diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzeidu" "b/data_all_eng_slimpj/shuffled/split2/finalzzeidu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzeidu" @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\n\nOne of the major achievements of HERA was the experimental evidence that \namong the whole set of $\\gamma^* p \\to X$ deep inelastic scattering events, almost 10\\% are diffractive (DDIS), of the form $\\gamma^* p \\to X Y$ with a rapidity gap between the proton remnants $Y$\nand the hadrons $X$\ncoming from the fragmentation region of the initial virtual photon~\\cite{Chekanov:2004hy-Chekanov:2005vv-Chekanov:2008fh,\nAktas:2006hx-Aktas:2006hy-Aaron:2010aa-Aaron:2012ad-Aaron:2012hua}.\nDiffraction can be theoretically described according to several approaches, important for phenomenological applications. The first approach involves\na {\\em resolved} Pomeron contribution (with a parton distribution function inside the Pomeron), while the second one\nrelies on a {\\em direct} Pomeron contribution involving the coupling of a Pomeron with the diffractive state. The diffractive states can be modelled in perturbation theory by a $q \\bar{q}$ pair (for moderate $M^2$, where $M$ is the invariant mass of the diffractively produced state $X$) or by higher Fock states as a $q \\bar{q} g$ state for larger values of $M^2$. Based on such a model, with\n a two-gluon exchange picture for the Pomeron, a good description of HERA data for diffraction could be achieved~\\cite{Bartels:1998ea}. One of the important features of this approach is that the $q \\bar{q}$ component with a longitudinally polarized photon plays a crucial role in the region of small\ndiffractive mass $M$, although it is a\ntwist-4 contribution.\nIn the direct components considered there, the $q \\bar{q} g$ diffractive state has been studied in two particular limits. The first one, valid for very large $Q^2$, corresponds to a collinear approximation in which the transverse momentum of the gluon is assumed to be much smaller than the transverse momentum of the emitter~\\cite{Wusthoff:1995hd-Wusthoff:1997fz}. \nThe second one~\\cite{Bartels:1999tn,Bartels:2002ri}, valid for very large $M^2$, is based on the assumption of a strong ordering of longitudinal momenta, encountered in BFKL equation~\\cite{Fadin:1975cb-Kuraev:1976ge-Kuraev:1977fs-Balitsky:1978ic}. Both these approaches were combined in order to describe HERA data for DDIS~\\cite{Marquet:2007nf}. \n\nBased on these very successful developments led at HERA\nin order to understand the QCD dynamics with diffractive events, \nit would be appropriate to look for similar hard diffractive events at LHC. \nThe idea there is to adapt the concept of photoproduction of diffractive jets, which was performed at HERA~\\cite{Chekanov:2007rh,Aaron:2010su}, now with a flux of\nquasi-real photons in ultraperipheral collisions (UPC)~\\cite{Baltz:2007kq-Baur:2001jj}, relying on the notion of equivalent photon approximation. In both cases, \n the hard scale is provided by the invariant mass of the tagged jets.\n\nWe here report on our computation~\\cite{Boussarie:2014lxa} of the $\\gamma^* \\to q \\bar{q} g$ impact factor at tree level with an arbitrary number of $t$-channel gluons described within the Wilson line formalism, also called QCD shockwave approach~\\cite{Balitsky:1995ub-Balitsky:1998kc-Balitsky:1998ya-Balitsky:2001re}. As an aside, we rederive the $\\gamma^* \\to q \\bar{q}$ impact factor. In particular, the \n$\\gamma^* \\to q \\bar{q} g$ transition is computed without any soft or collinear approximation for the emitted gluon, in contrast with the above mentioned calculations. These results provide necessary generalization of building blocks for inclusive DDIS as well as for two- and three-jet diffractive production. Since the results we derived can account for an arbitrary number of $t$-channel gluons, this could allow to include higher twist effects which are suspected to be rather important in DDIS for $Q^2 \\lesssim 5$ GeV$^2$~\\cite{Motyka:2012ty}. \n\n\n\n\n\n\\section{Formalism}\n\nAs stated before, we use Balitsky's shockwave formalism. \nIts application shows that this method is very powerful in determining evolution equations and impact factors at next-to-leading order for inclusive processes~\\cite{Balitsky:2010ze-Balitsky:2012bs}, at semi-inclusive level for $p_t$-broadening in $pA$ collisions~\\cite{Chirilli:2011km-Chirilli:2012jd} or in the evaluation of the triple Pomeron vertex beyond the planar limit~\\cite{Chirilli:2010mw}, when compared with usual methods based on summation of contributions of individual Feynman diagrams computed in momentum space. It is an effective way of estimating the effect of multigluon exchange. Its formulation in coordinate space makes it natural in view of describing saturation~\\cite{GolecBiernat:1998js-GolecBiernat:1999qd}.\nOne introduces Wilson lines as \n\\begin{equation}\nU_{i}=U_{\\vec{z}_{i}}=U\\left( \\vec{z}_{i},\\eta\\right) =P \\exp\\left[{ig\\int_{-\\infty\n}^{+\\infty}b_{\\eta}^{-}(z_{i}^{+},\\vec{z}_{i}) \\, dz_{i}^{+}}\\right]\\,.\n\\label{WL}%\n\\end{equation}\nThe operator $b_{\\eta}^{-}$ is the external shock-wave field built from slow gluons \nwhose momenta are limited by the longitudinal cut-off defined by the rapidity $\\eta$\n\\begin{equation}\nb_{\\eta}^{-}=\\int\\frac{d^{4}p}{\\left( 2\\pi\\right) ^{4}}e^{-ip \\cdot z}b^{-}\\left(\np\\right) \\theta(e^{\\eta}-|p^{+}|).\\label{cutoff}%\n\\end{equation}\nWe use the light cone gauge\n$\\mathcal{A}\\cdot n_{2}=0,$\nwith $\\mathcal{A}$ being the sum of the external field $b$ and the quantum field\n$A$%\n\\begin{equation}\n\\mathcal{A}^{\\mu} = A^{\\mu}+b^{\\mu},\\quad b^{\\mu}\\left( z\\right) =b^{-}(z^{+},\\vec{z}\\,) \\,n_{2}%\n^{\\mu}=\\delta(z^{+})B\\left( \\vec{z}\\,\\right) n_{2}^{\\mu}\\,,\\label{b}%\n\\end{equation}\nwhere\n$B(\\vec{z})$ is a profile function.\nThe dipole operator \n$\\mathbf{U}_{12}=\\frac{1}{N_{c}}\\rm{tr}\\left( U_{1}U_{2}^{\\dagger}\\right) -1$\nwill be used extensively. \n\n\n\\section*{Impact factor for $\\gamma\\rightarrow q\\bar{q}$ transition}\n\n\\begin{figure}\n\\scalebox{.89}{\\begin{tabular}{cc}\n\\raisebox{3.75cm}{\\includegraphics[scale=0.5]{qqbar.pdf}} &\n\\includegraphics[scale=0.65]{NLO.pdf}\n\\end{tabular}}\n\\caption{Left: diagram for $\\gamma\\rightarrow q\\bar{q}$ transition. Right: the 4 diagrams for $\\gamma\\rightarrow q\\bar{q}g$ transition.}\n\\label{Fig:diagrams}\n\\end{figure}\n\nFor $q \\bar{q}$ production one can write, after projection on the color singlet state and subtraction of the non-interacting term\n\\begin{equation}\nM_{0}^{\\alpha}=N_c \\int d\\vec{z}_{1}d\\vec{z}_{2}F\\left( p_{q},p_{\\bar{q}}%\n,z_{0},\\vec{z}_{1},\\vec{z}_{2}\\right) ^{\\alpha} \\mathbf{U}_{12}\\,.\n\\label{M0int}%\n\\end{equation}\nDenoting $Z_{12} = \\sqrt{x_{q}x_{\\bar{q}}\\vec{z}_{12}^{\\,\\,2}}$, we get for a longitudinal photon\n\\begin{eqnarray}\n\\label{FL}\nF\\left( p_{q},p_{\\bar{q}},k,\\vec{z}_{1},\\vec{z}_{2}\\right) ^{\\alpha\n}\\varepsilon_{L\\alpha}=\\theta(p_{q}^{+})\\,\\theta(p_{\\bar{q}}^{+})\\frac\n{\\delta\\left( k^{+}-p_{q}^{+}-p_{\\bar{q}}^{+}\\right) }{(2\\pi)^{2}}%\ne^{-i\\vec{p}_{q}\\cdot \\vec{z}_{1}-i\\vec{p}_{_{\\bar{q}}}\\cdot\\vec{z}_{2}}\n(-2i)\\delta_{\\lambda_{q},-\\lambda_{\\bar{q}}}\\,x_{q}x_{\\bar{q}}%\n\\,Q\\,K_{0}\\left(Q \\, Z_{12}\\right)\\,,\n\\end{eqnarray}\nand for a transverse photon\n\\begin{eqnarray}\n\\label{FT}\nF( p_{q},p_{\\bar{q}},k,\\vec{z}_{1},\\vec{z}_{2}) ^{j}%\n\\varepsilon_{Tj}\\!=\\theta(p_{q}^{+})\\,\\theta(p_{\\bar{q}}^{+})\\frac{\\delta(\nk^{+}\\!\\!-\\!p_{q}^{+}\\!-p_{\\bar{q}}^{+}\\!) }{(2\\pi)^{2}}e^{-i\\vec{p}_{q}\\cdot\\vec\n{z}_{1}-i\\vec{p}_{_{\\bar{q}}}\\cdot\\vec{z}_{2}}\n\\delta_{\\lambda_{q},-\\lambda_{\\bar{q}}}( x_{q}-x_{\\bar{q}%\n}+s\\lambda_{q}) \\frac{\\vec{z}_{12} \\cdot \\vec{\\varepsilon}_{T}}{\\vec{z}_{12}^{\\,\\,2}}\nQ \\,Z_{12} K_{1}(Q\\, Z_{12})\\,.\\!\\!\\!\\!\\!\n\\end{eqnarray}\n\n\\section*{Impact factor for $\\gamma\\rightarrow q\\bar{q}g$ transition}\n\nFor $q \\bar{q} g$ production, projecting on the color singlet state and subtracting the non-interacting term again, one can write\n\\begin{eqnarray}\n\\nonumber \nM^{\\alpha} &=& N_c^2 \\int d\\vec{z}_{1}d\\vec{z}_{2}d\\vec{z}_{3} \\, F_{1}\\left( p_{q},p_{\\bar{q}}%\n,p_{g},z_{0},\\vec{z}_{1},\\vec{z}_{2},\\vec{z}_{3}\\right) ^{\\alpha}\\frac{1}{2}\n\\left( \\mathbf{U}_{32} + \\mathbf{U}_{13} - \\mathbf{U}_{12} + \\mathbf{U}_{32}\\mathbf{U}_{13} \\right)\n\\\\\n&+& N_c \\int d\\vec{z}_{1}d\\vec{z}_{2} \\, F_{2}\\left( p_{q},p_{\\bar{q}},p_{g},z_{0}%\n,\\vec{z}_{1},\\vec{z}_{2}\\right) ^{\\alpha}\\frac{N_{c}^{2}-1}{2N_{c}} \\mathbf{U}_{12}\\,.