by assuming all Eu is produced by collapsars, which are a fraction of all core-collapse supernovae.\n\\label{fig:stellarscatter}}\n\\end{figure*}\n\n\nThe purpose of our model is to determine the distribution of $r$-process abundances (as measured by [Eu\/Fe]) in a fixed metallicity bin (as measured by [Fe\/H]). A schematic of the model can be seen in Figure \\ref{fig:stellarscatter}a. The novel feature of our model is that we explicitly study whether variable $r$-process yields from a single class of r-process events could produce observed abundance scatter. This is in contrast to previous models \\citep[e.g.,][]{Cescutti15,Ojima18,Shen15,vandeVoort15} which generally had a fixed yield per event and produced scatter through different $r$-process sites and\/or different galactic environments.\n\n\\subsection{Defining ``Collapsar''} \\label{subsec:definecollap}\n\nThe term ``collapsar'' typically refers to the collapse of a massive, rapidly-rotating star in which accretion onto a central black hole can produce a beamed jet, commonly evoked as the progenitors of long-duration $\\gamma$-ray bursts (LGRBs). In the model described below, we more broadly use the term to encompass a population of core-collapse supernovae that produce heavy r-process material with a power-law distribution of yields. \nThis definition is motivated by the traditional collapsar picture in which rapid accretion onto a compact object launches a collimated outflow wherein both the duration and luminosities of LGRBs are well-described by power laws \\citep{petropoulou17,sobacchi17}. Connecting jet properties to $r$-process production is inspired by the possible connection between the accretion phase during which $r$-process material is produced (due to a sufficiently high accretion rate that neutronizes the disk) and the phase during which the collapsar jet is launched. In particular, \\citet{siegel19} finds that the production of heavy r-process material requires $\\dot{M} \\gtrsim$ 10$^{-3}$ M$_\\odot$ s$^{-1}$, closely matched to the accretion rates required for jet production \\citep{MacFadyen99}. The results described below also hold for any prompt $r$-process site (e.g., occurring roughly concurrently with CCSN; this could potentially include fast-merging neutron stars) that lead to a power law distribution of heavy $r$-process material. We discuss other interpretations in Section \\ref{subsec:NSMvscollapsar}.\n\n\nIn addition, our model does not require that a jet successfully breaks out of the progenitor star. While the most extreme $r$-process producing events require large fallback accretion disks and are likely associated with LGRBs, our model also includes events that eject smaller amounts of $r$-process material. Such systems may produce weaker outflows\/jets and be observed as low-luminosity GRBs \\citep[ll-GRBs; e.g.,][]{Bromberg11,petropoulou17}, relativistic supernovae \\citep[e.g.,][]{Soderberg2010,Margutti2014}, or broad-lined Type Ic supernovae \\citep[Type Ic-BL; e.g.,][]{Milisavljevic2015,Modjaz2016}.\n\n\n\\subsection{Basic Physical Set-up and Model Parameters}\n\nWe analytically model the abundance distribution as arising from a burst of core-collapse supernova enrichment, some fraction of which are collapsars that produce non-zero $r$-process yields.\nEach core-collapse supernova produces an iron yield and each collapsar also produces an $r$-process yield that is independently and identically drawn from a power law distribution. We assume the typical star formed in these galaxies forms after the metal yields from all of the supernovae fall into and mix within the hydrogen gas of the system. During this process, some fraction of the metals are permanently lost from the galaxy due to gas outflows.\n\nThis model has five free parameters: \n\\begin{enumerate}\n\\item $N_{\\text{SN}}$: the number of core-collapse supernovae enriching the gas.\n\\item $\\avg{N_r}$: the average number of collapsars (i.e. supernovae that produce non-zero amounts of $r$-process material).\n\\item $M_{r,\\text{min}}$: the minimum mass of $r$-process material that can be produced by a collapsar. \n\\item $\\alpha$: the power law exponent for our power law distribution of $r$-process yield produced per collapsar.\n\\item $y_{\\rm{Fe,eff}}$: the effective iron yield per supernovae per unit gas mass. $y_{\\rm{Fe,eff}} = y_{\\text{Fe}}f_{\\text{retained}}\/M_{\\text{gas}}$, where $y_{\\text{Fe}}$ is the iron yield per supernova, $f_{\\text{retained}}$ is the fraction of iron retained in the galaxy and not carried out of the system by gas outflows, and $M_{gas}$ is the total gas mass in the system.\n\\end{enumerate}\n\nThese parameters combine to yield the mean gas metallicity, the the fraction of supernovae that are collapsars ($f_r$), and the average yield per collapsar ($\\avg{M_r}$), as described below.\n\n\\subsection{Gas Enrichment}\n\nWe determine the distribution of europium abundances produced by this model at a given metallicity. The mean metallicity of our stars is found from $M_{\\text{Fe}}\/M_{\\text{H}} = N_{\\text{SN}} y_{\\rm{Fe,eff}}$. The mass of iron and hydrogen is converted to [Fe\/H] using a mean molecular weight of $\\mu_{\\text{Fe}}=56$ for Fe, $\\log \\epsilon_\\odot(\\text{Fe}) = 7.50$, and $\\log \\epsilon_\\odot(\\text{H}) = 12.00$ \\citep{Asplund09}, where we use the stellar spectroscopist notation $\\mbox{[X\/Y]} \\equiv \\log N_{\\text{X}} \/ N_{\\text{Y}} - \\log \\left(N_{\\text{X}}\/N_{\\text{Y}}\\right)_\\odot = \\log(\\frac{M_{\\text{X}}\\mu_{\\text{Y}}}{M_{\\text{Y}}\\mu_{\\text{X}}}) - (\\log\\epsilon_\\odot(\\text{X}) - \\log\\epsilon_\\odot(\\text{Y}))$. \n\nIn order to determine the distribution of europium values, we enrich this gas with $N_r$ collapsars, which we draw stochastically from a Poisson distribution with mean $\\avg{N_r} = f_r N_{\\text{SN}}$.\nEach collapsar contributes an $r$-process yield $M_r$ that is independently drawn from a power law distribution.\n\n\\begin{equation}\n  p(M_r) \\propto \\left(\\frac{M_r}{M_{r,\\text{min}}}\\right)^{-\\alpha} \\quad M_r \\geq M_{r,\\text{min}}  \\label{eq:Mr}\n\\end{equation}\n\nThe $r$-process yield for a single explosion, $M_r$, can be converted to $M_{\\text{Eu}}$ by using the solar $r$-process mass fraction of europium compared to all nuclei with mass number A $> 70$. The mass fraction, $X_{\\text{Eu}}$, is approximately $10^{-3}$ ($1.75 \\times 10^{-3}$, \\citealt{Arnould07}; $9.77 \\times 10^{-4}$, \\citealt{Sneden08}). When converting total europium mass to [Eu\/Fe], we use a mean molecular weight of $\\mu_{\\text{Fe}}=152$ for Eu and $\\log \\epsilon_\\odot(\\text{Eu}) = 0.52$. We also assume that europium and iron have the same retention fraction, $f_{\\text{retained}}$, meaning the same fraction of both is lost from the galaxy.\n\nNote that if $\\alpha \\leq 2$, then the average yield produced per collapsar diverges and our model would also require an upper cutoff to the amount of r-process material that can be produced by a single collapsar, $M_{r,\\text{max}}$. However, when we compare to observed data in Section \\ref{sec:results}, it will turn out our results imply $\\alpha > 2$, in which case the average yield per collapsar is: \n\n\\begin{equation}\n\\avg{M_r} = M_{r,\\text{min}} \\frac{\\alpha-1}{\\alpha-2}\n\\end{equation}\n\nWhile in principle $y_{\\text{Fe}}$ is also stochastic, for simplicity we hold it constant. This is fine as long as $f_r$ is small, since variations in the Fe yield will average out.\n\nOperationally, we create a model [Eu\/Fe] distribution by considering several thousand instances of supernova enrichment. Each instance is a single data point in our modeled cumulative distribution function. For each instance, we draw an $N_r$ value and then draw $M_r$ for each of the $N_r$ collapsars. The total europium and iron masses retained in the galaxy in each instance are transformed into a [Eu\/Fe] measurement. We also add a 0.1 dex Gaussian uncertainty to mimic observational errors.\n\n\\subsection{Constraining Model Parameters: Literature Estimates for Effective Iron Yields} \\label{subsec:res_yfM}\n\nThe effective iron yield of core-collapse supernova per unit gas mass cannot be directly constrained from a sample of stellar abundance data. We constrain its value by combining estimates for each component parameter (recall $y_{\\rm{Fe,eff}} = y_{\\text{Fe}}f_{\\text{retained}}\/M_{\\text{gas}}$) from the literature. \n\nThe fraction of retained metals is set to $f_{\\text{retained}} = 10^{-2 \\pm 0.5}$,  assuming that metal-poor stars form early in small galaxies. Observationally, individual faint galaxies have $f_{\\text{retained}}$ in this range: the Milky Way's moderately faint dSphs (e.g., Ursa Minor) have kept less than 1\\% of their metals \\citep{Kirby11outflow}; while the faint but still star-forming galaxy Leo P has kept about 5\\% of its metals \\citep{McQuinn15}.\nTheoretically, retaining about 1\\% of metals in small galaxies reproduces the slope and normalization of the mass-metallicity relation \\citep[e.g.,][]{Dekel03,Robertson05}. The retention fraction is also borne out in hydrodynamic galaxy simulations \\citep[e.g.,][]{Emerick18}. \n\n$M_{\\text{gas}}$ is set by models of how supernovae dilute metals into a mixing mass of gas. For small, early galaxies that form metal-poor stars, the mixing mass is $M_{\\text{gas}} \\sim 10^6 \\, M_\\odot$ \\citep{2015MNRAS.454..659J}. The strict lower limit on this mass is the mass contained in a single final supernovae remnant, a minimum of around $\\sim10^{4.5}~M_\\odot$ \\citep[e.g., ][]{Magg20,Macias18}, with a range of average mixing masses for metal-poor stars of $10^{5}$ to $10^{8} M_\\odot$ of gas.\nFor systems with higher $M_{\\text{gas}}$, more metals are retained, resulting in a higher retention fraction $f_{\\text{retained}}$ (and vice versa).\n\nTo estimate an average iron yield from CCSN, we calculate a weighted average between observations of H-rich CCSN and H-poor CCSN. A detailed discussion can be found in Appendix~\\ref{subsec:yFe}, but find that the average yield is $y_{\\text{Fe}} \\approx 0.1 M_\\odot$, with the uncertainty in $f_{\\text{retained}}$ and $M_{\\text{gas}}$ far outweighing that of $y_{\\text{Fe}}$.\n\nAltogether, $y_{\\rm{Fe,eff}}$ has a wide range of possible values ($10^{-10} - 10^{-7}$), but our fiducial choice is $y_{\\rm{Fe,eff}} = 10^{-9}$. This choice is validated by an independent estimation of the frequency of $r$-process events in ultra-faint dwarf galaxies in Section \\ref{subsec:res_NSNfr}. We note, however, that there is tension between the values expected for $y_{\\rm{Fe,eff}}$ in very low mass galaxies based on the theoretical breakdown described in this section and comparisons to several external constraints. For example, the number of supernovae predicted in an ultra-faint dwarf galaxy using the Salpeter initial mass function suggests an effective iron yield closer to $\\sim10^{-7.5}$, and a simulation of extremely metal-poor ([Fe\/H] $=-3.42$) stars forming after a single supernova gives an estimated effective iron yield as high as $\\sim10^{-6.5}$ \\citep{Chiaki19}. This is not fully unexpected as $y_{\\rm{Fe,eff}}$ differs in different galaxies and the lowest mass galaxies will have the highest effective yields, but we note that this parameter remains uncertain and may trend higher than its fiducial value.\n\n\\subsection{Constraining Model Parameters: Fitting Stellar Abundance Data} \\label{subsec:model_stellar}\n\nAfter fixing the effective iron yield, stellar abundances are used to constrain the other model parameters. The stellar abundance data provides us effectively four observable quantities of interest:\n\n\\begin{enumerate}\n\\item the mean metallicity of the stars, $\\avg{\\text{[Fe\/H]}}$, \n\\item the mean $r$-process abundance, $\\avg{\\text{[Eu\/Fe]}}$, \n\\item the estimated fraction of stars that formed from gas not enriched by an $r$-process event, $f_0$, \n\\item the observed scatter in $r$-process abundance between stars, $\\text{IQR}_{\\text{Eu}}$. \n\\end{enumerate}\nRather than quantifying scatter with standard deviation, $\\sigma(\\text{Eu})$, we use the more robust interquartile range, a measure of statistical dispersion equal to the difference between the 75th and 25th percentiles, denoted $\\text{IQR}_{\\text{Eu}}$. Model parameters are then determined as follows: \n\n$\\avg{N_r}$ and $\\alpha$: The average number of collapsars, $\\avg{N_r}$, and the exponent of the $r$-process yield power law distribution, $\\alpha$, are determined by comparing the observed $f_0$ and $r$-process scatter, $\\text{IQR}_{\\text{Eu}}$, to those predicted by our models with varying $\\avg{N_r}$ and $\\alpha$. In a small metallicity bin, the shape of the [Eu\/Fe] distribution function is dependent on only these two parameters. The other potentially relevant parameters contribute only to shifting the distribution to higher or lower [Eu\/Fe]. By focusing on only the shape of the distribution, we can avoid making assumptions about any additional parameters when determining $\\avg{N_r}$ and $\\alpha$.\n\n$N_{SN}$ and $f_r$: The number of SN enriching the gas, $N_{SN}$, is determined from the mean metallicity of the stars and the effective iron yield using $\\avg{M_{\\text{Fe}}\/M_{\\text{H}}} = N_{SN} \\times y_{\\rm{Fe,eff}}$. The fraction of supernova that are collapsars, $f_r=\\avg{N_r}\/N_{SN}$, is then found by combining $N_{SN}$ and the average number of collapsars, $\\avg{N_r}$, from above.