\n\\label{F2tilde}%\n\\end{eqnarray}\nThe first and the second line of this equation correspond respectively to the two last diagrams of the first line and to the second line of diagrams of Fig.~\\ref{Fig:diagrams}.\nFor a longitudinally polarized photon, they read\n\\begin{eqnarray}\n&&\\hspace{-1cm}F_{1}\\left( p_{q},p_{\\bar{q}},p_{g},k,\\vec{z}_{1},\\vec{z}_{2},\\vec{z}%\n_{3}\\right) ^{\\alpha}\\varepsilon_{L\\alpha}=2\\,Q\\,g\\,\\delta(k^{+}-p_{g}^{+}-p_{q}%\n^{+}-p_{_{\\bar{q}}}^{+})\\theta(p_{g}^{+}-\\sigma)\\frac{e^{-i\\vec{p}_{q} \\cdot %\n\\vec{z}_{1}-i\\vec{p}_{_{\\bar{q}}} \\cdot \\vec{z}_{_{2}}-i\\vec{p}_{g} \\cdot \\vec{z}_{3}}}%\n{\\pi\\sqrt{2p_{g}^{+}}}\n\\nonumber \\\\\n&\\times&\\delta_{\\lambda_{q},-\\lambda_{\\bar{q}}}\\left\\{ (x_{_{\\bar{q}}}%\n+x_{g}\\delta_{-s_{g}\\lambda_{q}})x_{q}\\frac{\\vec{z}_{32} \\cdot \\vec{\\varepsilon}%\n_{g}^{\\,\\,\\ast}}{\\vec{z}_{32}^{\\,\\,2}}-(x_{q}+x_{g}\\delta_{-s_{g}%\n\\lambda_{\\bar{q}}})x_{_{\\bar{q}}}\\frac{\\vec{z}_{31} \\cdot \\vec{\\varepsilon}%\n_{g}^{\\,\\,\\ast}}{\\vec{z}_{31}^{\\,\\,2}}\\right\\} K_{0}(QZ_{123}) \\,,\\\\\n\\label{F1eL}%\n\\label{resF2L}\n&&\\hspace{-1cm}\\tilde{F}_{2}\\left( p_{q},p_{\\bar{q}},p_{g},k,\\vec{z}_{1},\\vec{z}_{2}\\right)\n^{\\alpha}\\varepsilon_{L\\alpha}=4ig \\, Q\\,\\theta(p_{g}^{+}-\\sigma)\\delta(k^{+}%\n-p_{g}^{+}-p_{q}^{+}-p_{_{\\bar{q}}}^{+})\\frac{e^{-i\\vec{p}_{q} \\cdot \\vec{z}%\n_{1}-i\\vec{p}_{_{\\bar{q}}} \\cdot \\vec{z}_{2}}}{\\sqrt{2p_{g}^{+}}}%\n\\nonumber \\\\\n&&\\times\\delta_{\\lambda_{q},-\\lambda_{\\bar{q}}}\\frac{x_{q}\\left( x_{g}%\n+x_{\\bar{q}}\\right) \\left( \\delta_{-s_{g}\\lambda_{q}}x_{g}+x_{\\bar{q}%\n}\\right) }{x_{\\bar{q}} \\, x_g }\\frac{\\vec{P}_{\\bar{q}} \\cdot %\n\\vec{\\varepsilon}_{g}^{\\,\\,\\ast}}{\\vec{P}_{\\bar{q}}^2}\\,e^{-i\\vec{p}_{g} \\cdot \\vec{z}_{2}}K_{0}%\n(QZ_{122})-\\left( q\\leftrightarrow\\bar{q}\\right) ,\n\\end{eqnarray}\nwhile for a transversally polarized photon, we have \n\\begin{eqnarray}\n&&\\hspace{-.7cm}F_{1}\\left( p_{q},p_{\\bar{q}},p_{g},k,\\vec{z}_{1},\\vec{z}_{2},\\vec{z}%\n_{3}\\right) ^{\\alpha}\\!\\varepsilon_{T\\alpha}=\\!-2i\\,g\\,Q\\delta(k^{+}-p_{g}^{+}%\n-p_{q}^{+}-p_{_{\\bar{q}}}^{+})\\theta(p_{g}^{+}-\\sigma)\n\\frac{e^{-i\\vec{p}_{q} \\cdot \\vec{z}_{1}-i\\vec{p}_{_{\\bar{q}}} \\cdot \\vec{z}_{_{2}%\n}-i\\vec{p}_{g} \\cdot \\vec{z}_{3}}}{\\pi Z_{123}\\sqrt{2p_{g}^{+}}}\\delta_{\\lambda\n_{q},-\\lambda_{\\bar{q}}}K_{1}(QZ_{123})\\\\\n&&\\hspace{-.8cm}\\times\\left\\{ \\frac{\\left( \\vec{z}%\n_{23} \\cdot \\vec{\\varepsilon}_{g}^{\\,\\,\\ast}\\right) \\left( \\vec{z}_{13} \\cdot %\n\\vec{\\varepsilon}_{T}\\right) }{\\vec{z}_{23}{}^{2}}x_{q}\\left( x_{q}%\n-\\delta_{s\\lambda_{\\bar{q}}}\\right) \\left( x_{\\bar{q}}+x_{g}\\delta\n_{-s_{g}\\lambda_{q}}\\right) +\\frac{\\left( \\vec{z}_{23} \\cdot \\vec{\\varepsilon}_{g}^{\\,\\,\\ast}\\right)\n\\left( \\vec{z}_{23} \\cdot \\vec{\\varepsilon}_{T}\\right) }{\\vec{z}_{23}{}^{2}}%\nx_{q}x_{\\bar{q}}\\left( x_{\\bar{q}}+x_{g}\\delta_{-s_{g}\\lambda_{q}}%\n-\\delta_{s\\lambda_{q}}\\right) \\right\\} -\\left( q\\leftrightarrow\\bar\n{q}\\right) \\, ,\\nonumber\\\\\n\\label{F1eT}%\n\\label{resF2tildeT}\n&&\\hspace{-.8cm}\\tilde{F}_{2}\\left( p_{q},p_{\\bar{q}},p_{g},k,\\vec{z}_{1},\\vec{z}_{2}\\right)\n^{\\alpha}\\varepsilon_{T\\alpha}=-4g\\,\\theta(p_{g}^{+}-\\sigma)\\,\\delta(k^{+}%\n-p_{g}^{+}-p_{q}^{+}-p_{_{\\bar{q}}}^{+})\\frac{e^{-i\\vec{p}_{q} \\cdot \\vec{z}%\n_{1}-i\\vec{p}_{_{\\bar{q}}} \\cdot \\vec{z}_{2}}}{\\sqrt{2p_{g}^{+}}}\\delta_{\\lambda\n_{q},-\\lambda_{\\bar{q}}}%\n\\nonumber \\\\\n&&\\times \n\\frac{\\left( \\delta\n_{\\lambda_{\\bar{q}}s}-x_{q}\\right) \\left( \\delta_{-s_{g}\\lambda_{q}}%\nx_{g}+x_{\\bar{q}}\\right)}{x_{\\bar{q}} \\, x_g} \n\\frac{\\vec{P}_{\\bar{q}} \\cdot %\n\\vec{\\varepsilon}_{g}^{\\,\\,\\ast}}{\\vec{P}_{\\bar{q}}^2}\n\\frac{\\vec{z}_{12} \\cdot \\vec{\\varepsilon}_{T}}{\\vec{z}_{12}^2} \n\\, Q \\, Z_{122}\nK_{1}(QZ_{122})e^{-i\\vec{p}_{g} \\cdot \\vec{z}_{2}%\n}-\\left( q\\leftrightarrow\\bar{q}\\right) \\,. \n\\end{eqnarray}\nWe denote \n$ F_{2}\\left( p_{q},p_{\\bar{q}},p_{g},z_{0},\\vec{z}_{1},\\vec{z}_{2}\\right)^{\\alpha}\\!=\\!\\tilde{F}_{2}\\left( p_{q},p_{\\bar{q}},p_{g},z_{0},\\vec{z}_{1}%\n ,\\vec{z}_{2}\\right) ^{\\alpha}\\!+\\!\\int d\\vec{z}_{3}\\,F_{1}\\left( p_{q},p_{\\bar{q}%\n },p_{g},z_{0},\\vec{z}_{1},\\vec{z}_{2},\\vec{z}_{3}\\right) ^{\\alpha}\\!.$\n\n\n\\section*{2- and 3-gluon approximation}\n\n\nLet us notice that the dipole operator $\\mathbf{U}_{ij}$ is of order $g^2$. Hence for only two or three exchanged gluons one can neglect the quadrupole term in the amplitude $M^{\\alpha}$ and get \n\\begin{eqnarray}\n\\label{M3gBis}\nM^{\\alpha} \\overset{\\mathrm{g^3}}{=} \\frac{1}{2}\\int d\\vec{z}_{1}d\\vec{z}%\n_{2} \\mathbf{U}_{12} \\left[ \\left( N_{c}^{2}-1\\right)\n\\tilde{F}_{2}\\left( \\vec{z}_{1},\\vec{z}%\n_{2}\\right) ^{\\alpha} + \\int d\\vec{z}_{3} \\left\\{ N_{c}^{2}F_{1}\\left(\n\\vec{z}_{1},\\vec{z}_{3},\\vec{z}_{2}\\right)^{\\alpha} +N_{c}^{2}F_{1}\\left( \\vec{z}_{3},\\vec{z}%\n_{2},\\vec{z}_{1}\\right) ^{\\alpha} - F_{1}\\left( \\vec{z}_{1},\\vec{z}_{2},\\vec{z}_{3}\\right) ^{\\alpha} \\right\\} \\right].\n %\n\\end{eqnarray}\nFor $\\vec{p}_q=\\vec{p}_g=\\vec{p}_{\\bar{q}}=\\vec{0}$, those integrals can be performed analytically. Otherwise they can be expressed as a simple convergent integral over $[0,1]$ that can be performed numerically for any future phenomenological study. \n\n\\section*{Conclusion}\n\nThe measurement of dijet production in DDIS was recently performed~\\cite{Aaron:2011mp}, and a precise comparison of \ndijet versus triple-jet production, which has not been performed yet at HERA~\\cite{Adloff:2000qi}, would be very useful to get a deeper understanding of the QCD mechanism underlying diffraction. Recent investigations of the azimuthal distribution of dijets in diffractive photoproduction performed by ZEUS~\\cite{Guzik:2014iba} show sign of a possible need for a 2-gluon exchange model, which is part of the shock-wave mechanism. Our calculation could be used for phenomenological studies of those experimental results.\nA similar and very complementary study could be performed at LHC with UPC events. One should note that getting a full quantitative first principle analysis of this would require an evaluation of virtual corrections to the $\\gamma^* \\rightarrow q\\bar{q}$ impact factor, which are presently under study~\\cite{Boussarie:prep}.\n\nDiffractive open charm production was measured at HERA~\\cite{Aktas:2006up} \nand studied in the large $M$ limit based on the direct coupling between a Pomeron and a $q \\bar{q}$ or a $q\\bar{q}g$ state, with massive quarks~\\cite{Bartels:2002ri}. Such a program could also be performed at LHC, \nagain based on UPCs and on\nthe extension of the above mentioned impact factors to the case of a massive quark. This could be further extended \nto $J\/\\Psi$ production, which are copiously produced at LHC.\n\n\n\\begin{theacknowledgments}\nA. V. G. acknowledges support of president grant MK-7551.2015.2 and\nRFBR grant 13-02-01023. This work was partially supported by the PEPS-PTI PHENODIFF,\nthe PRC0731 DIFF-QCD, the Polish Grant NCN No. DEC-2011\/01\/B\/ST2\/03915, the ANR PARTONS (ANR-12-MONU-0008-01), the COPIN-IN2P3 Agreement\nand the Joint Research Activity Study of Strongly Interacting Matter (acronym HadronPhysics3,\nGrant Agreement n.283286) under the Seventh Framework Programme of the\nEuropean Community\n\\end{theacknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:Introduction}\n\nOhta-Kawasaki (OK) model is introduced in \\cite{OhtaKawasaki_Macromolecules1986} and has been extensively applied for the study of phase separation of diblock copolymers, which\nhave generated much interest in materials science in the past years due to their remarkable ability for self-assembly into nanoscale ordered structures \\cite{Hamley_Wiley2004}. Diblock copolymers are chain molecules made by two different segment species, say $A$ and $B$ species. Due to the chemical incompatibility, the two species tend to be phase-separated; on the other hand, the two species are connected by covalent chemical bonds, which leads to the so-called microphase separation. The OK model can describe such microphase separation for diblock copolymers via a free energy functional:\n\\begin{align}\\label{functional:OK}\nE^{\\text{OK}}[\\phi] = \\int_{\\mathbb{T}^d} \\dfrac{\\epsilon}{2}|\\nabla\\phi|^2 + \\dfrac{1}{\\epsilon}W(\\phi)\\ \\text{d}x + \\dfrac{\\gamma}{2}\\int_{\\mathbb{T}^d} |(-\\Delta)^{-\\frac{1}{2}}(f(\\phi)-\\omega)|^2\\ \\text{d}x,\n\\end{align}\nwith a volume constraint\n\\begin{align}\\label{eqn:Volume}\n\\int_{\\mathbb{T}^d} (f(\\phi) - \\omega)\\ \\text{d}x = 0.\n\\end{align}\nHere $\\mathbb{T}^d = \\prod_{i=1}^d [-X_i, X_i] \\subset \\mathbb{R}^d, d = 2, 3$ denotes a periodic box and $0<\\epsilon \\ll 1$ is an interface parameter that indicates the system is in deep segregation regime. $\\phi = \\phi(x)$ is a phase field labeling function which represents the concentration of $A$ species. By the assumption of incompressibility for the binary system, the concentration of $B$ species can be implicitly represented by $1-\\phi(x)$. Function $W(\\phi) = 18(\\phi^2-\\phi)^2$ is a double well potential which enforces the phase field function $\\phi$ to be equal to 1 inside the interface and 0 outside the interface. Near the interfacial region, the phase field function $\\phi$ rapidly but smoothly transitions from 0 to 1. A new term of $f(\\phi) = 3\\phi^2 - 2\\phi^3$ is introduced in the free energy functional to resemble $\\phi$ as the indicator for the $A$ species. The first integral in (\\ref{functional:OK}) is a local surface energy which represents the short-range interaction between the chain molecules and favors the large domain; while the second integral in (\\ref{functional:OK}) is a term for the long-range (nonlocal) repulsive interaction with $\\gamma >0$ being the strength of the repulsive force. Finally, $\\omega\\in(0,1)$ is the relative volume of the $A$ species.\n\nTo study the microphase separation and the pattern formation of the diblock copolymer, we consider the $L^2$ gradient flow dynamics of the OK model. On the other hand, to relax the volume constraint (\\ref{eqn:Volume}), we can incorporate a penalty term into (\\ref{functional:OK}) and change it into an unconstrained one:\n\\begin{align}\\label{functional:pOK}\nE^{\\text{pOK}}[\\phi] = \\int_{\\mathbb{T}^d} \\dfrac{\\epsilon}{2}|\\nabla\\phi|^2 + \\dfrac{1}{\\epsilon}W(\\phi)\\ \\text{d}x + \\dfrac{\\gamma}{2}\\int_{\\mathbb{T}^d} |(-\\Delta)^{-\\frac{1}{2}}(f(\\phi)-\\omega)|^2\\ \\text{d}x + \\dfrac{M}{2}\\left[ \\int_{\\mathbb{T}^d} f(\\phi)-\\omega\\ \\text{d}x \\right]^2,\n\\end{align}\nwhere $M\\gg1$ is a penalty constant. Then we can consider the corresponding penalized $L^2$ gradient flow dynamics with given initial $\\phi(x,t=0) = \\phi_0$, which thereafter is called penalized Allen-Cahn-Ohta-Kawasaki (pACOK) equation:\n\\begin{align}\\label{eqn:pACOK}\n\\dfrac{\\partial}{\\partial t} \\phi = \\epsilon\\Delta\\phi - \\dfrac{1}{\\epsilon}W'(\\phi) - \\gamma(-\\Delta)^{-1}(f(\\phi)-\\omega)f'(\\phi) - M\\int_{\\mathbb{T}^d} (f(\\phi)-\\omega)\\ \\text{d}x\\cdot f'(\\phi).\n\\end{align}\n\nOur main contribution in this paper is threefold. Firstly, the new form of $f(\\phi)$ guarantees that the pACOK equation is maximum principle preserving (MPP), namely, if the initial data is bounded $0\\le\\phi_0\\le1$, then $0\\le\\phi(x)\\le1$ for any later time. Secondly, we adopt a linear splitting method to the pACOK equation and then apply a semi-implicit scheme for the numerical simulations. This scheme treats the linear terms implicitly but all the nonlinear and nonlocal terms explicitly, and with proper choice for the splitting constant, it inherits the MPP property at both time-discrete and fully-discrete levels. Besides, just as the energy dissipation law (energy stability) is obeyed by the continuous $L^2$ gradient flow (\\ref{eqn:pACOK}), the proposed numerical scheme also successfully inherits the energy stability at both time-discrete and fully-discrete levels. Thirdly, the error estimate analysis is carried and the rate of convergence is verified by numerical simulations.\n\nThe inclusion of a new nonlinear term $f(\\phi)$ in the OK model makes the key novelty in this paper. On one hand, this term $f(\\phi)$ accounts for mimicking the behavior of $\\phi$ as the indicator for the $A$-rich region, so it satisfies the condition:\n\\begin{align}\\label{eqn:f_condition}\nf(0) = 0, \\quad f(1) = 1.\n\\end{align}\nOn the other hand, in order to pin the phase field label $\\phi$ at 1 and 0 inside and outside the $A$-$B$ interface, respectively, we set an extra condition\n\\begin{align}\\label{eqn:f'_condition}\nf'(0) = 0, \\quad f'(1) = 0.\n\\end{align}\nso that the evolution of the pACOK dynamics (\\ref{eqn:pACOK}) only updates the phase field $\\phi$ near the interface but not in the away-from-interface region. This will help maintain $\\phi$ as a desired tanh profile better than simply taking $f(\\phi) = \\phi$. See \\cite{Zhao_2018CMS, WangRenZhao_CMS2019, XuZhao_JSC2019} for the numerical comparisons between linear and nonlinear choices of $f(\\phi)$. The polynomial of the smallest degree satisfying both conditions (\\ref{eqn:f_condition}) and (\\ref{eqn:f'_condition}) is\n\\begin{align}\\label{eqn:f}\nf(\\phi) = 3\\phi^2 - 2\\phi^3.\n\\end{align}\nIn some scenario, a polynomial of higher degree might be required. For instance, in the study of energy stable numerical scheme based on operator splitting, one needs to perform a linear extension for the nonlinearity $f$ up to second order continuous derivative, then the minimal degree has to be fifth \\cite{XuZhao_JSC2019}. The real magic that $f(\\phi)$ in (\\ref{eqn:f}) plays is that it preserves the maximum principle. The key observation is that $f'$ and $W'$ share a common factor $\\phi - \\phi^2$, such that any possible growth on $\\phi$ (which potentially breaks the MPP) can be safely killed by the double well potential term to save the MPP. See the proof of Theorem \\ref{theorem:MPP_continuous} for details.\n\n\n\nThe MPP is an important property held by the Allen-Cahn equation. It says that if the initial data is bounded between 0 and 1, then the solution remains between 0 and 1 for any later time. In recent years, efforts have been devoted to investigate the MPP numerical schemes for the Allen-Cahn equation. Tang and Yang studied a first order implicit-explicit scheme for the MPP property for the Allen-Cahn equation in \\cite{TangYang_JCM2016}. They further extended the results to the generalized Allen-Cahn equation in \\cite{ShenTangYang_CMS2016}. Later, some attempts have been made to study second order MPP schemes for fractional-in-space Allen-Cahn equation \\cite{HouTangYang_JSC2017} and nonlocal Allen-Cahn equation \\cite{DuJuLiQiao_SINA2019}. Recently some adaptive second order MPP schemes have been considered for the Allen-Cahn equation \\cite{LiaoTangZhou_SINA2019} and time-fractional Allen-Cahn equation \\cite{LiaoTangZhou_JCP2019}.\n\nIn this paper, as a first attempt, we will explore the MPP scheme for an Allen-Cahn type dynamics with a long-range interaction term. The new ingredient in the MPP scheme is inspired by the continuous MPP property, namely, the nonlinear function $f(\\cdot)$ which satisfies the condition (\\ref{eqn:f'_condition}). However two things are different between the continuous and the discrete settings. One is that $f(\\cdot)$ has to be linearly extended to 0 and 1 in the continuous case but not in the discrete case. See the equation (\\ref{eqn:f_extension}) and the related discussion. The other is that in the continuous case, $f(\\cdot)$ has to be of the smallest degree to satisfy (\\ref{eqn:f'_condition}) in order to be well controlled by the double well potential $W$, while in the discrete case, any polynomial $f(\\cdot)$ satisfying (\\ref{eqn:f'_condition}) (and (\\ref{eqn:f_condition})) would do the trick. Allowing weaker conditions for the discrete schemes is due to the fact that the MPP of $\\phi(t)$ in the continuous pACOK dynamics depends on the entire history before $t$; while the discrete MPP of $\\phi^{n}$ only depends on the $k$ previous states $\\{\\phi^{j}\\}_{j=n-k}^{n-1}$ (in this paper, we focus on the the case of $k=1$). See the proofs of Theorem \\ref{theorem:MPP_continuous}, Theorem \\ref{theorem:MPP_semi} and Theorem \\ref{theorem:MPP_fully} for details. This indeed provides much flexibility on choosing $f(\\cdot)$ to exploit various discrete MPP schemes for Allen-Cahn type dynamics with long-range interactions.