\n\n$M_{r,\\text{min}}$ and $\\avg{M_r}$: We first determine the average $r$-process yield produced per collapsar, $\\avg{M_r}$, using the relationship: $\\avg{M_r} f_{\\text{retained}} X_{\\text{Eu}} \\approx \\avg{M_{\\text{Eu}}} \/ \\avg{N_r}$. In this equation, $\\avg{M_{\\text{Eu}}}$ is found by considering the mean $r$-process abundance $\\avg{\\text{[Eu\/Fe]}}$ and mean metallicity $\\avg{\\text{[Fe\/H]}}$ in combination with $M_{\\text{H}} \\approx M_{\\text{gas}}$. The minimum $r$-process yield is then $M_{r,\\text{min}} = \\avg{M_r} \\frac{\\alpha-2}{\\alpha-1}$.\n\nWe do not attempt to model higher moments of the [Eu\/Fe] distribution beyond the mean and scatter because we expect selection effects in the data to dominate. We also do not attempt to model the shape of the distribution tails for both observational and theoretical reasons. Observationally, the low end of the [Eu\/Fe] distribution cannot be well known without a robust selection function. Theoretically, the low and high ends of our distribution are not robust due to our assumption that model stars form after all of the supernova yields have fallen into and mixed with the hydrogen gas. This is because our assumption precludes outlier stars that, for example, could have more or less europium due to inhomogeneous mixing.\n\nFuture work will address these concerns through a more detailed treatment of enrichment that incorporates our variable-yield work into a more complete picture that includes scatter due to differences in galaxy formation (e.g., different environments or metal mixing). This will allow for a better determination of whether our assumption of a power law is an appropriate shape for the $r$-process yield distribution and improved constraints on $\\alpha$ and $N_r$.\n\n\\section{Stellar Abundance Samples} \\label{sec:data}\n\n\\subsection{Sample Selection}\n\nWe use a stellar abundance sample from the $R$-process Alliance (RPA), a collection of detailed abundances of 601 halo stars \\citep{Hansen18,Sakari18,rpa3,RPA4}. The RPA stars are bright (V $<$ 13.5), metal-poor ([Fe\/H] $\\lesssim -2$) red giant stars in the Milky Way stellar halo. They were observed with a focus on obtaining a statistically complete sample of europium abundances. To verify the RPA data, we also consider a sample of 228 metal-poor red giant halo stars from \\citet{Roederer14b} \\citepalias[henceforth][]{Roederer14b}. Both of these samples report europium measurements or upper limits for every star.\n\nThe \\citetalias{Roederer14b} sample has [Eu\/Fe] abundances that are $0.22$ dex lower and [Fe\/H] abundances that are $0.19$ dex lower from other samples due to using a much cooler effective temperature scale and isochrone-based surface gravities \\citep{2014MNRAS.445.2946R}. We thus shift the reported measurements up by these amounts when plotting in Figure \\ref{fig:starCDF} and reporting values in Table \\ref{tab:CDF}.\n\nWe restrict most of our analysis to very metal-poor ([Fe\/H] $< -2.5$) stars, and the highest metallicity we consider is [Fe\/H] $< -1.75$ (when analyzing the evolution of the Eu scatter with increasing metallicity in Section \\ref{subsec:aN}). \nWe only consider stars with barium-to-europium abundance ratios that could be produced by the $r$-process ($-0.9 \\lesssim $ [Ba\/Eu] $ \\lesssim -0.4$). [Ba\/Eu] higher than $\\sim -0.4$ indicates contamination from the $s$-process, another nucleosynthetic process which forms europium. The solar $r$-process barium-to-europium ratio is [Ba\/Eu] $ \\approx -0.8$ \\citep{Sneden08}, and stars with much lower [Ba\/Eu] cannot be explained by the $r$-process pattern. We note that small variations in these purity cuts do not significantly change our results.\n\nTaking into account these restrictions (with [Fe\/H] $<-2.5$), the RPA sample includes 83 stars with Eu measurements and an additional 11 stars with Eu upper limits. The \\citetalias{Roederer14b} sample includes 36 stars with Eu measurements and 4 with Eu upper limits. The RPA sample (up to [Fe\/H] $<-1.75$) and its $\\text{IQR}_{\\text{Eu}}$ in different metallicity bins can be seen in Figure \\ref{fig:stellarscatter}b.\n\n\\subsection{Construction of Statistical Distributions}\n\n\n\\begin{figure*}[!htb]\n\\plotone{rpa_roederer_kme.pdf}\n\\caption{Cumulative distribution functions for the RPA and \\citetalias{Roederer14b} samples. Both CDFs are determined using the Kaplan-Meier estimator, which takes into account detections and upper limits to estimate the true distribution. The shaded regions show 95\\% confidence on the CDF estimate. Grey lines outline the interquartile range (25\\%-75\\%) for the RPA Kaplan-Meier CDF. The CDFs have been extended to the \\textit{y}-axis to show the estimated fraction of stars in each sample that have no $r$-process elements.\n\\label{fig:starCDF}}\n\\end{figure*}\n\n\\begin{table*}[!htb]\n\\centering\n\\caption{Interquartile ranges and fraction of stars formed from gas with no $r$-process enrichment for different [Eu\/Fe] CDFs from observational stellar samples with [Fe\/H] $< -2.5$. The distributions can be seen in Figure \\ref{fig:starCDF}. $\\text{IQR}_{\\text{Eu}}$ uncertainties are due to both KME confidence levels and uncertain observations. The $f_0$ values are upper limits as the distribution could continue to lower [Eu\/Fe] with lower $f_0$.\n} \\label{tab:CDF}\n\\begin{tabular}{c|c|c|c|c}\n\\tablewidth{0pt}\n\\hline\n\\hline\nStellar Abundance Sample & $\\text{IQR}_{\\text{Eu}}$ & $f_0$ & $\\avg{\\text{[Eu\/Fe]}}$ & $\\avg{\\text{[Fe\/H]}}$ \\\\\n\\hline\nRPA & $0.50^{+0.15}_{-0.10}$ & $0.04^{+0.10}_{-0.04}$ & $0.3^{+0.1}_{-0.1}$ & $-2.7^{+0.1}_{-0.1}$ \\\\\nR14 & $0.38^{+0.51}_{-0.14}$ & $0.04^{+0.11}_{-0.04}$ & $0.2^{+0.2}_{-0.2}$ & $-2.9^{+0.1}_{-0.1}$ \\\\\n\\hline\n\\end{tabular} \n\\end{table*}\n\n\nTo combine the mixture of measurements and upper limits into a statistical distribution of europium for each sample, we employ survival statistics, a branch of statistics that deals with censored datasets, e.g., upper limits. The most general single variate survival statistic is the Kaplan-Meier estimator (KME), which provides a non-parametric maximum likelihood estimate of a distribution from observed data. The Kaplan-Meier estimator and survival statistics have been used extensively in astronomical literature \\citep[e.g.][]{upperlimits1,upperlimits2,wardle86,Simcoe04}. We use the \\texttt{KaplanMeierFitter} from the survival analysis python package \\texttt{lifelines} \\citep{lifelines}. For this estimate to be valid, two assumptions about the distribution of upper limits must hold. First, the upper limits should be independent of each other, which is true here as the stars are independent. Second, the upper limits should be random -- i.e., the probability that a measurement will be censored should not correlate with the measurement value itself. This assumption may not hold because lower [Eu\/Fe] values are more likely to be censored.\nIdeally, we would fully forward model and censor our theoretical results, but that requires many additional assumptions including a completeness function (probability of measuring any value given [Eu\/Fe]), an error function (the value we measure for [Eu\/Fe] given its true value), and an upper limit function (the probability of setting a [Eu\/Fe] upper limit at a specific value given its true value). Fully forward modeling the observational sample is beyond the scope of this paper. We thus use the Kaplan-Meier estimate while keeping in mind that this may not be a perfect estimate.\n\nFigure \\ref{fig:starCDF} shows the [Eu\/Fe] cumulative distribution functions for the RPA and R14 samples. The interquartile range, $\\text{IQR}_{\\text{Eu}}$, differs slightly for the different samples but is consistent within the uncertainty. The mean [Eu\/Fe] and [Fe\/H] also differ slightly. The zero-limit $f_0$, the estimated fraction of stars that formed from gas that was not enriched by an $r$-process event, is the same in both samples. In our model, $f_0$ is the fraction of stars with no europium enrichment ([Eu\/Fe] $= -\\infty$), but we cannot identify if real stars have no $r$-process enrichment (and stars could receive trace amounts of europium enrichment through other processes despite the [Ba\/Eu] cuts we applied to purify our sample). We thus estimate $f_0$ in the data by taking the lowest CDF value from the observed distribution as estimated by survival statistics. This assumes that the CDF immediately plateaus at lower [Eu\/Fe] instead of continuing to decrease. Because the distribution could continue to decrease with lower [Eu\/Fe], the observed $f_0$ values are upper limits. Realistically, the real distribution certainly does not fully plateau even if our $f_0$ estimate is correct because of the possible other trace sources of europium, but for the purposes of this analysis and because we cannot estimate the CDF to extremely low [Eu\/Fe] regardless, we ignore those minor effects. These values are shown in Table \\ref{tab:CDF}.\n\n\\section{Results} \\label{sec:results}\n\nWe use the stellar abundance data to constrain the model parameters. The results are summarized in Table \\ref{tab:results}.\n\n\\begin{table*}[!htb]\n\\centering\n\\caption{Model parameters determined from observations. The wide ranges of $N_{\\text{SN}}$, $M_{r,\\text{min}}$, and $y_{\\rm{Fe,eff}}$ encompass broad uncertainty in the fraction of metals retained in each galaxy and each galaxy's gas mass. To be thorough we include these full ranges.\nWe also validate our fiducial values for $y_{\\rm{Fe,eff}}$, $f_r$, and $\\avg{N_r}$ (which also validates $N_{\\text{SN}}$, $M_{r,\\text{min}}$, and $\\avg{M_r}$). For $\\alpha$, the full range of values produce similar distribution shapes. Derived parameter values are shown below the double line.} \\label{tab:results}\n\\begin{tabular}{c|c|c|c}\n\\tablewidth{0pt}\n\\hline\n\\hline\n  & Description & Range & Fiducial Value \\\\\n\\hline\n$N_{\\text{SN}}$ & \nNumber of core-collapse supernovae & $30-30000$ & $3000$ \\\\\n\\hline\n$\\avg{N_r}$ & Average number of $r$-process collapsars & $2 - 4$ & 3 \\\\\n\\hline\n$M_{r,\\text{min}}$ & Minimum $r$-process yield produced per collapsar (see Eq. \\ref{eq:Mr}) & $3\\times10^{-4} - 3\\times10^{-1}$ & $3\\times10^{-2}$ \\\\\n\\hline\n$\\alpha$ & Power law exponent of $M_r$ distribution (see Eq. \\ref{eq:Mr}) & $2.2 - 6$ & $2.8$ \\\\\n\\hline\n$y_{\\rm{Fe,eff}}$\\footnote{determined from literature values} & Effective supernovae iron yield into the total gas mass, $y_{\\text{Fe}}f_{\\text{retained}}\/M_{\\text{gas}}$ & $10^{-10} - 10^{-7}$ & $10^{-9}$ \\\\\n\\hline\n\\hline\n$f_r$ & Fraction of supernovae that are collapsars, $\\avg{N_r}\/N_{\\text{SN}}$ & $10^{-4} - 10^{-1}$ & $10^{-3}$ \\\\\n\\hline\n$\\avg{M_r}$ & Average $r$-process yield produced per collapsar & $7\\times10^{-4} - 7\\times10^{-1}$ & $7\\times10^{-2}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{$\\avg{N_r}$ and $\\alpha$} \\label{subsec:aN}\n\nWe use the model described in Section \\ref{sec:model} to calculate theoretical cumulative distribution functions (CDF) of stellar [Eu\/Fe] abundances. CDFs resulting from different representative choices of $\\avg{N_r}$ and $\\alpha$ can be seen in Figure \\ref{fig:modelCDFs}. Each model CDF has an arbitrary offset that shifts the CDF left or right for plotting purposes. Recall that $\\avg{N_r}$ and $\\alpha$ can be constrained using only the shape of the distribution (i.e., the $\\text{IQR}_{\\text{Eu}}$ and $f_0$). A higher $\\avg{N_r}$ causes both a lower $f_0$ since fewer stars will form from un-enriched gas and a narrower distribution due to the central limit theorem. A higher $\\alpha$  also narrows the distribution by increasing the rarity of high $M_r$. When constraining these parameters with the $\\text{IQR}_{\\text{Eu}}$, a higher $\\avg{N_r}$ thus corresponds to a lower $\\alpha$ and vice versa.\n\n\\begin{figure*}[!htb]\n\\centering\n\\includegraphics[width=0.875\\linewidth]{model_CDF.pdf}\n\\caption{Stellar [Eu\/Fe] abundance cumulative distribution functions (colored lines) for models with different $\\avg{N_r}$ and $\\alpha$ values. The observed [Eu\/Fe] CDF for the RPA sample is shown in black with grey uncertainty. Our fiducial model, $\\avg{N_r}=3$ and $\\alpha=2.8$, is shown in solid blue in both plots. When either $\\avg{N_r}$ or $\\alpha$ is not specified, the fiducial value is used. The model CDFs have an arbitrary offset to shift the distribution left or right for plotting purposes, so only the shape (i.e., the IQR and zero-fraction) is relevant.\n\\label{fig:modelCDFs}}\n\\end{figure*}\n\n\\begin{figure*}[!htb]\n\\gridline{\n\t\\fig{model_f0s_labels.pdf}{0.4\\textwidth}{(a) Values of the zero-fraction, $f_0$, for different models. The heat map colors are normalized to $f_0 = 0.04^{+0.10}_{-0.04}$, the $f_0$ of the RPA stellar abundance sample (see Table \\ref{tab:CDF}). Red is higher than observed $f_0$, blue is lower. $\\avg{N_r}$ alone affects the $f_0$ value.}\n\t\\fig{model_IQRs_labels.pdf}{0.4\\textwidth}{(b) Values of the IQR for different models. The heat map colors are normalized to $\\text{IQR}_{\\text{Eu}} = 0.50^{+0.15}_{-0.10}$, the IQR of the RPA stellar abundance sample (see Table \\ref{tab:CDF}). Red is higher than observed IQR, blue is lower. For $\\avg{N_r}=1$, we determined the IQR by assuming symmetry and doubling the 50\\%-75\\% range (because the distribution is always above 25\\%).