\n\nFor the discrete MPP schemes, the Lemma \\ref{lemma:MPP_condition} (for the time-discrete case) and Lemma \\ref{lemma:MPP_condition2} (for the fully-discrete case) play the key role which will be crucial for the analysis of not only the first order MPP scheme in this paper but also potentially for other higher order ones. Plus, these two lemmas suggest that the nonlocal terms might have to be treated explicitly in order to satisfy the discrete MPP.\n\n\nOur work is by no mean an additional extension of the existing work on MPP by changing from one model to another. This work has potential wider impact on many other applications. Indeed, we further extend this model to binary systems with various long-range interactions. See Section \\ref{section:GeneralFramework} for the detailed discussion on the extension. Our work could provide a general framework to explore the MPP numerical schemes for other applications such as the micromagnetic model for garnet films \\cite{CondetteMelcherSuli_MathComp2010} , FitzHugh-Nagumo system\\cite{RenTruskinovsky_Elasticity2000} , implicit solvation model\\cite{Zhao_2018CMS} etc.\n\n\nSince discrete energy stability is a byproduct when exploring the MPP schemes, we briefly review some of the existing work for the energy stable numerical methods. The energy stable schemes, first studied by Du and Nicolaides in \\cite{DuNicolaides_SINA1991} for a second order accurate unconditionally stable time-stepping scheme for the Cahn-Hilliard equation, has been extensively studied for various $L^2$ and $H^{-1}$ gradient flow dynamics such as the standard Allen-Cahn and Cahn-Hilliard equations \\cite{ShenYang_DCDSA2010}, phase field crystal model \\cite{WiseWangLowengrub_SINA2009, HuWiseWangLowengrub_JCP2009}, modified phase field crystal model \\cite{WangWise_SINA2011}, and epitaxial thin film growth model \\cite{ChenCondeWangWangWise_JSC2012} etc. Several popular numerical schemes adopted by the community are listed below. One is the convex splitting method \\cite{Eyre_Proc1998} in which the double well potential $W(\\phi)$ is split into the sum of a convex function and a concave one, and the convex part is treated implicitly and the concave one is treated explicitly. However, a nonlinear system usually needs to be solved at each time step which induces high computational cost. Another widely adopted method is the stabilized semi-implicit method \\cite{XuTang_SINA2006,ShenYang_DCDSA2010} in which $W(\\phi)$ is treated explicitly. A linear stabilizing term is added to maintain the energy stability. Another recent method is the IEQ method \\cite{ChengYangShen_JCP2017, Yang_JCP2016} in which all nonlinear terms are treated semi-implicitly, the energy stability is preserved and the resulting numerical schemes lead to a symmetric positive definite linear system to be solved at each time step. A variation of the IEQ method, which is called SAV method, is well studied in the last couple of years \\cite{ShenXuYang_SIAMReview2019}. For a more comprehensive review on the topics of the modeling and numerical methods of phase field approach, we refer the interested readers to \\cite{DuFeng_Handbook2020}.\n\nSome conventional notations adopted throughout the paper are collected here. We will denote by $\\|\\cdot\\|_{L^p}$ and $\\|\\cdot\\|_{H^s}$ the standard norms for the periodic Sobolev spaces $L^p_{\\text{per}}(\\mathbb{T}^d)$ and $H^s_{\\text{per}}(\\mathbb{T}^d)$. The standard $L^2$ inner product will be denoted by $\\langle \\cdot, \\cdot \\rangle$. In order to make the MPP satisfied by the pACOK equation (\\ref{eqn:pACOK}), the nonlinear function $f$ needs to be extended to $\\tilde{f}$ as follows:\n\\begin{align}\\label{eqn:f_extension}\n\\tilde{f} =\n\\begin{cases}\n0, \\hspace{0.64in} s < 0; \\\\\n3s^2 - 2s^3, \\quad 0 \\le s \\le 1; \\\\\n1, \\hspace{0.64in} s > 1.\n\\end{cases}\n\\end{align}\nWe still use $f$ to denote such an extension for the brevity of notations. Indeed, the extension of $f$ is only used for the proof of the MPP property for the continuous pACOK equation (\\ref{eqn:pACOK}). For the time-discrete and fully-discrete pACOK equations, the unextended $f(s) = 3s^2 - 2s^3$ suffices to guarantee the MPP and energy stability, see the proofs of Theorem \\ref{theorem:MPP_continuous}, Theorem \\ref{theorem:MPP_semi} and Theorem \\ref{theorem:MPP_fully} for the details. We take\n\\[\nL_{W''}: = \\|W''\\|_{L^{\\infty}[0,1]}, \\quad L_{f''}: = \\|f''\\|_{L^{\\infty}[0,1]}, \\quad L_{f'} = \\|f'\\|_{L^{\\infty}[0,1]}.\n\\]\nNext, $\\|(-\\Delta)^{-1}\\|$ denotes the optimal constant such that $\\|(-\\Delta)^{-1}f\\|_{L^{\\infty}}\\le C \\|f\\|_{L^{\\infty}}$, namely, it is the norm of the operator $(-\\Delta)^{-1}$ from $L^{\\infty}(\\mathbb{T}^d)$ to itself. We will take $[\\![ n]\\!]$ to be the set of integers $\\{1,2,\\cdots, n\\}$. Lastly, we denote $\\tilde{\\omega} = \\max\\{\\omega, 1-\\omega\\}$.\n\nThe rest of the paper is organized as follows. In Section 2, we will prove the MPP property for the continuous pACOK dynamics. In Section 3, a first order time-discrete numerical scheme will be studied which inherits the MPP and energy stability. In Section 4, we will conduct analysis of MPP and energy stability for the fully-discrete scheme. The error estimate will be carried as well. The extension of the MPP to general binary systems with long-range interactions is discussed in Section 5. We will present some numerical results to support our theoretical findings in Section 6, followed by a summary in Section 7. In the appendix, we present the wellposedness of the pACOK equation and the $L^{\\infty}$ bound for the weak solution of the pACOK equation.\n\n\n\n\n\n\\section{MPP for the continuous pACOK dynamics}\n\nIn this section, we will prove that the continuous pACOK equation (\\ref{eqn:pACOK}) satisfies the MPP, and one can see the critical role that $f(\\phi)$ plays in the theory. Note that in this section, $f(\\cdot)$ represents the extended version (\\ref{eqn:f_extension}).\n\n\n\n\\begin{theorem}\\label{theorem:MPP_continuous}\nThe pACOK equation (\\ref{eqn:pACOK}) is MPP, namely, if $0\\le \\phi_0 \\le 1$, then $0\\le \\phi(t) \\le 1$ for any $t>0$, provided that $\\phi_0\\in H^1(\\mathbb{T}^d)$ and\n\\begin{align}\\label{eqn_MPPcondition_continuous}\n\\frac{\\epsilon\\tilde{\\omega}}{6}\\Big[ \\gamma\\|(-\\Delta)^{-1}\\| + M |\\mathbb{T}^d| \\Big] \\le 1.\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nMultiplying on the two sides of (\\ref{eqn:pACOK}) by $2\\phi-1$, one has\n\\begin{align*}\n\\dfrac{\\partial}{\\partial t} (\\phi^2-\\phi) =&\\ \\epsilon \\Delta(\\phi^2-\\phi) - 2\\epsilon|\\nabla\\phi|^2 - \\dfrac{36}{\\epsilon}(\\phi^2-\\phi)(2\\phi-1)^2 \\\\\n&- \\gamma(-\\Delta)^{-1}(f(\\phi)-\\omega)\\cdot 6(\\phi-\\phi^2)(2\\phi-1) - M\\int_{\\mathbb{T}^d} (f(\\phi)-\\omega)\\ \\text{d}x\\cdot 6(\\phi-\\phi^2)(2\\phi-1)\n\\end{align*}\nMultiplying on the two sides of the above equation by $(\\phi^2-\\phi)^+$ and taking integral over $\\mathbb{T}^d$, one has\n\\begin{align*}\n&\\dfrac{1}{2}\\dfrac{\\partial}{\\partial t} \\int_{\\mathbb{T}^d} |(\\phi^2-\\phi)^+|^2\\ \\text{d}x \\\\\n=&\\ -\\epsilon \\int_{\\mathbb{T}^d} |\\nabla(\\phi^2-\\phi)^+|^2\\ \\text{d}x - 2\\epsilon \\int_{\\mathbb{T}^d} |\\nabla\\phi|^2 (\\phi^2-\\phi)^+\\ \\text{d}x - \\dfrac{36}{\\epsilon}\\int_{\\mathbb{T}^d}|(\\phi^2-\\phi)^+|^2 (2\\phi-1)^2\\ \\text{d}x \\\\\n& + 6\\gamma \\int_{\\mathbb{T}^d} (-\\Delta)^{-1}(f(\\phi)-\\omega) |(\\phi^2-\\phi)^+|^2(2\\phi-1)\\ \\text{d}x + 6M\\int_{\\mathbb{T}^d} (f(\\phi)-\\omega)\\ \\text{d}x \\int_{\\mathbb{T}^d} |(\\phi^2-\\phi)^+|^2(2\\phi-1)\\ \\text{d}x \\\\\n=&\\ -\\epsilon \\int_{\\mathbb{T}^d} |\\nabla(\\phi^2-\\phi)^+|^2\\ \\text{d}x - 2\\epsilon \\int_{\\mathbb{T}^d} |\\nabla\\phi|^2 (\\phi^2-\\phi)^+\\ \\text{d}x - \\dfrac{36}{\\epsilon}\\int_{\\mathbb{T}^d}|(\\phi^2-\\phi)^+|^2 \\Big( (2\\phi-1)^2 - (A+B)(2\\phi-1) \\Big)\\ \\text{d}x\n\\end{align*}\nwhere\n\\[\nA = \\dfrac{\\gamma\\epsilon}{6}(-\\Delta)^{-1}(f(\\phi)-\\omega), \\quad B = \\dfrac{M\\epsilon}{6}\\int_{\\mathbb{T}^d}(f(\\phi)-\\omega)\\ \\text{d}x.\n\\]\nWhen $\\phi^2-\\phi\\ge0$, one has $|2\\phi-1| \\ge 1$. Note that the condition (\\ref{eqn_MPPcondition_continuous}) implies $\\|A\\|_{L^{\\infty}} + |B|\\le 1$, therefore $(2\\phi-1)^2 - (A+B)(2\\phi-1)\\ge0$, which implies that\n\\[\n\\dfrac{1}{2}\\dfrac{\\partial}{\\partial t} \\int_{\\mathbb{T}^d} |(\\phi^2-\\phi)^+|^2\\ \\text{d}x \\le 0.\n\\]\nTaking integral for time from 0 to $t$ leads\n\\[\n\\int_{\\mathbb{T}^d} |(\\phi^2-\\phi)^+|^2(t)\\ \\text{d}x \\le \\int_{\\mathbb{T}^d} |(\\phi^2-\\phi)^+|^2(0)\\ \\text{d}x.\n\\]\nIf $0\\le \\phi(0) \\le 1$, $\\int_{\\mathbb{T}^d} |(\\phi^2-\\phi)^+|^2(0)\\ \\text{d}x=0$, then\n\\[\n\\int_{\\mathbb{T}^d} |(\\phi^2-\\phi)^+|^2(t)\\ \\text{d}x \\le 0 \\Rightarrow 0\\le \\phi(t) \\le 1,\n\\]\nwhich completes the proof.\n\\end{proof}\n\n\n\n\\begin{remark}\nThe wellposedness of the pACOK equation (\\ref{eqn:pACOK}) can be well established by using the standard minimization movement scheme, see Theorem \\ref{theorem:wellposedness} in the Appendix for the related discussion.\n\\end{remark}\n\n\n\\begin{remark}\nFor the condition (\\ref{eqn_MPPcondition_continuous}), it is theoretically easy to achieve due to the smallness of the interfacial width $\\epsilon$, though the long-range repulsion strength $\\gamma$ and the penalty constant $M$ are supposed to be large.\n\\end{remark}\n\n\\begin{remark}\nThe extension of $f$ is critical in order to bound the $A$ term as\n\\[\n\\|A\\|_{L^{\\infty}} \\le \\frac{\\gamma\\epsilon}{6} \\|(-\\Delta)^{-1}\\|\\cdot \\|f(\\phi)-\\omega\\|_{L^{\\infty}} \\le \\frac{\\gamma\\epsilon}{6} \\|(-\\Delta)^{-1}\\|\\cdot \\tilde{\\omega}.\n\\]\nOn the other hand, in the 2d case, one can still have the MPP held for non-extended $f(\\phi)$ by showing that $\\|f(\\phi)\\|_{L^{\\infty}} \\le C$ for some generic constant $C$ which depends on $\\|\\phi_0\\|_{H^1(\\mathbb{T}^2)}, \\epsilon^{-1}, \\omega, \\gamma$ and $M$. See the Theorem \\ref{theorem: Linfity} in the Appendix for the $L^{\\infty}$ bound for the 2d weak solution $\\phi$. Then $A$ is still bounded as\n\\[\n\\|A\\|_{L^{\\infty}} \\le \\frac{C\\gamma\\epsilon}{6} \\|(-\\Delta)^{-1}\\|\\tilde{\\omega}.\n\\]\nHowever, to bound the quantity $\\|A\\|_{L^{\\infty}}+|B|$ by 1, one has to take sufficiently small value of $\\gamma$, which is theoretically acceptable but unrealistic in applications.\n\\end{remark}\n\n\n\n\n\\section{Time-discrete Scheme: MPP and Energy Stability}\n\n\n\n\n\nNow we will consider a semi-discrete scheme for the pACOK equation (\\ref{eqn:pACOK}), and show that such a scheme satisfies the MPP and energy stability under some conditions. Given time interval $[0,T]$ and an integer $N>0$, we take the uniform time step size $\\tau = T\/N$ and $t_n = n\\tau$ for $n=0,1,\\cdots,N$. Let $\\phi^n(x) \\approx \\phi(t_n,x)$ be the temporal semi-discrete approximation of the solution $\\phi$ at $t_n$. Given initial data $\\phi^0 = \\phi_0$ and a splitting constant (or stabilizer) $\\kappa > 0$, we consider the following stabilized time-discrete scheme:\n\\begin{align}\\label{eqn:pACOK_SemiImplicit}\n\\left(\\dfrac{1}{\\tau}+\\dfrac{\\kappa}{\\epsilon}\\right)(\\phi^{n+1}-\\phi^n) = &\\ \\epsilon\\Delta\\phi^{n+1} - \\dfrac{1}{\\epsilon}W'(\\phi^n) \\nonumber\\\\\n& - \\gamma(-\\Delta)^{-1}(f(\\phi^n)-\\omega)f'(\\phi^n) - M\\int_{\\mathbb{T}^d} (f(\\phi^n)-\\omega)\\text{d}x\\cdot f'(\\phi^n),\n\\end{align}\nwhich can be rewritten as\n\\begin{align}\\label{eqn:pACOK_SemiImplicit2}\n\\left(\\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right)I - \\tau\\epsilon\\Delta \\right)\\phi^{n+1} = &\\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right)\\phi^n - \\dfrac{\\tau}{\\epsilon}W'(\\phi^n) \\nonumber\\\\\n&- \\tau\\gamma(-\\Delta)^{-1}(f(\\phi^n)-\\omega)f'(\\phi^n) - \\tau M\\int_{\\mathbb{T}^d} (f(\\phi^n)-\\omega)\\text{d}x\\cdot f'(\\phi^n).\n\\end{align}\nA simple calculation reveals that the eigenvalues of the operator $(1+\\tau\\kappa\\epsilon^{-1})I - \\tau\\epsilon\\Delta$ on the left hand side of (\\ref{eqn:pACOK_SemiImplicit2}) are all positive. Therefore the scheme is unconditionally uniquely solvable.\n\n\n\n\n\\subsection{MPP for time-discrete scheme}\n\n\n\n\nIn this section, we will show that the scheme (\\ref{eqn:pACOK_SemiImplicit2}) is MPP. To this end, we begin with a lemma.\n\n\\begin{lemma}\\label{lemma:MPP_condition}\nLet\n\\[\n\\mathcal{F}(\\psi) = \\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right)\\psi - \\dfrac{\\tau}{\\epsilon}W'(\\psi) - \\tau\\gamma(-\\Delta)^{-1}(f(\\psi)-\\omega)f'(\\psi) - \\tau M\\int_{\\mathbb{T}^d} (f(\\psi)-\\omega)\\emph{d}x f'(\\psi).\n\\]\nIf $\\psi(x)\\in [0,1]$, then we have\n\\[\n\\max_{\\psi\\in[0,1]} \\mathcal{F}(\\psi) = 1+\\dfrac{\\tau\\kappa}{\\epsilon} ; \\quad \\min_{\\psi\\in[0,1]} \\mathcal{F}(\\psi) = 0,\n\\]\nprovided that\n\\begin{align}\\label{eqn:MPP_condition}\n\\frac{1}{\\tau}+ \\dfrac{\\kappa}{\\epsilon} \\ge \\dfrac{L_{W''}}{\\epsilon} + \\tilde{\\omega}L_{f''} \\Big(\\gamma\\|(-\\Delta)^{-1}\\|+M|\\mathbb{T}^d| \\Big).\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nNote that $f(\\cdot)$ satisfies $f'(0) = f'(1) = 0$, it follows that for $\\psi\\equiv 0$, $\\mathcal{F}(\\psi) = 0$; for $\\psi\\equiv 1$, $\\mathcal{F}(\\psi) = 1+\\tau\\kappa\/\\epsilon$. For any other $\\psi$ such that $0\\le \\psi \\le 1$ and any $x\\in\\mathbb{T}^d$, one has\n\\begin{align*}\n\\mathcal{F}(\\psi(x)) & = \\mathcal{F}(0) + \\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right)\\psi(x) - \\frac{\\tau}{\\epsilon}\\psi(x)W''(\\xi_0) \\\\\n& \\quad - \\tau\\gamma \\Big( (-\\Delta)^{-1}(f(\\psi)-\\omega) \\Big)(x) \\cdot \\psi(x) f''(\\eta_0) - \\tau M \\int_{\\mathbb{T}^d} (f(\\psi)-\\omega ) \\text{d}x \\cdot \\psi(x) f''(\\eta_0) \\\\\n& \\ge \\mathcal{F}(0) + \\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right)\\psi(x) - \\dfrac{\\tau}{\\epsilon}\\psi L_{W''} - \\tau\\gamma\\psi\\|(-\\Delta)^{-1}\\|\\tilde{\\omega}L_{f''} - \\tau M \\psi |\\mathbb{T}^d| \\tilde{\\omega} L_{f''} \\ge \\mathcal{F}(0),\n\\end{align*}\nwhere $\\xi_0, \\eta_0 \\in (0,\\psi(x)) \\subset (0,1)$ are constants obtained from Taylor expansion. On the other hand,\n\\begin{align*}\n\\mathcal{F}(1-\\psi(x)) & = \\mathcal{F}(1) - \\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right)\\psi(x) + \\frac{\\tau}{\\epsilon}\\psi(x)W''(\\xi_1) \\\\\n&\\quad + \\tau\\gamma \\Big( (-\\Delta)^{-1}(f(1-\\psi)-\\omega)\\Big)(x)\\cdot\\psi(x) f''(\\eta_1) + \\tau M \\int_{\\mathbb{T}^d}(f(1-\\psi)-\\omega)\\text{d}x\\cdot \\psi(x) f''(\\eta_1) \\\\\n& \\le \\mathcal{F}(1) -\\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right)\\psi(x) + \\dfrac{\\tau}{\\epsilon}\\psi(x) L_{W''} + \\tau\\gamma\\psi(x)\\|(-\\Delta)^{-1}\\| \\tilde{\\omega}L_{f''} - \\tau M \\psi |\\mathbb{T}^d| \\tilde{\\omega}L_{f''} \\le \\mathcal{F}(1),\n\\end{align*}\nwhere $\\xi_1, \\eta_1 \\in (1-\\psi(x), 1) \\subset (0,1)$ are constants by Taylor expansion. Consequently we have the desired bounds for $\\mathcal{F}(\\psi)$.\n\\end{proof}\n\nNow we present the MPP property for the scheme (\\ref{eqn:pACOK_SemiImplicit}) or (\\ref{eqn:pACOK_SemiImplicit2}).\n\n\\begin{theorem}\\label{theorem:MPP_semi}\nThe stabilized time-discrete semi-implicit scheme (\\ref{eqn:pACOK_SemiImplicit}) or (\\ref{eqn:pACOK_SemiImplicit2}) is MPP, namely\n\\[\n0\\le \\phi^0 \\le 1 \\Rightarrow 0\\le \\phi^n \\le 1, \\quad \\forall n \\in [\\![ N]\\!].\n\\]\nprovided that the condition (\\ref{eqn:MPP_condition}) holds.\n\\end{theorem}\n\\begin{proof}\nWe can prove the result by induction. Assume that $0\\le \\phi^n \\le 1$, and $\\phi^{n+1}$ is obtained by the scheme (\\ref{eqn:pACOK_SemiImplicit2}). Assume $\\phi^{n+1}$ reaches the maximal value at $x^*$, then $-\\tau\\epsilon\\Delta\\phi^{n+1}(x^*)\\ge0$, and\n\\[\n\\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right) \\phi^{n+1}(x^*) \\le \\mathcal{F}(\\phi^n(x^*)) \\le 1+\\dfrac{\\tau\\kappa}{\\epsilon} \\Rightarrow \\phi^{n+1} \\le 1.\n\\]\nSimilarly let $x_*$ be a minimal point for $\\phi^{n+1}$, then $-\\tau\\epsilon\\Delta\\phi^{n+1}\\le0$, and\n\\[\n\\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right) \\phi^{n+1}(x_*) \\ge \\mathcal{F}(\\phi^n(x_*)) \\ge 0\\Rightarrow \\phi^{n+1} \\ge 0.\n\\]\nwhich completes the proof.\n\\end{proof}\n\n\\begin{remark}\nNote that the condition (\\ref{eqn:MPP_condition}) holds for sufficiently large stabilizer $\\kappa$ no matter what value of $\\tau>0$. Therefore the stabilized semi-implicit scheme (\\ref{eqn:pACOK_SemiImplicit}) or (\\ref{eqn:pACOK_SemiImplicit2}) is unconditionally MPP for sufficiently large $\\kappa$.