}}\n\\caption{Heat maps showing how the cumulative distribution IQR and zero-fractions of our model vary with $\\avg{N_r}$ and $\\alpha$. Black boxes outline the parameter combinations that can explain the stellar data within observational uncertainty (using the RPA sample; see Table \\ref{tab:CDF}). Note: The plotted $\\alpha$ values increase by 0.2 until $\\alpha = 3.0$, at which point they increase by 0.5 due to increasingly slower variation in $\\text{IQR}_{\\text{Eu}}$. \\label{fig:modelHeatmaps}\n}\n\\end{figure*}\n\n$\\avg{N_r}$, the average number of $r$-process collapsars enriching our stellar population, is constrained by the estimated fraction of stars which formed from gas not enriched by an $r$-process event, $f_0$. For $f_0 = 0.04$, $\\avg{N_r} = 3$ is the best fit value. Figure \\ref{fig:modelHeatmaps}a shows how the value of $f_0$ changes with $\\avg{N_r}$, independently of $\\alpha$. In this figure, the black boxes outline the parameter values which explain the observed $f_0$ or $\\text{IQR}_{\\text{Eu}}$.\n$\\avg{N_r} = 2$ to 4 can also explain observations. Note that the observed $f_0$ is an upper limit as the distribution could smoothly continue to lower [Eu\/Fe] with a lower $f_0$. The constraint on $\\avg{N_r}$ from $f_0$ is thus a lower bound.\n\n\\begin{figure*}[!htb]\n\\gridline{\\fig{scatter_evolution_Nr_3_lines.pdf}{0.45\\textwidth}{(a) The evolution of the Eu scatter with increasing metallicity can be well explained by our fiducial model choices of $\\avg{N_r}=3$ at $\\avg{\\text{[Fe\/H]}} = -2.7$ (the mean metallicity of our RPA sample) and $\\alpha = 2.8$.}\n\t\\fig{scatter_evolution_Nr_30_lines.pdf}{0.45\\textwidth}{(b) If $\\avg{N_r}$ is increased to $\\avg{N_r} = 30$ at $\\avg{\\text{[Fe\/H]}} = -2.7$, no single choice of $\\alpha$ well explains the evolution of the Eu scatter. Higher $\\avg{N_r}$ choices result in poorer matches to observations.}}\n\\caption{The decrease in the [Eu\/Fe] scatter with higher metallicity seen in the data (hollow circles and squares) is reproduced by our model (colored dots). Each [Fe\/H] bin of 0.3 dex corresponds to approximately a factor of 2 increase in supernovae, hence why we double the number of $r$-process collapsars in each bin. Reproducing the evolution in the scatter at higher metallicity as well as low metallicity increases confidence in our fiducial model choices of $\\avg{N_r}$ and $\\alpha$. \\label{fig:scatterevol}\n}\n\\end{figure*}\n\nTo validate our fiducial value of $\\avg{N_r}=3$, we also reproduce the evolution in europium scatter with increasing metallicity. This gives an upper bound to the constraint. We examine the RPA stellar abundance sample in several metallicity bins (up to [Fe\/H]  $= -1.65$; see Figure \\ref{fig:scatterevol}). We compare model scatter to the scatter of the RPA distribution as determined by both the Kaplan-Meier estimator (which takes into account europium detections and upper limits) and as determined by only europium detections. The \\citetalias{Roederer14b} sample is excluded from this plot because it has too few stars in each bin to determine distributions.\n\nAs metallicity increases, $\\avg{N_r}$ should increase linearly, but the scatter should decrease with $\\sqrt{\\avg{N_r}}$. Reproducing the $\\text{IQR}_{\\text{Eu}}$ in several metallicity bins thus suggests our model uses the correct $\\avg{N_r}$. When binning on metallicity, our model with $\\avg{N_r}=3$ at $\\avg{[\\text{Fe\/H}]} =-2.7$ well reproduces the observed decrease in scatter. If we increase $\\avg{N_r}$ by a factor of 10 or more, our model no longer well reproduces the observed decrease in scatter unless $\\alpha$ is allowed to vary with metallicity. Considering all uncertainty from the Kaplan Meier Estimator, the upper bound is $\\avg{N_r} \\approx 50$, but the model favors a much lower $\\avg{N_r}$. This suggests our fiducial $\\avg{N_r}=3$ is roughly correct despite being a lower bound.\n\nTo be thorough, we also explore the extreme case where \\textit{all} core-collapse supernovae result in $r$-process collapsars -- i.e., all core-collapse supernovae form an accretion disk that is able to synthesize a non-zero amount of $r$-process material ($\\avg{N_r} = N_{\\text{SN}}$ and $f_r=1$). We consider the case where $\\avg{N_r}=3000$, where 3000 is our fiducial value of $N_{\\text{SN}}$ (see Section \\ref{subsec:res_NSNfr}). In this extreme case, the vast majority of collapsars would produce extremely small amounts of $r$-process material. As seen in Figure~\\ref{fig:CDFfr1}, this extreme case can explain the observed $\\text{IQR}_{\\text{Eu}}$ with a scatter of $\\text{IQR}_{\\text{Eu}}=0.45$ for $\\alpha = 1.8$. The upper bound set on $\\avg{N_r}$ by the evolution of scatter with metallicity (Figure \\ref{fig:scatterevol}) disfavors this model, however. The situation where all core-collapse supernovae produce $r$-process is only favored if $N_{\\text{SN}}$ is below 30, lower than even our most extreme $N_{\\text{SN}}$ value.\nThis extreme model also does not reproduce the observed distribution at low [Eu\/Fe] as well as the fiducial model, though we note that the tails of the observed distribution are less trustworthy than the $\\text{IQR}_{\\text{Eu}}$. We thus keep $\\avg{N_r}=3$ for our results.\n\n$\\alpha$, the exponent of the $r$-process yield power law distribution, is constrained by the $\\text{IQR}_{\\text{Eu}}$ value of the distribution, which varies with both $\\alpha$ and $\\avg{N_r}$ as shown in Figure \\ref{fig:modelHeatmaps}b. For $\\avg{N_r} = 3$ and $\\text{IQR}_{\\text{Eu}} = 0.50^{+0.15}_{-0.10}$, the constrained value is $\\alpha = 2.8^{+4.2}_{-0.6}$.\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{CDF_fr1.pdf}\n\\caption{Distribution for an extreme model where all core-collapse supernovae produce some $r$-process material ($f_r=1$). This extreme model can explain the observed $\\text{IQR}_{\\text{Eu}}$. It cannot well explain the evolution of scatter with increasing metallicity, however (e.g., Figure \\ref{fig:scatterevol}). It also cannot well explain the low [Eu\/Fe] tail. For this model, the minimum amount of $r$-process material per collapsar is extremely small, $M_{r,\\text{min}} \\approx 10^{-6} M_\\odot$. \\label{fig:CDFfr1}}\n\\end{figure}\n\n\\subsection{$N_{\\text{SN}}$ and $f_r$} \\label{subsec:res_NSNfr}\n\nThe number of supernovae, $N_{\\text{SN}}$, is linearly related to $y_{\\rm{Fe,eff}}$. To explain the mean metallicity $\\avg{\\text{[Fe\/H]}}=-2.7 \\pm 0.1$ (using the method described in Section \\ref{subsec:model_stellar}), on the extreme ends of $y_{\\rm{Fe,eff}}$ values we need anywhere from 30 to 30,000 supernovae; lower $y_{\\rm{Fe,eff}}$ corresponds to higher $N_{\\text{SN}}$ as more supernovae are needed to explain the mean metallicity. For our fiducial value of $y_{\\rm{Fe,eff}} = 10^{-9}$, $N_{\\text{SN}} \\approx 3000$.\n\nWith values for $\\avg{N_r}$ (from Section \\ref{subsec:aN}) and $N_{\\text{SN}}$, we can determine the fraction of supernovae that result in $r$-process material producing collapsars, $f_r = \\avg{N_r}\/N_{\\text{SN}}$. Considering the extremes of the possible values of $y_{\\rm{Fe,eff}}$, $f_r \\approx 0.0001$ to $0.1$. For our fiducial values ($\\avg{N_r} = 3$ and $y_{\\rm{Fe,eff}} = 10^{-9}$), $f_r \\approx 0.001$.\n\nTo validate our fiducial choice of $y_{\\rm{Fe,eff}}$, we also estimate $f_r$ using observations of ultra-faint dwarf galaxies around the Milky Way. There are now high-resolution spectroscopic abundances for stars in 19 surviving ultra-faint dwarfs.\nOf these, three of the dwarfs (Grus~II, Reticulum~II, and Tucana~III) exhibit $r$-process enrichment \\citep{Hansen20_Grus,Ji16b,Hansen17}. Since these are extremely small systems, we assume each of these three dwarfs experienced one $r$-process event (as in \\citealt{Ji16b,Brauer19}), and then estimate the total number of supernovae that contributed to all of their stellar populations to estimate $f_r$. We combined literature values of their absolute magnitudes $M_V$ \\citep{Munoz18,Torrealba18,Drlica15,Bechtol15,Mutlu18} with a Salpeter individual mass function that predicts $0.02 L_0$ supernovae where $L_0$ is the present-day luminosity in $L_\\odot$ \\citep{Ji16b}. The ultra-faint dwarfs cumulatively experienced about $1800$ supernovae. The fraction of supernovae that result in $r$-process material producing collapsars is thus $f_r \\sim 3\/1800 = 0.002$. This validates our fiducial model values of $f_r \\approx 0.001$ and $y_{\\rm{Fe,eff}} \\approx 10^{-9}$. We note again, however, that there is tension between our fiducial estimate of $y_{\\rm{Fe,eff}}$ and several external constraints in very low mass galaxies, as discussed in Section \\ref{subsec:res_yfM}, so we continue to report the full uncertainty in these parameters.\n\n\\subsection{$M_{r,\\text{min}}$}\n\nThe minimum $r$-process yield produced per collapsar, $M_{r,\\text{min}}$ (see Eq. \\ref{eq:Mr}), depends on $\\alpha$ and varies linearly with $y_{\\rm{Fe,eff}}$. To transform between total $r$-process yield (nuclei with A $\\geq 70$) and europium yield, we use the solar $r$-process europium mass fraction $X_{\\text{Eu}} \\approx 10^{-3}$. To explain the observed mean europium-iron abundance ratio $\\avg{\\text{[Eu\/Fe]}}$, on the extreme ends of $y_{\\rm{Fe,eff}}$ values, we find that $M_{r,\\text{min}} \\approx 0.0003 - 0.3 M_\\odot$; lower $y_{\\rm{Fe,eff}}$ corresponds to higher $M_{r,\\text{min}}$ both because a lower $f_{\\text{retained}}$ causes less europium to be retained in the galaxy and because higher $M_{\\text{gas}}$ requires a higher mass of iron and europium to explain the mean [Fe\/H] and [Eu\/Fe] abundances. For our fiducial values of $y_{\\rm{Fe,eff}} = 10^{-9}$, $\\avg{N_r} = 3$, and $\\alpha = 2.8$, we find $M_{r,\\text{min}} \\approx 0.03 M_\\odot$, or a mean $r$-process yield per collapsar of $\\avg{M_r} \\approx 0.07 M_\\odot$.\n\nIn the extreme case where all core-collapse supernovae produce a nonzero amount of $r$-process material (Figure \\ref{fig:CDFfr1}), the minimum amount of $r$-process material per collapsar would be extremely small, $M_{r,\\text{min}}\\approx 10^{-6} M_\\odot$ for $\\avg{N_r} \\approx 3000$. This situation is disfavored because it does not reproduce the observed decrease of Eu scatter with increasing metallicity unless $y_{\\rm{Fe,eff}}$ is much higher than our fiducial value.\n\n\\section{Discussion} \\label{sec:disc}\n\nUsing stellar abundance data to constrain parameters in our stochastic collapsar chemical enrichment model produces a self-consistent physical picture, which was not guaranteed a priori. We now discuss this in more detail and place our results in context with other potentially physically relevant values. We also discuss the limitations of this model in Section \\ref{subsec:caveats}.\n\n\\begin{figure*}[t!]\n\\plotone{constraints_new.pdf}\n\\caption{Constraints on $\\avg{M_r}$, $f_r$, and $\\alpha$ from our model (blue) in context of potentially physically relevant values (black dotted lines). For descriptions of the reference values, see Section \\ref{sec:disc}. Our fiducial model values are plotted as a blue dot, while the dark blue shaded region represents an order of magnitude uncertainty around our fiducial values and the light blue shaded region represents the full uncertainty.  \\label{fig:constraints}}\n\\end{figure*}\n\n\n\\subsection{Implications of $f_r$: The Fraction of CCSN that Produce Collapsars} \\label{subsec:disc_fr}\n\nWhat fraction of core-collapse supernovae (CCSN) produce collapsars? Recall that our definition of collapsar is motivated by physical picture in which a rapid fallback accretion onto a black hole simultaneously produces heavy r-process material via accretion disk winds and launches a collimated outflow, but does \\emph{not} require that a jet successfully break out of the progenitor star. The power law distribution yields adopted in Section~\\ref{sec:model} naturally includes less extreme explosions that produce smaller amounts of $r$-process material via disk outflows.\n\nTo reproduce the observed scatter in [Eu\/Fe] abundances at low metallicity with a collapsar-like model, we require that the fraction of CCSN producing $r$-process material is between 10$^{-4}$ and 10$^{-1}$ with a fiducial value of 10$^{-3}$. We now compare these values to the observed rates for various classes of transients that have previously been proposed to be powered by collapsars or jet-driven explosions.\n\nWe begin with long-duration $\\gamma$-ray bursts (LGRBs). Current measurements of the local (z=0) rate for LGRBs beamed towards Earth from the \\emph{Swift} satellite range from $1.3^{+0.6}_{-0.7}$ Gpc$^{-3}$ yr$^{-1}$ for L$>$10$^{50}$ ergs \\citep{Wanderman2010} to $0.42^{+0.9}_{-0.4}$ Gpc$^{-3}$ yr$^{-1}$ accounting for complex \\emph{Swift} trigger criteria \\citep{Lien2014}. Correcting these for a canonical beaming factor of 50 \\citep{Guetta2005}, results in inferred intrinsic rates of $65^{+0.30}_{-0.35}$ Gpc$^{-3}$ yr$^{-1}$ and $21^{+4.5}_{-2}$ Gpc$^{-3}$ yr$^{-1}$, respectively. Comparing these to the local CCSN rate from the Lick Observatory SN Search (LOSS) of 0.705 ($\\pm$0.089) $\\times$10$^{-4}$ Mpc$^{-3}$ yr$^{-1}$ \\citep{Li2011} yields R$_{\\rm{LGRB}}$\/R$_{\\rm{CCSN}}$ values of (2$-$9)$\\times$10$^{-4}$. This range of values is only slightly lower than our fiducial value of $f_r$. LGRBs could thus be linked to $r$-process production. Our uncertainty on $f_r$ errs toward a higher value for the $r$-process fraction, though, so $f_r$ could very well be larger than $f_{\\text{LGRB}}$. In that case, we would require that massive stars beyond those that launch successful GRBs form accretion disks with physical conditions capable of producing heavy $r$-process material.\n\nIn particular, Type Ic-BL supernovae are a class of hydrogen-poor SN that display high ejecta velocities (hence ``broad-lined'') and kinetic energies ($\\sim$10$^{52}$ ergs) for which central engines are commonly evoked. While the nature of the central engine is still debated \\citep{Thompson2004,MacFadyen99,Barnes2018}, the fact that all SN observed in association with LGRBs have been Type Ic-BL supernovae has lead to the hypothesis that all events of this class are powered by jets. Differences in the detailed manifestation of these explosions (LGRBs, low-luminosity GRBs, relativistic SN, or ``ordinary'' Ic-BL SN) would then be driven by a distribution of engine timescales or progenitor radii \\citep[e.g.][]{Lazzati2012,Margutti2014}. We therefore compare our constraints on $f_r$ to the rates of Type Ic-BL SN to investigate if they are consistent with all Type Ic-BL SN harboring collapsar engines.\n\nBased on the full LOSS sample, \\citet{2017PASP..129e4201S} find that Type Ic-BL SN account for a fraction of (1.1 $\\pm$ 0.8) $\\times 10^{-2}$ of CCSN.  However, the LOSS sample was a targeted survey, biased towards high metallicity galaxies and it is well established that Type Ic-BL SN show a preference for low metallicity environments \\citep[e.g.][]{Modjaz2020}. It is therefore possible that Type Ic-BL SN represent a higher fraction of all CCSN at low metallicity, which is what our parameter $f_r$ actually constrains. Unfortunately, to date, there has been no untargeted, volume-limited study that examines the fraction of CCSN that are Type Ic-BL at low-metallicity. \\citet{Graur2017b} and \\citet{Arcavi2010} examine relative rates of different core collapse SN subtype in ``high'' and ``low'' mass galaxies for the LOSS and early PTF samples, respectively. \\citet{Graur2017b} find no significant difference in the Type Ic-BL fraction (1-2\\%), while \\citet{Arcavi2010} find that Type Ic-BL may make up a significantly higher fraction of all SN ($\\sim 10-13\\%$) in low luminosity galaxies. We caution, however, that both samples contain only 2-3 Type Ic-BL events and are therefore dominated by low number statistics. More recently, \\citet{2020arXiv200805988S} investigate the host galaxies of the full sample of 888 SN identified by PTF, including 36 Type Ic-BL. They find that Type Ic-BL production is significantly stifled above a galaxy mass of $\\log{M\/M_\\odot} = 10$, with Type Ic-BL comprising $\\gtrsim 5$\\% of their observed CCSN sample below this threshold compared to $\\lesssim 2$\\% above.\n\nR$_{\\rm{Ic-BL}}$\/R$_{\\rm{CCSN}}$ values of 0.01--0.1 fall within the range of $f_r$ found by our model (see Figure~\\ref{fig:constraints}). However, the latter is at the extreme high end, implying that while our model is consistent with all Type Ic-BL SNe producing europium, it favors a scenario in which $\\lesssim$10\\% do. We note that this would not preclude the possibility that all Type Ic-BL SNe harbor jets, but rather require that some lack the accretion disk properties necessary for the production of \\emph{heavy} r-process material. This could imply that a subset of Type Ic-BL SNe (a) harbor accreting black holes, but do not reach sufficiently high accretion rates ($> 10^{-3}$ M$_\\odot$ s$^{-1}$) to proceed past $^{56}$Ni-rich outflows \\citep{siegel19}, or (b) harbor magnetar central engines for which neutrino irradiation can limit neucleosynthesis from disk ejecta to the light r-process \\citep[e.g.][]{Margalit17,Radice18}.\n\nFor comparison, we also calculate a rough effective rate of neutron star mergers per CCSN. The cosmic NSM rate from the second LIGO-Virgo gravitational wave transient catalog is $320^{+490}_{-240}$ Gpc$^{-3}$ yr$^{-1}$ \\citep{2020arXiv201014533T}. When comparing this to the LOSS rate of galactic CCSN, the estimated NSM fraction is $4.5^{+7.0}_{-3.4} \\times 10^{-3}$. This is higher than our fiducial value of $f_r$, but within model uncertainties. The LIGO rate of NSMs could thus potentially account for the rate of $r$-process events required by our model to explain metal-poor star abundances, though it is not favored by our fiducial results. We also note that the rate of NSMs in the early universe likely differs from the rate found by LIGO, and that NSMs would need to be fast-merging to be described by our model.\n\n\n\\subsection{Implications of $\\avg{M_r}$: The Amount of $r$-Process Yield Produced per Collapsar}\n\nOur determination of the minimum and average amounts of $r$-process yield produced per collapsar ($M_{r,\\text{min}} \\approx 0.03 M_\\odot$ and $\\avg{M_r} \\approx 0.07 M_\\odot$, respectively) is based entirely on our analysis of RPA stellar abundance data, independent of any previous estimates in literature of the amount of $r$-process material that might be produced by such events. To place our results in context, we compare them to several reference estimates of $r$-process yields from single events (see Figure \\ref{fig:constraints}). Note again that we define $r$-process yield as the yield of nuclei with mass number A $\\geq 70$.\n\n\\citet{siegel19} demonstrated that accretion disk outflows in collapsars could produce significant amounts of $r$-process material. For different presupernova models, they found the amount of europium varied from $6.0 \\times 10^{-6} \\, M_\\odot$ to $5.8 \\times 10^{-4} \\, M_\\odot$, or $M_r = 0.006$ to $0.579 \\, M_\\odot$ (for our definition $M_r$). Their fiducial model corresponds to $M_r = 0.27  M_\\odot$. Their fiducial yield is about four times larger than our fiducial average yield, but our $M_{r,\\text{min}}$ and $\\avg{M_r}$ values fall within their range of yields.\nIn Figure \\ref{fig:constraints}, the shaded region correspond to the spread of $r$-process yields found by the \\citet{siegel19} simulations.\n\nFurthermore, if we assume that the isotropic energy of a $\\gamma$-ray burst roughly traces the amount of $r$-process yield, we can compare the energies of LGRBs to that of GW170817 to estimate the $M_r$ from collapsars in which the associated jet successfully breaks out of the progenitor star. This assumption predicates on the ideas that (1) the same physical processes act in both short and long GRBs and (2) the accretion phase during which europium is produced roughly coincides with the phase during which the GRB occurs in the source frame, matching assumptions of \\citet{siegel19}. \\citet{cote18} infer that $\\sim 3-15 \\times 10^{-6} M_\\odot$ of europium was ejected from the post-merger accretion disk of GW170817. This translates to $\\sim 0.01 M_\\odot$ of heavy $r$-process material for a europium mass fraction of $X_{\\text{Eu}}$~$=$~$10^{-3}$.\nThe istropic $\\gamma$-ray energy of GW170817 was $E_{\\gamma,iso,GW170817} = 2.1^{+6.4}_{-1.5} \\times 10^{52}$ ergs \\citep{GW170817Eiso}, and from a sample of 468 LGRBs, the mean istropic energy of LGRBs is $E_{\\gamma,iso,LGRBs} \\approx 2.6^{+2.7}_{-0.5} \\times 10^{53}$ ergs \\citep{LGRB_stats}. With these values:\n$$M_{r,collapsar}\\sim M_{r,GW170817}\\frac{E_{\\gamma,iso,LGRBs}}{E_{\\gamma,iso,GW170817}} \\sim 0.1 M_{\\odot}$$ \n(see also \\citealt{Siegel20}). This aligns with our fiducial value of $\\avg{M_r}$. In particular, our fiducial results lie near the intersection of the $r$-process yield expected per LGRB and the fraction of LGRBs per CCSN (see Figure \\ref{fig:constraints}). This supports the possibility that LGRBs are linked to $r$-process production.\n\nFor the final reference mass, we compare to the amount of $r$-process yield that was produced in the $r$-process event that enriched the ultra-faint dwarf galaxy Reticulum II \\citep{Ji16b}. This galaxy preserves $r$-process enrichment from a single prolific event in the early universe. To explain the europium abundances of its stars, it likely experienced an event with a europium yield of $10^{-4.3}$ to $10^{-4.6} M_\\odot$ \\citep{Ji16b}. With $X_{\\text{Eu}}=10^{-3}$, this corresponds to $M_r \\sim 0.04 M_\\odot$. Our $\\avg{M_r}$ value is only slightly higher than this mass. This yield is also consistent with that expected for neutron star mergers \\cite[e.g., the yield estimated from GW170817,][]{siegel2019b,cote18}.\n\n\\subsection{Implications of $\\alpha$: Learning About Collapsar Properties from $r$-Process Abundance Scatter} \\label{subsec:disc_a}\n\nUnfortunately, the current precision on the shape of the [Eu\/Fe] distribution does not provide tight constraints on $\\alpha$, the exponent of our $r$-process yield power law distribution. For $\\avg{N_r} = 3$, any $\\alpha = 2.2-6.0$ can  explain the observed scatter. Our fiducial value of $\\alpha = 2.8$ best fits the data, but the full range of possible values produces similar distribution widths (see Figure \\ref{fig:modelHeatmaps}b).\n\nThe $\\alpha$ constraints from metal-poor stars can be compared to power law distributions of long $\\gamma$-ray burst (LGRB) engine duration, engine luminosity, and isotropic energy. Figure \\ref{fig:constraints} shows our constraints on $\\alpha$ in context with the exponents from these distributions.\n\n\\citet{petropoulou17} modeled the central engines which power LGRBs, determining power law distributions for both the engine luminosities and engine activity times:  $p(L_{engine})\\propto L^{-\\alpha_L}$ and $p(t_{engine})\\propto t^{-\\alpha_t}$. By assuming that more powerful engines can more quickly break out of the collapsing star to produce $\\gamma$-ray signals (with a breakout time that scales with jet luminosity as $L^{-\\chi}$), they show that the shape of the $\\gamma$-ray duration distribution can be uniquely determined by the observed GRB luminosity function. In particular, they determine the power law indexes of the $L_{engine}$ and $t_{engine}$ distributions by connecting them with the observed distributions of luminosities and durations of LGRBs. For $\\chi=1\/3$, \\citet{petropoulou17} find $\\alpha_L=2.4$ and $\\alpha_t=3.5$, while for $\\chi=1\/2$, they constrain $\\alpha_L=2.4$ and $\\alpha_t=4.6$. In addition, by assuming a single breakout time, \\citet{sobacchi17} find a power law distribution for $t_{engine}$ consistent with $\\alpha_t\\sim4$. \n\nFurthermore, we can determine the isotropic energy distribution of LGRBs since $E \\propto L \\times t$. Because both $L_{engine}$ and $t_{engine}$ draw from power law distributions, the distribution of their product follows the distribution of the variable with a smaller $\\alpha$, in this case $\\alpha_L = 2.4$.\n\nOur $\\alpha$ constraint overlaps with all of these values, with the fiducial value falling closer to $L_{engine}$ or $E_{iso}$. Any of these properties could therefore potentially trace the $r$-process yield.\nFor a better constraint on $\\alpha$, we need a significantly lower uncertainty on the observed $\\text{IQR}_{\\text{Eu}}$. Figure \\ref{fig:IQRuncertainty} shows how tightly $\\text{IQR}_{\\text{Eu}}$ must be measured for the stellar samples to improve the $\\alpha$ constraint. This plot was constructed assuming the $\\text{IQR}_{\\text{Eu}}$ is centered on $\\text{IQR}_{\\text{Eu}}=0.50$, as found for the RPA sample. To differentiate between the distributions for $t_{engine}$ and $L_{engine}$ or $E_{iso}$, the $\\text{IQR}_{\\text{Eu}}$ must be measured with uncertainty $<0.05$ dex. This abundance precision is better than what current measurements can achieve in metal-poor stars, though it may become achievable in the future as stellar spectroscopy methods improve.\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{IQR_uncertainty.pdf}\n\\caption{To improve our constraint on $\\alpha$, we must improve our measurement of the stellar $\\text{IQR}_{\\text{Eu}}$ for metal-poor stars. Here we show how the $\\alpha$ constraint is improved for several different $\\text{IQR}_{\\text{Eu}}$ uncertainties. \\label{fig:IQRuncertainty}}\n\\end{figure}\n\nThe $\\text{IQR}_{\\text{Eu}}$ is a robust but very inefficient estimator of the distribution shape. Alternatively, we could use the full distribution shape. This requires a reliable selection function, but would likely not demand 0.05 dex precision.\n\n\\subsection{Neutron Star Mergers vs. Collapsars} \\label{subsec:NSMvscollapsar}\n\nHere we focused on collapsars, demonstrating that a $r$-process production site with a power law distribution inspired by LGRB jet properties can self-consistently reproduce the abundances and scatter observed in metal-poor stars. These results would also apply to another prompt $r$-process site ejecting a Solar $r$-process abundance pattern that scales with a power law, however.\n\nThe possibility that collapsars produce $r$-process material is debated. For example, earlier semi-analytic work on collapsar disk winds by \\citet{Surman06} found that collapsar outflows are too neutron-poor to produce heavy $r$-process isotopes. A recent study by \\citet{Miller19} that investigated the \\citet{siegel19} results with more detailed modeling of neutrino transport also found that collapsar outflows are incapable of producing third peak $r$-process material. Furthermore, \\citet{Macias19} found that any $r$-process site that also produces large amounts of iron is disfavored by observations of metal-poor stars. Collapsars that do not produce large amounts of iron (e.g., LGRBs without an associated supernovae; \\citealt{2006Natur.444.1047F}) would avoid the dilution problems discussed by Macias \\& Ramirez-Ruiz, but the topic is unsettled.\n\nNeutron star mergers are a demonstrated source of $r$-process thanks to GW170817 and, in principle, their europium yields can vary as well with different neutron star binary masses, mass ratios \\citep{Korobkin12,Bauswein13,Hotokezaka13,dietrich15,Sekiguchi16}, and eccentricities \\citep{Chaurasia18,Papenfort18}.