\n\\end{remark}\n\n\n\n\n\n\n\n\\subsection{Energy stability for time-discrete scheme}\n\nWhile the stabilized semi-discrete scheme (\\ref{eqn:pACOK_SemiImplicit2}) is MPP, it is also energy stable as shown in the following theorem.\n\\begin{theorem}\\label{theorem:EnergyStability}\nAssume the initial $\\phi^0$ satisfies $0\\le \\phi^0\\le 1$, then the stabilized semi-implicit scheme (\\ref{eqn:pACOK_SemiImplicit}) or (\\ref{eqn:pACOK_SemiImplicit2}) is unconditionally energy stable in the sense that\n\\begin{align}\nE^{\\emph{pOK}}[\\phi^{n+1}] \\le E^{\\emph{pOK}}[\\phi^{n}]\n\\end{align}\n provided that\n\\begin{align}\\label{eqn:ES_condition}\n\\frac{\\kappa}{\\epsilon} \\ge \\frac{L_{W''}}{\\epsilon} + (L_{f'}^2+\\tilde{\\omega}L_{f''})\\Big(\\gamma\\|(-\\Delta)^{-1}\\| + M |\\mathbb{T}^d|\\Big).\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nTaking the $L^2$ inner product with $\\phi^{n+1}-\\phi^{n}$ on the two sides of (\\ref{eqn:pACOK_SemiImplicit}) , we have\n\\begin{align}\\label{eqn:estimate1}\n&\\dfrac{1}{\\tau} \\|\\phi^{n+1} - \\phi^{n}\\|_{L^2}^2 \\nonumber\\\\\n= &\\ - \\dfrac{\\kappa}{\\epsilon}\\|\\phi^{n+1} - \\phi^{n}\\|_{L^2}^2 \\underbrace{-\\epsilon \\langle \\nabla\\phi^{n+1}, \\nabla\\phi^{n+1} - \\nabla\\phi^{n} \\rangle }_{\\text{I}} \\underbrace{ - \\epsilon^{-1} \\langle W'(\\phi^{n}), \\phi^{n+1} - \\phi^{n}\\rangle }_{\\text{II}} \\nonumber\\\\\n& \\underbrace{ -\\gamma \\left\\langle (-\\Delta)^{-1}(f(\\phi^n)-\\omega)f'(\\phi^n), \\phi^{n+1}-\\phi^{n} \\right\\rangle }_{\\text{III}} \\underbrace{ -M \\textstyle{\\int_{\\mathbb{T}^d}} (f(\\phi^n)-\\omega)\\text{d}x \\left\\langle f'(\\phi^n), \\phi^{n+1}-\\phi^{n} \\right\\rangle }_{\\text{IV}}.\n\\end{align}\nUsing the identity $a\\cdot (a-b) = \\frac{1}{2}|a|^2 - \\frac{1}{2}|b|^2 + \\frac{1}{2}|a-b|^2$ and $b\\cdot (a-b) = \\frac{1}{2}|a|^2 - \\frac{1}{2}|b|^2 - \\frac{1}{2}|a-b|^2$, we have:\n\\begin{align*}\n\\text{I} =&\\ - \\frac{\\epsilon}{2} \\left( \\|\\nabla\\phi^{n+1}\\|_{L^2}^2 - \\|\\nabla\\phi^{n}\\|_{L^2}^2 + \\|\\nabla\\phi^{n+1}-\\nabla\\phi^n\\|_{L^2}^2 \\right); \\\\\n\\text{II} =& - \\epsilon^{-1}\\left\\langle 1, W'(\\phi^n)(\\phi^{n+1}-\\phi^n) \\right\\rangle = -\\epsilon^{-1}\\left\\langle 1, W(\\phi^{n+1}) \\right\\rangle + \\epsilon^{-1} \\left\\langle 1, W(\\phi^{n}) \\right\\rangle + (2\\epsilon)^{-1} W''(\\xi^n) \\|\\phi^{n+1}-\\phi^n\\|_{L^2}^2 ; \\\\\n \\text{III} =&\\ -\\gamma \\Big\\langle (-\\Delta)^{-1}(f(\\phi^{n})-\\omega), f(\\phi^{n+1}) - f(\\phi^{n}) \\Big\\rangle + \\dfrac{\\gamma}{2} \\Big\\langle (-\\Delta)^{-1}(f(\\phi^{n})-\\omega), f''(\\eta^n)(\\phi^{n+1} - \\phi^{n})^2 \\Big\\rangle \\\\\n \\quad=&\\ -\\dfrac{\\gamma}{2} \\left( \\|(-\\Delta)^{-\\frac{1}{2}}(f(\\phi^{n+1})-\\omega)\\|_{L^{2}}^2 - \\|(-\\Delta)^{-\\frac{1}{2}}(f(\\phi^{n})-\\omega)\\|_{L^{2}}^2 - \\|(-\\Delta)^{-\\frac{1}{2}}(f(\\phi^{n+1})-f(\\phi^n))\\|_{L^{2}}^2 \\right) \\\\\n& \\ + \\dfrac{\\gamma}{2} \\Big\\langle (-\\Delta)^{-1}(f(\\phi^{n})-\\omega), f''(\\eta^n)(\\phi^{n+1} - \\phi^{n})^2\\Big\\rangle ;\\\\\n \\text{IV} = & -\\dfrac{M}{2}\\left( \\left| \\textstyle{\\int_{\\mathbb{T}^d}} ( f(\\phi^{n+1}) - \\omega ) \\text{d}x \\right|^2 -\n \\left| \\textstyle{\\int_{\\mathbb{T}^d}} ( f(\\phi^{n}) - \\omega ) \\text{d}x \\right|^2 -\n \\left| \\textstyle{\\int_{\\mathbb{T}^d}} ( f(\\phi^{n+1}) - f(\\phi^n) \\text{d}x \\right|^2\n \\right)& \\\\\n& + \\dfrac{M}{2} \\left( \\textstyle{\\int_{\\mathbb{T}^d}} ( f(\\phi^{n}) - \\omega ) \\text{d}x\\right) f''(\\eta^n) \\|\\phi^{n+1}-\\phi^n\\|_{L^2}^2 .\n\\end{align*}\nwhere $\\xi^n$ and $\\eta^n$ are between $\\phi^n$ and $\\phi^{n+1}$. Note that the condition (\\ref{eqn:ES_condition}) implies (\\ref{eqn:MPP_condition}), Theorem \\ref{theorem:MPP_semi} gives $\\phi^n,\\phi^{n+1} \\in [0,1]$, consequently $\\xi^n, \\eta^n \\in (0,1)$. Therefore, we do not need the extension of $f$ as in (\\ref{eqn:f_extension}) to perform Taylor expansion above. Finally, inserting the equalities for I--IV back into (\\ref{eqn:estimate1}) and noting that $|f'|0$ and an integer $N>0$ such that $\\tau = \\frac{T}{N}$ and $t_n = n\\tau$ for $n=0,1,\\cdots, N$. Assume that $\\phi_t \\in L^2(0,T; L^2)$ and $\\phi_{tt} \\in L^2(0,T; H^{-1})$, then for $\\kappa$ satisfying the energy stability condition (\\ref{eqn:ES_condition}), if the time step size $\\tau \\le \\epsilon\/(10\\kappa)$, we have\n\\begin{align}\\label{eqn:errorestimate_time}\n\\|\\phi(t_n) - \\phi^n\\|_{L^2} \\le \\tilde{C}\\tau, \\quad \\forall n \\in [\\![ N]\\!]\n\\end{align}\nwhere $\\tilde{C} = e^{6T\\kappa\/\\epsilon} \\Big( \\frac{1+\\frac{|\\mathbb{T}^d|}{4\\pi^2}}{3\\epsilon\/2} \\|\\phi_{tt}\\|_{L^2(0,T; H^{-1})}^2 + \\frac{4\\kappa}{\\epsilon} \\|\\phi_t\\|_{L^2(0,T;L^2)}^2 \\Big)^{\\frac{1}{2}} $ is a constant independent of $\\tau$ and $N$.\n\\end{theorem}\n\n\n\n\n\n\n\n\\section{Fully-discrete Scheme: Maximum Principle Preservation and Energy Stability}\n\nIn this section, we propose a fully-discrete scheme by discretizing the spatial operators by a second order finite difference approximation. To this end, we adopt some notations for the finite difference approximation. For the brevity of notations, we will focus the discussion on the 2D case, which can be easily extended to 3D formulation.\n\n\\subsection{Second order finite difference scheme for spatial discretizaiton}\n\nWe consider $\\mathbb{T}^2 = \\prod_{i=1}^2 [-X_i,X_i] \\subset \\mathbb{R}^2$. Let $N_1, N_2$ be positive even integers. Take $h_i = \\frac{2X_i}{N_i}, i=1,2$ and $\\mathbb{T}^2_h = \\mathbb{T}^2\\ \\cap (\\otimes_{i=1}^2 h_i\\mathbb{Z})$. We define the index set:\n\\begin{align*}\nS_h &= \\left\\{ (k_1,k_2)\\in\\mathbb{Z}^2 | 1\\le k_i \\le N_i, i=1,2 \\right\\}.\n\\end{align*}\nDenote by $\\mathcal{M}_h$ the collection of periodic grid functions on $\\mathbb{T}^2_h$:\n\\begin{align*}\n\\mathcal{M}_h = \\left\\{ f: \\mathbb{T}^2_h\\rightarrow\\mathbb{R} | f_{k_1+m_1N_1, k_2+m_2N_2} =f_{k_1,k_2}, \\forall (k_1,k_2)\\in S_h, \\forall (m_1,m_2)\\in \\mathbb{Z}^2 \\right\\}.\n\\end{align*}\nFor any $f,g\\in\\mathcal{M}_h$ and $\\textbf{f} = (f^1,f^2)^T, \\textbf{g} = (g^1,g^2)^T\\in\\mathcal{M}_h\\times\\mathcal{M}_h$, we define the discrete $L^2$ inner product $\\langle\\cdot,\\cdot\\rangle_h$, discrete $L^2$ norm $\\|\\cdot\\|_{h,L^2}$ and discrete $L^{\\infty}$ norm $\\|\\cdot\\|_{h, L^{\\infty}}$ as follows:\n\\begin{align*}\n\\langle f, g\\rangle_h &= h_x h_y \\sum_{(i,j)\\in S_h} f_{ij} g_{ij}, \\quad \\|f\\|_{h,L^2} = \\sqrt{\\langle f, f \\rangle_h}, \\quad \\|f\\|_{h, L^{\\infty}} = \\max_{(i,j)\\in S_h} |f_{ij}| ; \\\\\n\\langle \\textbf{f}, \\textbf{g}\\rangle_h &= h_x h_y \\sum_{(i,j)\\in S_h} \\left( f_{ij}^1 g_{ij}^1 + f_{ij}^2 g_{ij}^2 \\right), \\quad \\|\\textbf{f}\\|_{h,L^2} = \\sqrt{\\langle \\textbf{f}, \\textbf{f}\\rangle_h}.\n\\end{align*}\nLet $\\mathring{\\mathcal{M}}_h = \\{f\\in\\mathcal{M}_h | \\langle f, 1 \\rangle_h = 0\\}$ be the collections of all periodic grid functions with zero mean.\n\nWe define the second order central difference approximation of the Laplacian operator $\\Delta$ as a discrete linear operator $\\Delta_h: \\mathring{\\mathcal{M}}_h \\rightarrow \\mathring{\\mathcal{M}}_h$\n\\begin{align}\n\\Delta_h u = f: \\Delta_h u_{ij} = \\frac{1}{h_1^2}(u_{i-1,j} - 2u_{ij} + u_{i+1,j}) + \\frac{1}{h_2^2}(u_{i,j-1} - 2u_{ij} + u_{i,j+1})\n\\end{align}\nwhere the periodic boundary condition applies when the the indices $i \\notin [\\![ N_1]\\!]$ or $j \\notin [\\![ N_2]\\!]$. Note that $\\Delta_h: \\mathring{\\mathcal{M}}_h \\rightarrow \\mathring{\\mathcal{M}}_h$ is one-to-one, it is safe to define its inverse $(\\Delta_h)^{-1}: \\mathring{\\mathcal{M}}_h \\rightarrow \\mathring{\\mathcal{M}}_h$\n\\begin{align}\n(\\Delta_h)^{-1} f = u \\quad \\text{if and only if} \\quad \\Delta_h u = f.\n\\end{align}\nWe denote by $\\|(-\\Delta_h)^{-1}\\|$ the optimal constant such that $\\|(-\\Delta_h)^{-1}f\\|_{h,L^\\infty}\\le C \\|f\\|_{h,L^\\infty}$, namely, the norm of the operator $(-\\Delta_h)^{-1}$ from $L^{\\infty}(\\mathring{\\mathcal{M}}_h)$ to itself.\n\n\nGiven the discrete Laplacian operator $\\Delta_h$ defined above, and denote $\\Phi^{n}\\approx \\phi(x,t_n)|_{\\mathbb{T}^2_h}$ the numerical solution, we arrive at the following first order fully-discrete semi-implicit scheme for the pACOK equation (\\ref{functional:pOK}): for $\\forall n \\in [\\![ N]\\!]$, find $\\Phi^{n+1} = (\\Phi_{ij}^{n+1}) \\in \\mathcal{M}_h$ such that\n\\begin{align}\\label{eqn:pACOK_FullDiscrete}\n\\left(\\dfrac{1}{\\tau}+\\dfrac{\\kappa_h}{\\epsilon}\\right)(\\Phi^{n+1}-\\Phi^n) = \\epsilon\\Delta_h\\Phi^{n+1} - \\dfrac{1}{\\epsilon}W'(\\Phi^n) &- \\gamma(-\\Delta_h)^{-1}(f(\\Phi^n)-\\omega) \\odot f'(\\Phi^n) \\nonumber \\\\\n&- M \\langle f(\\Phi^n)-\\omega, 1 \\rangle_h \\text{d}\\mathbf{x} f'(\\Phi^n),\n\\end{align}\nwith $\\Phi^0 = (\\Phi_{ij}^0) = \\phi_0|_{\\mathbb{T}^2_h}$ being the given intial data, and $\\kappa_h$ the stabilization constant. Here $\\text{d}\\mathbf{x} = h_1h_2$ and $\\odot$ represents pointwise multiplication. The scheme can be reformulated as\n\\begin{align}\\label{eqn:pACOK_FullDiscreteII}\n\\left(\\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right)I - \\tau\\epsilon\\Delta_h\\right)\\Phi^{n+1} = \\left(1+\\dfrac{\\tau\\kappa}{\\epsilon}\\right)\\Phi^{n} - \\dfrac{\\tau}{\\epsilon}W'(\\Phi^n) & - \\tau\\gamma(-\\Delta_h)^{-1}(f(\\Phi^n)-\\omega)\\odot f'(\\Phi^n) \\nonumber \\\\\n&- \\tau M \\langle f(\\Phi^n)-\\omega, 1 \\rangle_h \\text{d}\\mathbf{x} f'(\\Phi^n),\n\\end{align}\nfrom which the unconditional unique solvability can be guaranteed by realizing the positivity of all the eigenvalues of the operator\n$\\left( 1+ \\tau\\kappa\\epsilon^{-1} \\right) I - \\tau \\epsilon \\Delta_h$ on the left hand side of (\\ref{eqn:pACOK_FullDiscreteII}).\n\n\n\n\\subsection{Maximum principle preservation for fully-discrete scheme}\n\nIn this section, we will show that the full-discrete scheme (\\ref{eqn:pACOK_FullDiscrete}) is MPP under a condition similar to (\\ref{eqn:MPP_condition}). To this end, a discrete counterpart of Lemma \\ref{lemma:MPP_condition} is needed.\n\n\n\\begin{lemma}\\label{lemma:MPP_condition2}\nLet $\\Psi\\in\\mathcal{M}_h$ be such that $0\\le\\Psi\\le 1$, and define $\\mathcal{F}_h: \\mathcal{M}_h \\rightarrow \\mathcal{M}_h$ as follows:\n\\[\n\\mathcal{F}_h(\\Psi) = \\left(1+\\dfrac{\\tau\\kappa_h}{\\epsilon}\\right)\\Psi - \\dfrac{\\tau}{\\epsilon}W'(\\Psi) - \\tau\\gamma(-\\Delta_h)^{-1}(f(\\psi)-\\omega)\\odot f'(\\Psi) - \\tau M \\langle f(\\psi)-\\omega,1 \\rangle\\emph{d}\\mathbf{x} f'(\\Psi),\n\\]\nthen we have\n\\[\n\\max_{0\\le\\Psi\\le 1} \\{\\mathcal{F}_h(\\Psi)\\}= 1+\\dfrac{\\tau\\kappa_h}{\\epsilon} ; \\quad \\min_{0\\le\\Psi\\le 1} \\{ \\mathcal{F}_h(\\Psi) \\} = 0,\n\\]\nprovided that\n\\begin{align}\\label{eqn:MPP_condition2}\n\\frac{1}{\\tau} + \\dfrac{\\kappa_h}{\\epsilon} \\ge \\dfrac{L_{W''}}{\\epsilon} + \\tilde{\\omega}L_{f''}\\Big(\\gamma\\|(-\\Delta_h)^{-1}\\| + M|\\mathbb{T}^2| \\Big).\n\\end{align}\n\\end{lemma}\n\nThe proof of Lemma \\ref{lemma:MPP_condition2} is similar to that of Lemma \\ref{lemma:MPP_condition}. The only difference is that the Laplacian operator $-\\Delta$ is replaced by a discrete Laplacian operator $-\\Delta_h$, and the integral term $\\int (f(\\psi)-\\omega) \\text{d}x$ is replaced by the Riemann sum. We therefore omit the details.\n\nNow we present the MPP for the fully-discrete sheme (\\ref{eqn:pACOK_FullDiscrete}) or (\\ref{eqn:pACOK_FullDiscreteII}).\n\n\\begin{theorem}\\label{theorem:MPP_fully}\nThe stabilized fully-discrete scheme (\\ref{eqn:pACOK_FullDiscrete}) or (\\ref{eqn:pACOK_FullDiscreteII}) is MPP provided that the condition (\\ref{eqn:MPP_condition2}) holds.\n\\end{theorem}\n\\begin{proof}\nAssume that $0\\le \\Phi^n \\le 1$, and $\\Phi^{n+1}$ is obtained by the scheme (\\ref{eqn:pACOK_FullDiscrete}). Assume $\\Phi^{n+1}$ reaches the maximal value at the index $(i^*,j^*)$, then\n\\[\n(\\Delta_h\\Phi^{n+1})_{i^*j^*} = \\frac{(\\Phi^n)_{i^*-1,j^*}+(\\Phi^n)_{i^*+1,j^*}-2(\\Phi^n)_{i^*j^*}}{h_1^2} + \\frac{(\\Phi^n)_{i^*,j^*-1}+(\\Phi^n)_{i^*,j^*+1}-2(\\Phi^n)_{i^*j^*}}{h_2^2} \\le 0,\n\\]\nand\n\\[\n\\left(1+\\dfrac{\\tau\\kappa_h}{\\epsilon}\\right) (\\Phi^{n+1})_{i^*j^*} \\le \\Big(\\mathcal{F}_h((\\Phi^n)\\Big)_{i^*j^*} \\le 1+\\dfrac{\\tau\\kappa}{\\epsilon} \\Rightarrow \\Phi^{n+1} \\le 1.\n\\]\nSimilarly let $(i_*, j_*)$ be the index for the smallest component of $\\Phi^{n+1}$, then\n\\[\n(\\Delta_h\\Phi^{n+1})_{i_*j_*} = \\frac{(\\Phi^n)_{i_*-1,j_*}+(\\Phi^n)_{i_*+1,j_*}-2(\\Phi^n)_{i_*j_*}}{h_1^2} + \\frac{(\\Phi^n)_{i_*,j_*-1}+(\\Phi^n)_{i_*,j_*+1}-2(\\Phi^n)_{i_*j_*}}{h_2^2} \\ge 0,\n\\]\nand\n\\[\n\\left(1+\\dfrac{\\tau\\kappa_h}{\\epsilon}\\right) (\\Phi^{n+1})_{i_*j_*} \\ge \\Big(\\mathcal{F}_h((\\Phi^n)\\Big)_{i_*j_*} \\ge 0 \\Rightarrow \\Phi^{n+1} \\ge 0,\n\\]\nwhich completes the proof.\n\\end{proof}\n\n\n\n\n\\subsection{Energy stability for fully-discrete scheme}\n\nWhile the stabilized fully-discrete scheme (\\ref{eqn:pACOK_FullDiscrete}) is MPP, it is also energy stable for the discrete OK energy functional defined below:\n\\begin{align}\\label{eqn:discreteEnergy}\nE_h^{\\text{pOK}}[\\Phi] = & -\\frac{\\epsilon}{2} \\langle \\Delta_h\\Phi,\\Phi \\rangle_h + \\frac{1}{\\epsilon} \\langle W(\\Phi),1\\rangle_h + \\frac{\\gamma}{2} \\Big\\langle (-\\Delta_h)^{-1}(f(\\Phi)-\\omega), (f(\\Phi)-\\omega) \\Big\\rangle_h \\nonumber \\\\\n& + \\frac{M}{2} \\Big( \\langle f(\\Phi^n)-\\omega, 1 \\rangle_h \\text{d}\\mathbf{x} \\Big)^2.\n\\end{align}\n\n\n\n\\begin{theorem}\nAssume the initial $\\Phi^0$ satisfies $0\\le \\Phi^0\\le 1$, then the stabilized fully-discrete semi-implicit scheme (\\ref{eqn:pACOK_FullDiscrete}) or (\\ref{eqn:pACOK_FullDiscreteII}) is unconditionally energy stable in the sense that\n\\begin{align}\nE_h^{\\emph{pOK}}[\\Phi^{n+1}] \\le E_h^{\\emph{pOK}}[\\Phi^{n}]\n\\end{align}\n provided that\n\\begin{align}\\label{eqn:discreteES_condition}\n\\frac{\\kappa_h}{\\epsilon} \\ge \\frac{L_{W''}}{\\epsilon} + (L_{f'}^2+\\tilde{\\omega}L_{f''})\\Big(\\gamma\\|(-\\Delta_h)^{-1}\\| + M |\\mathbb{T}^2|\\Big).\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nThe proof is similar to that of Theorem \\ref{theorem:EnergyStability}. To see how the discrete operators apply in the proof, we will still show it in details. Taking the discrete $L^2$ inner product with $\\Phi^{n+1}-\\Phi^{n}$ on the two sides of (\\ref{eqn:pACOK_FullDiscrete}), we have\n\\begin{align}\\label{eqn:estimate2}\n&\\dfrac{1}{\\tau} \\|\\Phi^{n+1} - \\Phi^{n}\\|_{h, L^2}^2 \\nonumber\\\\\n= &\\ - \\dfrac{\\kappa_h}{\\epsilon}\\|\\Phi^{n+1} - \\Phi^{n}\\|_{h,L^2}^2 +\\underbrace{\\epsilon \\langle \\Delta_h\\Phi^{n+1}, \\Phi^{n+1} - \\Phi^{n} \\rangle_h }_{\\text{I}} \\underbrace{ - \\epsilon^{-1} \\langle W'(\\Phi^{n}), \\Phi^{n+1} - \\Phi^{n}\\rangle_h }_{\\text{II}} \\nonumber\\\\\n& \\underbrace{ -\\gamma \\left\\langle (-\\Delta_h)^{-1}(f(\\Phi^n)-\\omega)f'(\\Phi^n), \\Phi^{n+1}-\\Phi^{n} \\right\\rangle_h }_{\\text{III}} \\underbrace{ -M \\langle f(\\Phi^n)-\\omega, 1\\rangle\\text{d}\\textbf{x} \\left\\langle f'(\\Phi^n), \\Phi^{n+1}-\\Phi^{n} \\right\\rangle_h }_{\\text{IV}}.\n\\end{align}\nUsing the identity $a\\cdot (a-b) = \\frac{1}{2}|a|^2 - \\frac{1}{2}|b|^2 + \\frac{1}{2}|a-b|^2$ and $b\\cdot (a-b) = \\frac{1}{2}|a|^2 - \\frac{1}{2}|b|^2 - \\frac{1}{2}|a-b|^2$, we have:\n\\begin{align*}\n\\text{I} =&\\ \\frac{\\epsilon}{2} \\Big( \\Big\\langle \\Delta_h\\Phi^{n+1}, \\Phi^{n+1} \\Big\\rangle_h - \\Big\\langle \\Delta_h\\Phi^{n}, \\Phi^{n} \\Big\\rangle_h + \\Big\\langle \\Delta_h(\\Phi^{n+1}-\\Phi^{n}), \\Phi^{n+1}-\\Phi^n \\Big\\rangle_h \\Big); \\\\\n\\text{II} =& -\\epsilon^{-1}\\left\\langle 1, W(\\Phi^{n+1}) \\right\\rangle_h + \\epsilon^{-1} \\left\\langle 1, W(\\Phi^{n}) \\right\\rangle_h + (2\\epsilon)^{-1} W''(\\xi^n) \\|\\Phi^{n+1}-\\Phi^n\\|_{h, L^2}^2 ; \\\\\n \\text{III} =&\\ -\\gamma \\Big\\langle (-\\Delta_h)^{-1}(f(\\Phi^{n})-\\omega), f(\\Phi^{n+1}) - f(\\Phi^{n})\\Big\\rangle_h + \\dfrac{\\gamma}{2} \\Big\\langle (-\\Delta_h)^{-1}(f(\\Phi^{n})-\\omega), f''(\\eta^n)(\\Phi^{n+1} - \\Phi^{n})^2\\Big\\rangle_h \\\\\n \\quad=&\\ -\\dfrac{\\gamma}{2} \\Big( \\Big\\langle (-\\Delta_h)^{-1}(f(\\Phi^{n+1})-\\omega), f(\\Phi^{n+1})-\\omega \\Big\\rangle_{h} - \\Big\\langle (-\\Delta_h)^{-1}(f(\\Phi^{n})-\\omega), f(\\Phi^{n})-\\omega \\Big\\rangle_{h} \\\\\n& \\hspace{0.4in} - \\Big\\langle (-\\Delta_h)^{-1}(f(\\Phi^{n+1})-f(\\Phi^{n})),f(\\Phi^{n+1})-f(\\Phi^{n}) \\Big\\rangle_{h} \\Big) \\\\\n& \\ + \\dfrac{\\gamma}{2} \\Big\\langle (-\\Delta_h)^{-1}(f(\\Phi^{n})-\\omega), f''(\\eta^n)(\\Phi^{n+1} - \\Phi^{n})^2\\Big\\rangle_h ;\\\\\n \\text{IV} = & -\\dfrac{M}{2}\\Big( \\left(\\langle f(\\Phi^{n+1}) - \\omega, 1 \\rangle_h \\text{d}\\textbf{x} \\right)^2 -\n \\left(\\langle f(\\Phi^{n}) - \\omega, 1 \\rangle_h \\text{d}\\textbf{x} \\right)^2 -\n \\left(\\langle f(\\Phi^{n+1}) - f(\\Phi^{n}), 