\nFurthermore, the $M_{r,\\text{min}}$ and $\\avg{M_r}$ values in our model are roughly consistent with the $r$-process yield estimated for GW170817 \\citep{siegel2019b,cote18}. Because of this, variable-yield neutron star mergers could also potentially explain the $r$-process scatter in metal-poor stars via a similar model to that presented in this paper. More work is needed to determine a reasonable distribution of $r$-process effective yields from neutron star mergers, combining input distributions of binary neutron star properties and yields (e.g., those from the numerical simulations cited above) and kick velocities \\cite[e.g.,][]{Tarumi20, Safarzadeh19b, Bonetti19}.\n\n\\subsection{Limitations of Initial Model} \n\\label{subsec:caveats}\n\nOur initial model is purposefully simple in order to act as a focused exploration of variable-yield collapsars. In particular, the model assumes all abundance scatter is due to variable stellar populations. This assumption allows us to expressly investigate variable yields as a source of scatter, but it does not consider possible effects due to differences in galaxy formation. Real dwarf galaxies have differences in their hierarchical assembly, small amounts of cross-pollution, and experience inhomogeneous mixing \\cite[e.g.,][]{Venn04,Ji15,Griffen18}. Abundance scatter is likely affected by these complexities. Inhomogeneous enrichment has been included in some previous models \\cite[e.g.,][]{Cescutti15,Wehmeyer15}, but it is not a solved problem. In this very metal-poor regime, theoretical work has not yet given a simple way to model the amount of scatter from galaxy formation effects.\n\nThis model also assumes that each star probes an independent gas reservoir. For every star in our model, we assume that it originates from a different dwarf galaxy in which a number of SN exploded over some time, the metals fell back down into the galaxy and fully mixed, and then our model star formed from the mixed gas. This approximates the average star that formed in a given gas reservoir. Real stellar samples likely contain stars that originated together, though. Observational work that studies the accretion origin of stars through, for example, analysis of stellar streams and kinematic clustering will inform the quality of this assumption in the future.\n\nTo transcend the limitations of this initial model, future models will consider scatter due to differences in galaxy formation and include a more detailed treatment of chemical enrichment and star formation. We are currently developing high-resolution hydrodynamic simulations of dwarf galaxy evolution that will study these effects and further explore the origins of $r$-process material.\n\n\\section{Conclusions} \\label{sec:conc}\n\nWe have produced a self-consistent model in which collapsars synthesize all of the $r$-process material in the early universe. By assuming the $r$-process material in metal-poor ([Fe\/H] $< -2.5$) stars was formed exclusively in collapsars with stochastic yields, we can reproduce the observed distribution of europium abundances with parameter values that are consistent with other independently determined reference values. This was not guaranteed a priori.\n\nThis is not evidence that collapsars dominantly produce $r$-process material in the early universe, however. Neutron star mergers with variable effective europium yields may also be able to explain the $r$-process scatter. More work is needed on the effective europium yields of neutron star mergers. In particular, the retention fraction is important for the collapsar model, but it becomes even more important for neutron star mergers with different natal kick velocities and coalescence times.\n\nAbundance scatter of metal-poor stars is an important window into the different mechanisms producing $r$-process elements. Individual mechanisms can produce scatter without the need for multiple sources. In this paper, we assume a power law distribution of collapsar $r$-process yields. The range of constrained values for the exponent $\\alpha$ is comparable to those of the distributions for long $\\gamma$-ray burst isotropic energies, engine luminosities, and engine times (Section \\ref{subsec:disc_a}). Improved constraints on $\\alpha$ could allow us to investigate which, if any, of these collapsar properties trace $r$-process yield. \n\nLastly, in our model, the fraction of core-collapse supernovae that result in $r$-process collapsars, $f_r$, is comparable to the fraction of core-collapse supernovae that result in long $\\gamma$-ray bursts. This could indicate a link between LGRBs and $r$-process. The uncertainty in our model errs to higher $f_r$, though, and if $f_r$ is higher then we would require a significant number of $r$-process collapsars that do not produce long $\\gamma$-ray bursts. Our model also favors a scenario in which $\\lesssim 10$\\% of Type Ic-BL supernovae produce europium. This does not preclude all Ic-BL SNe from harboring choked jets, but would imply that some Ic-BL SNe lack the accretion disk properties to synthesize heavy $r$-process isotopes.\n\n       \n\\acknowledgments\nWe thank Daniel Siegel for providing nucleosynthesis products from his group's collapsar simulations and for helpful discussions.\nWe also thank Paz Beniamini Tony Piro, \nEnrico Ramirez-Ruiz, Phillip Macias, and Anne Kolborg for helpful discussions.\nKB acknowledges support from the United States Department of Energy grant DE-SC0019323.\nAPJ acknowledges support by NASA through Hubble Fellowship grant HST-HF2-51393.001, awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.\nAPJ also acknowledges a Carnegie Fellowship and the Thacher Research Award in Astronomy.\nMRD acknowledges support from the NSERC through grant RGPIN-2019-06186, the Canada Research Chairs Program, the Canadian Institute for Advanced Research (CIFAR), and the Dunlap Institute at the University of Toronto.\nAF acknowledges support from NSF grant AST-1716251, the Silverman (1968) Family Career Development Professorship and thanks the Wissenschaftskolleg zu Berlin for their wonderful Fellow's program and generous hospitality.\n\n\n\\section{} \n\n\\textit{Research Notes of the \\href{https:\/\/aas.org}{American Astronomical Society}}\n(\\href{http:\/\/rnaas.aas.org}{RNAAS}) is a publication in the AAS portfolio\n(alongside ApJ, AJ, ApJ Supplements, and ApJ Letters) through which authors can \npromptly and briefly share materials of interest with the astronomical community\nin a form that will be searchable via ADS and permanently archived.\n\nThe astronomical community has long faced a challenge in disseminating\ninformation that may not meet the criteria for a traditional journal article.\nThere have generally been few options available for sharing works in progress,\ncomments and clarifications, null results, and timely reports of observations\n(such as the spectrum of a supernova), as well as results that wouldn't\ntraditionally merit a full paper (such as the discovery of a single exoplanet\nor contributions to the monitoring of variable sources). \n\nLaunched in 2017, RNAAS was developed as a supported and long-term\ncommunication channel for results such as these that would otherwise be\ndifficult to broadly disseminate to the professional community and persistently\narchive for future reference.\n\nSubmissions to RNAAS should be brief communications - 1,000 words or fewer\n\\footnote{An easy way to count the number of words in a Research Note is to use\nthe \\texttt{texcount} utility installed with most \\latex\\ installations. The\ncall  \\texttt{texcount -incbib -v3 rnaas.tex}) gives 57 words in the front\nmatter and 493 words in the text\/references\/captions of this template. Another\noption is by copying the words into MS\/Word, and using ``Word Count'' under the\nTool tab.}, and no more than a single figure (e.g. Figure \\ref{fig:1}) or table\n(but not both) - and should be written in a style similar to that of a\ntraditional journal article, including references, where appropriate, but not\nincluding an abstract.\n\nUnlike the other journals in the AAS portfolio, RNAAS publications are not\npeer reviewed; they are, however, reviewed by an editor for appropriateness\nand format before publication. If accepted, RNAAS submissions are typically\npublished within 72 hours of manuscript receipt. Each RNAAS article is\nissued a DOI and indexed by ADS \\citep{2000A&AS..143...41K} to create a\nlong-term, citable record of work.\n\nArticles can be submitted in \\latex\\ (preferably with the new \"RNAAS\"\nstyle option in AASTeX v6.2), MS\/Word, or via the direct submission in the\n\\href{http:\/\/www.authorea.com}{Authorea} or\n\\href{http:\/\/www.overleaf.com}{Overleaf} online collaborative editors.\n\nAuthors are expected to follow the AAS's ethics \\citep{2006ApJ...652..847K},\nincluding guidance on plagiarism \\citep{2012AAS...21920404V}.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.85,angle=0]{aas.pdf}\n\\caption{Top page of the AAS Journals' website, \\url{http:\/\/journals.aas.org},\non October 15, 2017.  Each RNAAS manuscript is only allowed one figure or\ntable (but not both). Including the\n\\href{http:\/\/journals.aas.org\/\/authors\/data.html\\#DbF}{data behind the figure}\nin a Note is encouraged, and the data will be provided as a link in the\npublished Note.\\label{fig:1}}\n\\end{center}\n\\end{figure}\n\n\n\\acknowledgments\n\nAcknowledge people, facilities, and software here but remember that this counts\nagainst your 1000 word limit.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Intro}\n\nNew physics searches are often accomplished by looking for excess in cross-section pertaining to specific decay channels of new physics particles. Constraining the measured cross-section to specific decay channels can often reduce the background significantly enough to identify the excess with high confidence level. Thus, it is usually cleaner to search for new physics particles in the leptonic or photonic channels as they offer very small background at the Large Hadron Collider (LHC). However, the cleaner channels often have extremely small cross-section that it makes the searches for new physics difficult or requires a large amount of data to get a significant sample of the signal events. Additionally, leptonic searches can be difficult if the physics or decay channels of the new particle are not known. Further difficulties may arise if there is a missing transverse energy in a leptonic channel. Similar difficulties also arise for the Standard Model (SM) heavy particles, namely, $W,~Z$ bosons, Higgs boson \nand top quarks, except that the physics of these particles is well known. These heavy particles dominantly decay to light quarks and gluons (collectively partons) which then shower and fragment into jets of hadrons. Since jets are invariably produced at the LHC due to hard scattering of the constituent partons, heavy SM particles are often faked by light quark\/gluon radiations and radiations coming from other sources, such as initial state radiation, underlying event\\cite{Barnafoldi:2011ad, Cacciari:2009dp} and pile-up \\cite{Krohn:2013lba}.\nAt present collider energies, these heavy SM particles are often produced with a large Lorentz boost factor and their hadronic decay products are realised as a single collimated 'fat jet'~\\cite{Adams:2015hiv, Altheimer:2013yza, Abdesselam:2010pt, Altheimer:2012mn}. In such cases, the mass of the heavy particle should be reflected in the mass of the fat jet. One may expect to disentangle the signal from the large background of light parton jets, using the jet mass cut\\footnote{In this work, we mainly focus on gluon jets as the background since their cross-section is much larger.}, but as it happens to be the case, gluon jet cross-section has a significant long tail even after accounting for underlying event \\cite{Barnafoldi:2011ad, Cacciari:2009dp} and pile-up \\cite{Krohn:2013lba}, which can be dealt with effectively using the jet grooming techniques, such as soft drop~\\cite{Larkoski:2014wba, Marzani:2017kqd, Dreyer:2018tjj}, pruning \\cite{Ellis:2009su, Ellis:2009me}, trimming \\cite{Krohn:2009th} and mass drop\/filtering \\cite{Dasgupta:2013ihk, Butterworth:2008iy, Marzani:2017mva}.\nFurther discussion on underlying event and pile-up is beyond the scope of this work. As far as this paper is concerned, we will assume that we start with a well-groomed jet. Since a simple mass cut is not effective in identifying the particle originating the jet inspite of grooming, thus a great deal of effort has been made in jet identification by constructing observables that can reject the parton jets and reduce the background~\\cite{Altheimer:2013yza, Abdesselam:2010pt, Altheimer:2012mn, Adams:2015hiv, Larkoski:2017jix, Marzani:2019hun}. The emphasis of such observables is to improve the signal rate to mistag rate ratio. This will also be the spirit of our work. The SM heavy particles provide a perfect playground for improving on such jet identification strategies as their decay channels are well understood. The purpose of our work is to improve upon these strategies by studying a new non-linear jet observable {\\em zest} dependent only on the transverse momentum distribution of particles in the jet, similar to transverse-zeal introduced in the context of jet quenching studies \\cite{Gavai:2015pka}.\n\n\nThe remainder of the paper is organized as follows. In Sec.~\\ref{Zest}, we define zest and discuss its properties. In Sec.~\\ref{sec:gluzest}, we outline the features exhibited by zest distribution of gluon-initiated jets that make it suitable for vetoing the gluon jets. In Sec.~\\ref{Simulation}, we discuss the details of our simulations and provide the principal results of our analysis in Sec.~\\ref{zestFilter}. In Sec.~\\ref{bibFilter}, we introduce another substructure observable {\\it bib} which also depends only on the transverse momentum but linearly and is largely uncorrelated to zest. We contrast the two observables and perform a bi-variate analysis to improve the discrimination ability in Sec.~\\ref{bivariate}. In Sec.~\\ref{pzest}, we  generalize zest by introducing a real parameter $p$ that can be tuned to modify the contribution of the soft particles in the jet. The parameter $p$ can be optimized to provide improved discrimination for the heavy SM particle jets from the gluon background, as discussed in Sec.~\\ref{Z} and Sec.~\\ref{top}. We show that for the top quark-initiated jets, the discrimination provided by generalized zest is in close comparison to recent machine learning (ML) approaches. We propose that studying such infrared and collinear (IRC) unsafe observables may help to uncover the hidden physics behind ML approaches. Since, zest is collinear unsafe and is computable for hadronic final states only, we study its dependence on different hadronization models in Sec.~\\ref{hadModel}. We conclude in Sec.~\\ref{Conclusion}.\n\n\n\\section{Zest of a Jet}\n\\label{Zest}\n\nGiven a well-groomed jet composed of hadrons, reconstructed through a suitable jet algorithm like the anti-k$_t$ clustering algorithm~\\cite{Cacciari:2008gp}, {\\em zest} of the jet is defined as\n\n\\begin{eqnarray}\n\\label{eq:zest}\n\\zeta = \\frac{-1}{\\log \\big (\\sum_{i \\in  \\rm{Jet}} e^{-P_T\/\\vert  {\\bf p}_{T i} \\vert}\\big )} \\, ,\n\\end{eqnarray}\n\nwhere $ P_{T} = \\sum_{i \\in \\rm{Jet}} \\vert \\textbf{p}_{T i} \\vert $ and $\\textbf{p}_{T i}$ is the transverse momentum of the $i^{th}$ particle in the jet with respect to the jet axis.  \nNote that zest is composed purely out of the transverse momenta of the final state particles. For the extreme case of a jet composed of a single energetic particle, it is straightforward to see that $\\zeta$ reduces to $1$. Similarly, for two leading particles in the jet carrying equal fractions of energy, i.e. $P_{T} = 2\\, p_{T}$, we get\n\\begin{equation}\n\\label{eq:zest2pt}\n\\zeta = \\frac{-1}{\\log(2)-2} \\approx 0.765\\, ,\n\\end{equation}  \nThis value remains roughly the same even if the two particles do not carry equal fractions of energy, i.e. for a generic break up of $P_{T} = x\\, p_{T} + (1-x)\\, p_{T}$ with $0<x<1$.\nThis is confirmed also by looking at FIG.~\\ref{fig:2ptzest} where we have plotted the normalized distribution of $10^5$ randomly generated pairs of particles with total $P_{T}=1$. From the plot, we see that the two particle distribution has a sharp peak at about $\\sim 0.77$. A similar analysis done for only three particles in a jet gives a slightly broader multiplicity peak at $\\zeta \\sim 0.53$. Therefore, we see that the zest distribution captures multiplicity peaks whenever there are a few energetic particles in the jet. These multiplicity peaks shift towards smaller and smaller value as the number of particles contributing to the jet increases. In the special case of a jet containing $n$ particles each carrying equal $\\vert  {\\bf p}_{T i} \\vert$, $\\zeta$ reduces to $\\frac{1}{n - \\log n}$. Thus we expect zest to be strongly correlated to {\\em multiplicity of a jet}. Later we will generalize zest such that multiplicity will emerge as a limiting case.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.65]{Zest2pt}\n\\caption{Zest distribution for two leading particles in a jet with a fixed $P_{T}$. We see a dominant two-particle peak at a zest value of 0.76.}\n\\label{fig:2ptzest}\n\\end{figure} \n\nZest has certain interesting properties which makes it a useful observable to study jet substructure : (1) It is invariant under boosts made along the jet axis as it is composed entirely of transverse momentum components of the jet constituents, (2) It is sensitive to transverse momentum distribution of energetic particles while it is largely insensitive to the soft particles in the jet since their contribution to the observable is exponentially suppressed, and (3) Zest is also mostly insensitive to the global color flow of partons. In other words, the zest distribution of the gluon-initiated jets (or any other colored particle) is stable against the change of color flow direction of the colored particles forming the jet. A colored parton that evolves into a color neutral hadron jet recoils against another colored parton. The color neutral final state is obtained due to the exchange of several soft partons. Changing the direction of the recoiling colored partner effects only the exchange of a few soft particles between the partners. Since zest is stable against a few changes in the soft sector, it also becomes stable to the color recoil.\n\nWhile zest is a non-linear and collinear unsafe jet observable due to which it cannot be calculated in perturbation theory, it can be computed using Monte Carlo based event generators~\\cite{Sjostrand:2007gs, Bellm:2015jjp}. Although collinear unsafe, zest may provide a new perspective into the jet substructure that may not be accessible through IRC safe jet observables. In particular, the jet observables based on machine learning techniques, while not calculable through perturbation theory, provide the best discrimination ability~\\cite{Kasieczka:2019dbj,Amram:2020ykb,Bradshaw:2019ipy,Dolen:2016kst,Chen:2019uar,Moreno:2019bmu,CMS:2019gpd}. The physical features extracted by such observables that enable the discrimination remains unknown. We anticipate that studying zest and other IRC unsafe observables may shed some light on the black box offered by neural-network-based observables.  It goes without saying that zest is hadronization model dependent, however we expect that its ability to discriminate between the originating particles will be independent of hadronization models used in a Monte Carlo simulator, as is demonstrated later in Sec.~\\ref{hadModel}.\n\n\\section{Zest distribution of gluon-initiated jets}\n\\label{sec:gluzest}\n\nAs gluon jets, on an average, consist of a larger number of particles, the zest distribution of gluon-initiated jets is expected to peak at smaller values of the observable.  \n\\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.45\\textwidth]{Gluon_Zest_v3}\n \\caption{Zest distribution of gluon-initiated jets with a jet mass around $W$ boson mass, $Z$ boson mass and top quark mass.}\n \\label{zplot}\n \\end{figure}\nWe validate this in FIG.~\\ref{zplot} by plotting zest distribution for jets initiated by offshell gluons of masses around $W,~Z$ boson and top quark masses. \nWe see that the zest distribution for gluon jets peaks at small $\\zeta \\sim$ 0.1, as expected. Interestingly, we also see that these zest distributions are nearly independent of the jet mass. To investigate this, we have shown the ratio of the mean of $n$ maximum $p_{T i }$ to $P_T$ for $n=1,~2,~5$ and $10$ for various jet masses in TABLE~\\ref{tab:veto}. We find that these ratios are nearly independent of jet mass, which explains why the zest distribution for gluons is insensitive to the jet mass.\n\n\n\n\\begin{table*}\n\\begin{tabular}{|m{2cm}||*{7}{c|}}\\hline\n\\backslashbox[1.5cm]{$m_{J}$}{$n$}\n&\\makebox[5em]{1}&\\makebox[3em]{2}&\\makebox[3em]{5}\n&\\makebox[3em]{10}\\\\\\hline\\hline\n\\center gluons around W mass &0.121 $\\pm$ 0.049 &0.105 $\\pm$ 0.035 &0.081 $\\pm$ 0.018 &0.0617 $\\pm$ 0.0091\\\\\\hline\n\\center gluons around Z mass &0.120 $\\pm$ 0.048 &0.104 $\\pm$ 0.035 &0.079 $\\pm$ 0.018 &0.0603 $\\pm$ 0.0094\\\\\\hline\n\\center gluons around top mass &0.116 $\\pm$ 0.051 &0.099 $\\pm$ 0.037 &0.074 $\\pm$ 0.019 &0.0552 $\\pm$ 0.0102\\\\\\hline\n\\end{tabular}\n\\\\~\\\\\n\\caption{The entries in the table give the ratio of mean of $n$ maximum $\\vert p_{T i}\\vert$ to $P_{T}$ of the jet, that is $\\frac{\\langle max_{n}\\vert p_{T i}\\vert \\rangle}{P_{T}}$. The first set of numbers specify the mean value and the second is the standard deviation calculated over a set of 10000 jets with the specified jet mass.}\n\\label{tab:veto}\n\\end{table*}\n\n\\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.44\\textwidth]{SoftvsHard}\n  \\includegraphics[width=0.43\\textwidth]{ColorFlow}\n \\caption{(a) Zest distribution of gluon-initiated jets with a hard gluon of energy about 300 GeV when the recoiling gluon is soft (blue) and hard (pink). (b) Zest distribution of gluon-initiated jets of energy about 300 GeV when the soft recoiler is taken at $30^{\\circ}$ (green), $60^{\\circ}$ (red), $90^{\\circ}$ (light blue), $180^{\\circ}$ (black) vs. a hard recoiler at $180^{\\circ}$ (purple). } \n \\label{colorflow}\n \\end{figure}\n \n \\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.44\\textwidth]{Gluon_Zest_incl_excl}\n \\caption{Zest distribution of gluon-initiated jets around top quark mass by including or excluding a few soft particles $< 3$ GeV to the jet.}\n \\label{softphysics}\n \\end{figure}\n\n\nWe also verify color flow independence of zest, discussed in the previous section, by looking at the zest-distribution of gluon jets recoiling against complementary color gluon at various angles. The result is shown in FIG.~\\ref{colorflow}(a) and \\ref{colorflow}(b). In FIG.~\\ref{colorflow} we have considered jets of jet energy about 300 GeV, for two different cases: (i) a hard gluon of energy 300 GeV recoiling a soft gluon of energy 1 MeV at $180^\\circ$, and (ii) a hard gluon of energy 300 GeV recoiling a hard gluon of energy 200 GeV at $180^\\circ$. From the plot in FIG.~\\ref{colorflow}(a), we note that the choice of the recoiling partner energy does not strongly effect the zest distribution. Furthermore, from  FIG.~\\ref{colorflow} (b), we observe the color stability of gluon-initiated jets against the direction of its recoiling color neutralising partner. Since, the energy as well as the direction of the recoiling colored particle does not largely effect the analysis, for the rest of the paper, we will take the recoiling gluon as soft and back-to-back to the primary hard gluon.\n\nAs zest is largely insensitive to soft particles in the jet, its distribution curve is not effected by including or excluding a few soft particles in the jet. We verify this in FIG.~\\ref{softphysics} where we have plotted the zest distribution curves for gluon-initiated jets of mass around the top quark mass with following three variations: (i) including all the particles in the jet, (ii) excluding all the soft particles less than $3$ GeV from the jet (iii) including few extra randomly generated soft particles less than $3$ GeV in the jet. We see that the zest distribution shifts only very slightly. \n\n\nThe stability of the gluon zest distribution against jet mass variation, color flow direction, and inclusion\/exclusion of few soft particles makes it a potentially useful observable for vetoing gluon jets at the LHC.\nFor instance, a zest cut at about $\\zeta \\sim 0.3$ is expected to largely cut out the gluon background to heavy SM particles and hopefully also for heavier new physics resonances.\n\n\n\\section{Simulation Details}\n\\label{Simulation}\n\nThe details of our Monte Carlo simulations are presented below.\n\n\\begin{enumerate}\n\n\\item All jets are simulated using {\\sc Pythia} 8 \\cite{Sjostrand:2007gs} by inserting $W$ and $Z$ bosons with energy 500 GeV along the central axis (y-axis) and allowing them to decay to hadronic modes only.\\footnote{For the study presented here, we have turned off all the strong, weak and electromagnetic decays of the primary hadrons formed from showering and subsequent hadronization. \nWe do not study detector analysis in this paper.\n} \n\n\\item We collect all the particles in the forward hemisphere and call it a jet. Apart from hemisphere jets, sometimes we also construct anti-k$_t$ jets with different jet radii, R = \\{0.5, 0.7, 0.9, 1.2\\}, using the {\\sc Fastjet} package~\\cite{Cacciari:2011ma} in {\\sc Pythia}.\n\n\\item For simulating a color-singlet top quark jet, we perform an $e^+\\,e^-$ annihilation and allow the intermediate $Z$ boson to decay to a top-antitop pair at 500 GeV each, travelling back-to-back. We further constrain the top decay channels to hadrons only. \nJets are constructed by dividing the event into two hemispheres using the thrust axis.\n\n\\item To shower a color singlet gluon-initiated jet, we insert an energetic gluon of energy 500 GeV and offshellness same as the heavy particle mass in the event record along the y-axis, while a soft onshell gluon of energy 1 MeV with opposite color is inserted in the opposite direction to the +y-axis. The direction of the color conserving gluon partner of the hard gluon does not effect the analysis, as zest is stable to global color flow of the partons (see FIG.~\\ref{colorflow}). For convenience, we choose the direction of the recoiling gluon opposite to that of hard gluon.\n\n\\item The energetic gluon jet is identified by constructing the thrust axis and then taking all particles in the forward hemisphere.\n\n\\item The resulting gluon jets are accepted if their masses are within 10 GeV of the corresponding heavy particle mass. Although the energy of the jet does not effect zest, we observe that the selected gluons jets have energy ranging between 485 GeV to 500 GeV. \n\n\n\n \\begin{figure*}\n\\centering\n  \\includegraphics[width=0.62\\textwidth]{zest1}\n  \\caption{Zest distribution for $W,~Z$ boson and top quark initiated jets along with gluon jets of corresponding masses.}\n \\label{zestplot}\n \\end{figure*}\n\n\\item  In order\nto test the sensitivity of our observable to non-perturbative physics, we also generate jets in two distinct ways: (a) by varying the default parameters of the hadronization model implemented in {\\sc Pythia}, thereby producing a new set of final state events, and (b) by utilizing a different model for non-perturbative hadronization effects by using the {\\sc Herwig}~\\cite{Bellm:2015jjp} event generator. We study this non-perturbative physics dependence for $Z$ boson-initiated jet along with corresponding gluon background in Sec.~\\ref{hadModel}.\n\n\\item The {\\sc Herwig} event is sampled using the process $e^+\\, e^- \\to {\\rm Z} \\to {\\rm jets}$ at a center of mass energy equal to the mass of the $Z$ boson, i.e., $E_{\\rm cm} = m_Z$. We then boost all the final state particles along a direction, which we take as the +y-axis. The resulting boosted jet has a mass about the $Z$ boson and energy around 500 GeV.\n\n\\item In this work, we will not study Higgs-initiated jets because a Higgs boson dominantly decays to a $b\\,\\bar{b}$ pair which receives a large background from $Z\\rightarrow b\\,\\bar{b}$ and $g\\rightarrow b\\,\\bar{b}$, in which case the analysis proposed here does not offer a substantially good discrimination ability.\n\\end{enumerate}\n\n\\section{Zest as Filter}\n\\label{zestFilter}\n\n \nAs noted earlier, the zest distribution of gluon-initiated jets peaks at small values of the observable in comparison to their heavy particle counterparts. This accounts for an appreciable distinguishing criterion offered by zest in characterization of heavy particle jets from the background gluon jets.\nWe analyze this discriminating ability provided by zest by looking at the $\\zeta$-distribution curves for $W,~Z$ boson and top quark-initiated jets along with the corresponding gluon-initiated jets with a mass about the heavy particle masses. This comparison is shown in FIG.~\\ref{zestplot}. From the plot, we see that the zest distribution of the gluon-initiated jets peaks at relatively smaller values of $\\zeta$ compared to their heavy particle counterparts. In addition, the zest distribution of gluon-initiated jets is found to be narrow, independent of jet energy and has a little overlap with a similar distribution of most other heavy particle jets. Moreover, we find that for the top quark-initiated jet, the zest distribution peaks at a slightly higher value than $W$ or $Z$ boson-initiated jets.\n\nThe tagging efficiency of a zest-based binary classifier can be further evaluated by comparing the performance curves of these jets. In FIG.~\\ref{ROC}, we present the relative operating characteristic (ROC) curves for the zest-based filter. We note that zest provides good signal statistics along with a high background rejection rate. For example, we note that for all the heavy particle jets considered, nearly 80 to 90\\% of the signal stays in the accepted sample after applying a zest cut to remove 90\\% of the gluons.\n\n\\begin{figure}\n\\centering\n \\includegraphics[scale=0.35]{zestROC1}\n  \\caption{ROC curves for $W,~Z$ boson and top quark-initiated jets using {\\it zest} as the binary classifier.}\n \\label{ROC}\n\\end{figure}\n\n \\begin{figure}\n\\centering\n  \\includegraphics[width=0.47\\textwidth]{ZJets_VaryingR}\n  \\caption{Zest distribution for $Z$ boson-initiated jets with different jet radii. Here jets are constructed using the anti-kT clustering algorithm for different $R$ values.}\n \\label{ZJetsR}\n \\end{figure}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{bibplot}\n\\caption{$bib$-distribution of heavy particle initiated jets and gluon jets forming the background.}\n\\label{bibplot}\n\\end{figure*}\n\n\n\n\nIn experiments, jets are often constructed by a suitable jet algorithm employing a cut on the jet cone radius $R$. At large jet radius, the wide angle soft particles contribute to the jet, while as the jet radius decreases, the observable receives contribution from collinear particles in the jet and a few soft particles. Since zest is not significantly effected by the inclusion or exclusion of a few soft particles, we anticipate the zest distribution to be stable against change in the jet cone radius. This is confirmed by FIG.~\\ref{ZJetsR}, where $\\zeta$-distribution curves with different jet radii, namely R = \\{0.5, 0.7, 1.0, 1.2\\}, are presented for $Z$-boson initiated jets. From the plot, we also note that the two particle peak at $\\zeta \\approx 0.77$ gets enhanced when the jet radius is sufficiently small, aligned with our expectation.\n\\section{Boost-invariant Broadening as Filter}\n\\label{bibFilter}\n\nTo contrast zest with an observable linearly dependent on transverse momentum $\\vert {\\bf{p}}_{T}\\vert$ of the particles, we introduce a simple observable \\textit{boost-invariant broadening} or $bib$ (for brevity) which is defined by the equation,\n\\begin{equation}\nbib = \\frac{1}{m_{J}} \\sum_{i\\epsilon Jet} \\vert {\\bf p}_{T i} \\vert = \\frac{P_T}{m_J}\\,.\n\\label{bib}\n\\end{equation}\nHere as well, all transverse momenta are measured with respect to the jet axis and $P_T$ is the same quantity as defined in the context of zest. It is interesting to note that $bib$ is similar to the event shape observable \\textit{broadening}~\\cite{Dokshitzer:1998kz}, however the use of $ m_{J}$ instead of $E_{J}$ renders it invariant under boosts made along the jet axis. Zest and $bib$ have boost-invariance along the jet axis and jet energy independence in common with each other. Although there can be other observables which can be chosen for a bi-variate analysis, $bib$ provides a simple contrast to zest : (i) $bib$ is a linear function of the transverse momentum of the particles while zest is not, (ii) $bib$ is calculable using standard perturbative techniques\\footnote{As $bib$ is defined by the ratio of two IRC safe observables, it is essentially Sudakov safe~\\cite{Larkoski:2013paa, Larkoski:2015lea} which implies that even though the observable does not has a valid fixed order expansion in $\\alpha_s$, it can still be calculated using all-order resummation. This ensures that the singular region of the phase space for the observable is exponentially suppressed.}, while zest is collinear unsafe and relies on a suitable Monte Carlo generator that encodes a definite model to capture the hadronization effects for its computation, and (iii) $bib$ is sensitive to the global color flow of partons while zest is insensitive to it. The results presented in this section and for the  bi-variate analysis in the next section, are strictly for the case in which the soft gluon is back-to-back with the primary hard gluon.   \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47\\textwidth]{bibROC1}\n\\caption{ROC curves for heavy particle jets with $bib$ as the classifier.}\n\\label{bibROC}\n\\end{figure}\n\nThe physics of the primary splitting of heavy particles leading to partonic decays is different from that of branching of quarks and gluons. Parton splittings dictated by Quantum Chromodynamics (QCD) are dominated by soft and collinear radiation having small transverse momenta, thus we expect gluon initiated jets to have a peak at a relatively smaller value in $bib$-distribution compared to jets initiated by heavy particles at the same jet mass.\nThe comparison of $bib$-distributions for heavy particles and corresponding gluon jets is shown in FIG.~\\ref{bibplot}.\nWe also present the ROC curves for $bib$ when used as a binary classifier in FIG.~\\ref{bibROC}. From the plot, we find that $bib$ acts as an appreciable classifier for $W,~Z$ boson-initiated jets but does not offer good discrimination for heavier top quark-initiated jets. This is in contrast with the zest ROC curves of FIG.~\\ref{ROC}. Overall, zest provided better classifier performance compared to $bib$. Nonetheless, $bib$ also contrasts zest in the stability of gluon distributions with respect to the jet mass. We have also verified that the two observables do not have a significant linear correlation. Thus, we expect that the two observables in conjunction can be used to improve the signal to background ratio. We present this bi-variate analysis in the next section.\n\n\n\n\\section{Bi-variate Analysis}\n\\label{bivariate}\n\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.48\\textwidth]{scatterPlot}\\\\\n  \\includegraphics[width=0.35\\textwidth]{scatterplot1}\n \\caption{(a)-(c) Scatter plots in $bib$-zest plane for heavy particle and corresponding gluon jets.}\n \\label{scatter}\n\\end{figure}\n\n\nDiscrimination for the originating particle can be further improved with a multivariate analysis. Recently, multivariate approaches are becoming extremely popular as they are expected to provide a more detailed characterization of the QCD radiation pattern within a jet which can be exploited to further enhance the new physics searches at the LHC~\\cite{Larkoski:2017jix, Marzani:2019hun}. For the bi-variate analysis proposed here, we generate scatter plots in the \\textit{bib}-zest plane for gluon initiated jets and heavy particle jets, as shown in FIG.~\\ref{scatter} (a)-(c).  \nWe note that, in general, gluon jets occupy the extreme left corner of the $bib$-zest plane, while heavy particle jets are mostly concentrated towards the upper right region in the plot, thus providing statistical discrimination between the two types of jets.\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.48\\textwidth]{ExclInclW.pdf}\n \n \\caption{Scatter plot in $bib$-zest plane for $W$-boson jet and the corresponding gluon jet, with the 90\\% Gaussian contour and inclusion and exclusion-zone cuts specified on it.}\n \\label{Wscatter}\n\\end{figure}\n\n\n\\begin{figure*}\n\\centering\n \\includegraphics[width=0.7\\textwidth]{fig1b}\n \\caption{ROC curves for heavy particle jets with zest, inclusion and exclusion$-$filter based binary classifiers.}\n \\label{allRoC}\n\\end{figure*}\n\nThe zest-$bib$ distributions shown in FIG.~\\ref{scatter} are reasonably well-approximated by an asymmetric two-dimensional Gaussian distribution. We fit both the heavy particle and gluon jet distributions with an asymmetric Gaussian distribution. On the fitted Gaussian distributions, we draw contours of different heights. An example for the 90\\% contour, illustrated with a dashed line, is shown in FIG.~\\ref{Wscatter} for the $W$-boson initiated jets along with the corresponding gluon background. By 90\\% contour, we refer to the closed curve that includes 90\\% of the jets inside it. The tangents are drawn parallel to $bib$-axis and zest-axis on the gluon distribution contour to intercept with the plot frame at the bottom and to the left, respectively. The solid line represents an open curve made by combining the tangents and right-side of the contour meeting the tangents. The background region is defined as the area to the left of the solid curve. Similar exercise is also done on the heavy particle initiated jet distribution. The signal region is defined as the region to the right of the solid curve on the heavy particle distribution. We define two new cuts (1) the  `inclusion-zone' cut and (2) the `exclusion-zone' cut. The `inclusion-zone' cut is the solid curve on the signal while the `exclusion-zone' cut is the solid curve on the background. We note that for the case of a $W$ boson-initiated jet with 90\\% of the signal as the desired value, an exclusion-zone cut provides better statistics over an inclusion-zone cut.\n\n\n\n\n\nThe relative performance of these bi-variate cuts in comparison to zest alone is studied through ROC curves in FIG.~\\ref{allRoC}. We present three types of ROC curves for each type of heavy particle jet (a) zest based cuts only (represented by dashed lines), (b) cuts based on inclusion-zone obtained by including the percentage of events contained within the primary contour (represented by dotted lines), and (c) cuts based on exclusion-zone obtained by excluding the percentage of events contained within the primary contour (represented by solid lines). From the plot, we note that for a high signal rate, exclusion-zone statistics provides slightly better discrimination than only zest-based cuts for the $W,~Z$ bosons, however inclusion\/exclusion-zone provide no significant improvement in comparison to a zest only filter for the top quark-initiated jet.\nOn the other hand, inclusion-zone statistics is more efficient when high gluon rejection is required and small signal rate is acceptable.\n\n\n\\section{Generalized Zest ($p$-zest)}\n\\label{pzest}\nAs pointed out earlier in Sec.~\\ref{Zest}, in the special case of $n$-equal transverse momenta particles in the jet, zest is a function of {\\it particle multiplicity} in a jet. In this section, we aim to generalize the observable such that the particle multiplicity emerges as a limiting case of generalized zest. We define the generalization of zest (or, `$p$-zest' for brevity) as follows:\n\n\\begin{equation}\n\\zeta_p = -\\frac{1}{\\log\\Big(\\sum_i\\, e^{-P_T^{(p)}\/\\vert {\\bf p}_{T i}^{p}\\vert}\\Big)}\\, .\n\\end{equation}\nHere the sum is over all the final state particles in the jet, as earlier, and $P_{T}^{(p)}$ is defined as\n\\begin{equation}\nP_T^{(p)} = \\sum_i \\vert {\\bf p}_{T\\,i}^{\\,p} \\vert \\, ,\n\\end{equation}\nwhere $p > 0$. The ability to tune the parameter $p$ may also provide a new way of looking at the jet substructure by allowing us to vary the net contribution of the soft sector.\n\nFor $p > 1$, the contribution of the soft particles to the determination of $P_T^{(p)}$ will reduce. \nThus, mostly collinear or energetic particles will contribute to $p$-zest.\n\n \\begin{figure}\n \\centering\n  \\includegraphics[width=0.475\\textwidth]{zvsmult_Mass_1}\n  \\includegraphics[width=0.48\\textwidth]{zvsmult_Mass_2}\n \\caption{(a)Zest distribution of gluon jets at various heavy particle mass windows, and (b) $1\/n$ distribution of gluon jets at various heavy particle mass windows.}\n \\label{zvsmult_Mass}\n \\end{figure}  \n \n\nIn the extreme limit of $p \\to \\infty$, only the leading parton contributes and $\\zeta_{\\infty}$ approaches 1. Therefore, this limit offers no useful jet discrimination ability.   \nIn contrast, in the limit $p \\to 0$, all particles contribute equally giving\n\\begin{equation}\n\\zeta_0 = \\frac{1}{n-\\log n} \\, ,\n\\label{eq:mult}\n\\end{equation}\nwhere $n$ is the number of particles in the jet. For a large enough $n$ such that $n \\gg \\log n$, the above expression reduces simply to the inverse of particle multiplicity.  \nThus, we note that closer the value of $p$ is to 0, the stronger is the correlation of $p$-zest and multiplicity. As $p$ is increased, the correlation becomes smaller and vanishes for the extreme limit of $p \\to \\infty$.\nThough the observables zest (with $p=1$) and multiplicity are correlated,  \nzest exhibits interesting properties such as stability against change in the jet mass and stability against change in the jet radius for gluon-initiated jets, which are absent in the $1\/n$ distribution. This is confirmed by the plots in FIG.~\\ref{zvsmult_Mass} and FIG.~\\ref{zvsmult_R}. In FIG.~\\ref{zvsmult_Mass}(a) and~\\ref{zvsmult_Mass}(b), the zest and the $1\/n$ distribution curves for gluon-initiated jets with masses about heavy particle mass, are presented.   \nFrom the plots, we see that the $1\/n$ distribution shifts to the left as the mass of the jet is increased while the zest distribution remains more or less unchanged. \\footnote{Note that this is not to say that $1\/n$ does not offer gluon jet vetoing ability, instead it may very well provide the ability to veto gluon jets like zest. From the trend observed in FIG.~\\ref{zvsmult_Mass}(b), we expect the heavier jets to shift further to the left and \nhence a cut at around 0.04 may cut out the gluon jets largely. However, as we will show in Sec.~\\ref{top}, for the heavy top quark-initiated jets, zest offers an appreciable improvement over multiplicity.}\n \\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.48\\textwidth]{zvsmult_R_1}\n  \\includegraphics[width=0.47\\textwidth]{zvsmult_R_2}\n \\caption{(a)Zest distribution of $Z$ boson-initiated jets for different $R$ values, and (b) $1\/n$ distribution of $Z$ boson-initiated jets for varying jet radii.}\n \\label{zvsmult_R}\n \\end{figure}  \nSimilarly, the stability of the zest distribution curves to varying the jet radius is shown in FIG.~\\ref{zvsmult_R}(a) for a $Z$ boson-initiated jet of jet energy 300 GeV while the corresponding $1\/n$ curves are shown in FIG.~\\ref{zvsmult_R}(b)\\footnote{Here we have used jets of initial energy 300 GeV to demonstrate this difference. This is because a $Z$ boson at 500 GeV is extremely collimated and $R = 0.5$ is sufficient to capture all the radiations originating from this extremely boosted $Z$ boson.}. From the plots, we clearly see that the zest distribution curves are mostly stable to the change in jet radius while for the $1\/n$ distribution, the curve for $R=0.5$ appears to be shifted away from other $R$ values. This is expected as zest is mostly uneffected by the inclusion or exclusion of a few soft particles while the multiplicity curves are somewhat sensitive to it.\n\nNoting these differences, we investigate the discrimination ability offered by $p$-zest over particle multiplicity in a jet and zest ($p$=1). It will be interesting to find $p$ that provides the best discrimination. For this,  we consider two type of heavy particle jets : (a) a $Z$ boson-initiated jet, and (b) a top quark-initiated jet. \nThe detailed analysis of these results is presented in the following subsections.\n\n\n\\subsection{$Z$ boson-initiated jet}\n\\label{Z}\nFor the $Z$ boson-initiated jet, we generate the ROC curves for $p \\!=\\!\\{0.3, 0.5, 1, 1.5, 2, 3\\}$ and particle multiplicity, as shown in FIG.~\\ref{ZFilter}. The solid cyan line with open squares on it illustrates the multiplicity distribution while the other solid lines with the corresponding legends represent the $p$-zest distributions. From the plots, we find that multiplicity provides a better discrimination in comparison to zest ($p$ = 1) while $p$-zest with $p = 0.3-0.5$ provides the optimal discrimination.\nThis is also confirmed by looking at FIG.~\\ref{ZFilter1} where with 80\\% of signal efficiency as the accepted value, we plot the curve for the background jet rejection offered by $p$-zest with varying values of parameter $p$. From the plot, we find that $p=0.3-0.5$ provides the best discriminating ability for a $Z$-boson initiated jet, although it is not significantly better than just multiplicity.\n\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.5\\textwidth]{ZFilters}\\\\\n\\caption{ROC curves for $Z$ boson-initiated jets with various choices of $p$ compared against the inverse of multiplicity as a filter.}\n \\label{ZFilter}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47\\textwidth]{ZDiscriminate}\n\\caption{Discrimination power offered by $p$-zest at fixed signal efficiency of 80\\% for $Z$-boson initiated jets.}\n \\label{ZFilter1}\n\\end{figure}\n\n\\subsection{top quark-initiated jet}\n\\label{top}\n\nThe ROC curves for top-initiated jets with $p$-zest and $1\/n$ as the observables are presented in FIG.~\\ref{TopFilter}. Interestingly, here we find that $p$-zest for $p= 0.3 - 1.0$ performs significantly better than the $1\/n$ distribution. The same is confirmed further by the background rejection efficiency curve with respect to $p$ at fixed signal efficiency of 80\\%, as shown in FIG.~\\ref{TopDiscriminate}.\n\n\nFrom this analysis, we find that $p$-zest allows us to find the $p$ value that optimizes the discrimination. We note that a $p$ value of about $0.3$ to $0.5$ is optimal for the heavy particles studied here. \n\n\nIt is interesting to note that the discrimination ability offered by our preliminary $p$-zest study for the top quark-initiated jets is comparable with that of a class of ML-based top taggers ~\\cite{Kasieczka:2019dbj}. Although our study does not include proton-proton collisions and detector effects considered in~\\cite{Kasieczka:2019dbj}, we are not claiming a strict comparison of our results with ML-based techniques. While these ML algorithms offer the highest discrimination ability for heavy particle jets, the particular features that offer this improvement remain largely unknown. $p$-Zest may offer some insight into the physics of ML-based taggers. Therefore, we propose that studying IRC unsafe observables based on physical principles may provide a window into the physics hidden in ML based techniques.\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.5\\textwidth]{TopFilters}\n\\caption{ROC curves for top quark-initiated jets with various choices of $p$ compared against the inverse of multiplicity as a filter.}\n \\label{TopFilter}\n\\end{figure} \n\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.47\\textwidth]{TopDiscriminate}\n\\caption{Discrimination power offered by $p$-zest at fixed signal efficiency of 80\\% for top quark initiated jets.}\n \\label{TopDiscriminate}\n\\end{figure}\n\n\\section{Hadronization Model dependence}\n\\label{hadModel}\n\nAs pointed out earlier, $p$-zest is collinear unsafe and cannot be calculated by standard perturbative techniques, therefore we rely on Monte Carlo event generators that incorporate a suitable hadronization model for its computation. In this section, we study the effect of varying hadronization models on zest. This will allow us to understand how the observed value of zest is sensitive to the modeling of non-perturbative effects. This check is important as the observable can only be computed for hadronic final states.\n\n\\begin{figure*}\n\\centering\n \\includegraphics[width=0.7\\textwidth]{hadronPlot}\\\\\n \\includegraphics[width=0.7\\textwidth]{hadron}\n\n \\caption{(a) Zest distribution curves for $Z$ boson-initiated jets with various choices of modelling the non-perturbative physics, and (b) ROC curves for $Z$ boson-initiated jets by varying the hadronization parameters in {\\sc Pythia}.}\n \\label{hadronization}\n\\end{figure*}\nTo study the hadronization model dependence of zest, we vary the non-perturbative physics modelling by: a) changing the hadronization parameters in {\\sc Pythia}, and b) changing the hadronization model by using the {\\sc Herwig} event generator to simulate the event. This will modify the spectrum of the final state primary hadrons, therefore allowing us to study the hadronization model dependence for zest.\n\nIn FIG.~\\ref{hadronization} (a), we present the zest-distribution curves for a $Z$ boson-initiated jet with different clustering mechanisms for implementing the long distance physics. The {\\sc Pythia} 8 event generator that we use for simulating the events, incorporates the Lund string model of hadronization \\cite{Andersson:1983ia} to cluster final state partons into hadrons. This model is built upon a ``string\" analogy, i.e. as the separation between the two partons increases, the potential energy stored in the string rises linearly. At some point, the potential energy stored is so large that the string breaks forming a $q\\bar{q}$ pair leading to individual hadrons, with the `fragmentation function' given by,\n\\begin{equation}\n  f(z) \\propto z^{-1} (1-z)^{a} \\exp(-b\\, m_{\\perp}^{2}\/z)\\, ,\n  \\label{lund}\n\\end{equation}\nwhere $z$ is the energy fraction carried by the hadron and $m_{\\perp}$ is its transverse mass while $a, b$ are some adjustable parameters.   \nChanging $a$ and\/or $b$ in the above equation, changes the way fragmentation happens, hence modifying the final state spectrum.\\footnote{The default values for the parameters $a$ and $b$ used in {\\sc Pythia} 8 are 0.3 and 0.8 respectively. Here\nwe will vary these values to their extremum limits by setting $a$ to 1.8 and $b$ to 2.} The zest-distribution shifts as shown in FIG.~\\ref{hadronization}(a). Moreover, we observe a similar shift for the gluon zest distribution (background) and the performance curves change only very slightly, as shown in FIG.~\\ref{hadronization}(b). For {\\sc Herwig}, we cannot simulate single offshell gluons, thus we do not have corresponding curve for the gluon zest and ROC. However, from the heavy particle distribution for the {\\sc Herwig} simulated event shown by the solid brown curve (with filled triangles) in FIG.~\\ref{hadronization}(a), we see that the zest distribution lies well within the extreme bounds of hadronization parameters used in {\\sc Pythia}, shown by the solid magenta (filled squares) and green (filled diamonds) curves in FIG.~\\ref{hadronization}(a). \nThus, from our analysis we conclude that although zest is not calculable perturbatively, but zest based discrimination is stable against different hadronization models.\n\nSince zest is collinear unsafe, its discrimination ability may depend upon the resolution of the detector. We have also verified that while the zest curves shift if we coarse-grain the angular resolution between the particles for both the gluon-initiated and heavy-particle-initiated jets, they shift systematically such that the discrimination ability of the observable remains unaffected.\n\n\n\\section{Conclusion}\n\\label{Conclusion}\n\nWe have presented a new jet substructure observable, {\\em zest}, and discussed its potential for discriminating standard model heavy particles forming jets from the QCD background of gluon-initiated jets. We have shown that the zest distribution of gluon-initiated jets is stable against the change in jet mass, change of global color flow of the partons and inclusion or exclusion of a few soft particles into\/from the jet. These properties make it a suitable observable for vetoing the large gluon background at the colliders. Though zest is a non-linear and collinear unsafe observable, we have demonstrated that zest-based discrimination is largely insensitive to different hadronization models. We have shown that zest can be generalized through a real parameter, $p$, which can be optimized to further enhance the discrimination ability. A $p$ value between 0.3 to 0.5 is found to provide the optimal discrimination ability for all the heavy particle jets. We have also shown that for a top quark-initiated jet, $p$-zest provides a significant improvement over particle multiplicity in a jet. The discrimination provided by $p$-zest (with $p=0.3-0.5$) for the top quark-initiated jets approaches ML-based results, and we propose that looking at other IRC unsafe jet observables may help to uncover the physics hidden through such ML-based techniques.\n\n\\begin{acknowledgments}\nWe thank Tuhin Roy for discussions on some of the aspects of this paper and comments on the manuscript.\n\\end{acknowledgments}\n\n\n\\bibliographystyle{unsrt}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}