1 \\rangle_h \\text{d}\\textbf{x} \\right)^2\n \\Big)& \\\\\n& + \\dfrac{M}{2} \\langle f(\\Phi^{n}) - \\omega, 1 \\rangle_h \\text{d}\\textbf{x} f''(\\eta^n) \\|\\Phi^{n+1}-\\Phi^n\\|_{h,L^2}^2,\n\\end{align*}\nwhere $\\xi^n$ and $\\eta^n$ are between $\\Phi^n$ and $\\Phi^{n+1}$ due to the smoothness up to 2nd order derivative for $f$ and $W$. Note that the condition (\\ref{eqn:discreteES_condition}) implies (\\ref{eqn:MPP_condition2}), owing to Theorem \\ref{theorem:MPP_fully}, $\\Phi^n,\\Phi^{n+1} \\in [0,1]$, therefore $\\xi^n, \\eta^n \\in (0,1)$. Finally inserting the equalities for I--IV back into (\\ref{eqn:estimate2}) and noting that $|f'|0$ and an integer $N>0$ such that $\\tau = T\/N$ and $t_n = n\\tau$ for $n=0,1,\\cdots,N$. Assume the initial value $\\phi_0$ is smooth, periodic and bounded $0\\le \\phi_0 \\le 1$, and the exact solution $\\phi(x,t)$ is sufficiently smooth. Let the stabilization constant $\\kappa_h$ satisfy the condition (\\ref{eqn:discreteES_condition}). We denote by $\\{\\Phi^n\\}_{n=1}^N = \\{(\\Phi_{ij}^n)\\}_{n=1}^N$ the approximate solution calculated by the scheme (\\ref{eqn:pACOK_FullDiscrete}) with $\\Phi^0 = \\phi_0|_{\\mathbb{T}^2}$. If the step size $\\tau$ is sufficiently small, we have\n\\begin{align}\n\\|\\phi(t_n) - \\Phi^{n}\\|_{h,L^2} \\le C (\\tau + h_1^2 + h_2^2), \\quad n \\in [\\![ N]\\!],\n\\end{align}\nwhere $C>0$ is some generic constant which depends on $\\phi, T, \\kappa_h, \\epsilon, \\gamma, M, |\\mathbb{T}^2|$ but is independent of $\\tau, h_1, h_2$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\\section{MPP schemes for a general Allen-Cahn type model}\\label{section:GeneralFramework}\n\nOur study on MPP can be extended into a more general setting. Consider a general Allen-Cahn type dynamics:\n\\begin{align}\\label{eqn:generalAC}\n\\dfrac{\\partial}{\\partial t} \\phi = \\epsilon\\Delta\\phi - \\dfrac{1}{\\epsilon}W'(\\phi) - \\mathcal{L}(f(\\phi)-\\omega)f'(\\phi) - M\\int_{\\mathbb{T}^d} (f(\\phi)-\\omega)\\ \\text{d}x\\cdot f'(\\phi),\n\\end{align}\nwhere $\\mathcal{L}$ is a positive semi-definite linear operator from $L^{\\infty}(\\mathbb{T}^d)$ to $L^{\\infty}(\\mathbb{T}^d)$ with the norm denoted by $\\|\\mathcal{L}\\|$. The last term on the right hand size counts for a possible volume constraint (\\ref{eqn:Volume}) when necessary. For instance, if $\\mathcal{L} = (-\\Delta)^{-1}$, then the volume constraint (\\ref{eqn:Volume}) is necessary, and it recovers the pACOK dynamics (\\ref{eqn:pACOK}). If $\\mathcal{L} = (I - \\gamma^2\\Delta)^{-1}$, then the volume constraint is unnecessary, and we set $M = 0$. This general dynamics can be viewed as the $L^2$ gradient flow dynamics associated to the free energy functional\n\\begin{align}\\label{eqn:generalframework}\nE^{\\text{ge}}[\\phi] = \\int_{\\mathbb{T}^d} \\frac{\\epsilon}{2}|\\nabla\\phi|^2 + \\frac{1}{\\epsilon}W(\\phi)\\ \\text{d}x + \\int_{\\mathbb{T}^d} |\\mathcal{L}^{\\frac{1}{2}}(f(\\phi)-\\omega)|^2\\ \\text{d}x + \\frac{M}{2}\\Big( \\int_{\\mathbb{T}^d} f(\\phi)-\\omega\\ \\text{d}x \\Big)^2,\n\\end{align}\nwhere depending on the different form of the operator $\\mathcal{L}$, we might or might not need the volume constraint.\n\nHere are some examples which fit into the general framework described above.\n\\begin{itemize}\n\\item In the micromagnetic model for garnet films \\cite{CondetteMelcherSuli_MathComp2010} with $d = 2$, the operator $\\mathcal{L}$ is being characterized by its eigenvalues $\\lambda(k) = \\frac{1-\\exp(-\\delta|k|)}{\\delta|k|}$. Here $\\delta>0$ corresponds to the relative film thickness. There is no volume constraint in this model.\n\n\\item A nonlocal geometric variational problem studied by \\cite{RenTruskinovsky_Elasticity2000} takes $\\mathcal{L} = (I - \\gamma^2\\Delta)^{-1}$ with no volume constraint. This problem can lead to the FitzHugh-Nagumo system \\cite{ChenChoiHuRen_SIMA2018}.\n\n\\item We can consider a positive semi-define linear operator $\\mathcal{L}$ which is the inverse of the following nonlocal operator $\\mathcal{K}: L^2(\\mathbb{T}^d) \\rightarrow L^2(\\mathbb{T}^d)$\n\\begin{align*}\n\\mathcal{K}: v(x) \\longmapsto \\int_{\\mathbb{T}^d} K(x-y) (v(x) - v(y))\\ \\text{d}y,\n\\end{align*}\nin which the kernel $K$ is nonnegative, radial, $\\mathbb{T}^d$-periodic with bounded second moment \\cite{Du_Book2020}. This can be viewed as a nonlocal OK model for the diblock copolymer system.\n\n\\item One example that cannot fit into the general framework (\\ref{eqn:generalframework}) but still satisfy the MPP property is the phase field variational implicit solvation model (pVISM), in which the free energy is formulated as \\cite{Zhao_2018CMS}:\n\\[\nE^{\\text{pVISM}}[\\phi] = \\int_{\\mathbb{T}^d} \\frac{\\epsilon}{2}|\\nabla\\phi|^2 + \\frac{1}{\\epsilon}W(\\phi)\\ \\text{d}x + \\int_{\\mathbb{T}^d} f(\\phi(x)) U(x;X)\\ \\text{d}x.\n\\]\nHere $X = (x_1,\\cdots, x_m)$ are the locations of the $m$ solute atoms, and $U$ is the potential between the solute atoms $X$ and solvent molecules $x$ (for instance, water). The phase field $\\phi$ labels the solvent so that the nonlocal interaction by $U$ takes integral only in the solvent region. The potential $U$ in pVISM typically consists of two parts, the solute-solvent van der Waals interaction and the electrostatic interaction. Additionally, the potential $U$ is cut off as a constant near the solute atoms $X$ so that it remains bounded. See \\cite{Zhao_2018CMS} and the references therein for the detailed discussion. In the next section, we will use pVISM as an example to show that choosing $f(\\phi) = 3\\phi^2 - 2\\phi^3$ make the MPP while $f(\\phi) = \\phi$ violates the MPP. See Subsection \\ref{subsection:pVISM} for the details.\n\\end{itemize}\n\nIn this general setting, the $L^2$ gradient flow dynamics always hold the MPP as in the following theorem.\n\\begin{theorem}\\label{theorem:generalMPP_continuous}\nThe general $L^2$ gradient flow dynamics (\\ref{eqn:generalAC}) is maximum principle preserving, namely, if $0\\le \\phi_0 \\le 1$, then $0\\le \\phi(t) \\le 1$ for any $t>0$, provided that\n\\begin{align}\\label{eqn_MPPcondition_continuous_generalsetting}\n\\frac{\\epsilon\\tilde{\\omega}}{6}\\Big[ \\|\\mathcal{L}\\| + \\tilde{M} |\\mathbb{T}^d| \\Big] \\le 1,\n\\end{align}\nwhere $\\tilde{M} = M$ if there is a volume constraint (\\ref{eqn:Volume}), and $\\tilde{M} = 0$ if there is no volume constraint.\n\\end{theorem}\nThe proof is identical to that of theorem \\ref{theorem:MPP_continuous} by replacing $\\gamma(-\\Delta)^{-1}$ by $\\mathcal{L}$, so we omit it.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Numerical simulations}\n\n\nIn this section, some numerical examples will be presented to validate the proposed schemes. Moreover some interesting patterns arising from the OK model will be shown. To begin with, let us briefly explain how to implement the numerical scheme (\\ref{eqn:pACOK_FullDiscreteII}). The implementation is as follows:\n\n\\begin{enumerate}\n\\item At the $n$-th step, take the Discrete Fourier Transform(DFT) on the right hand side $\\widehat{RHS}_{jk}$;\n\\item Calculate the DFT of $\\Phi^{n+1}$ as $\\hat{\\Phi}^{n+1}_{jk} = \\frac{\\widehat{RHS}_{jk}}{\\left(1+\\frac{\\tau\\kappa}{\\epsilon}\\right) + \\frac{4}{h_1^2}\\sin^2\\left(\\frac{j\\pi h_1}{2X_1}\\right) + \\frac{4}{h_2^2}\\sin^2\\left(\\frac{k\\pi h_2}{2X_2}\\right) }$;\n\\item Take the inverse DFT of $\\hat{\\Phi}^{n+1}_{jk}$ to obtain $\\Phi^{n+1}$ and move to the $(n+1)$-th step;\n\\end{enumerate}\n\nNow we solve the pACOK equation (\\ref{eqn:pACOK_FullDiscreteII}) coupled with periodic boundary condition. In this section, we fix $\\mathbb{T}^2 = [-1,1)^2 \\subset \\mathbb{R}^2$ and $N = N_1 = N_2 = 256$ unless stated otherwise. We set the stopping criteria for the time iteration by:\n\\begin{align}\n\\dfrac{\\|\\Phi^{n+1}-\\Phi^{n}\\|_{h,L^{\\infty}}}{\\tau} \\le \\text{TOL} = 10^{-3}.\n\\end{align}\nThe penalty constant is taken to be sufficiently large $M \\gg 1$. We take a sufficiently large value of $\\kappa_h>0$ to fulfill the energy stability condition (\\ref{eqn:discreteES_condition}) (and therefore fulfill the MPP condition (\\ref{eqn:MPP_condition2})), say $\\kappa_h = 2000$. Other parameters such as $\\epsilon, \\gamma, \\tau, \\omega$ might vary for different simulations.\n\n\\subsection{Rate of convergence}\n\nWe first of all test the convergence rates and the spatial accuracy of the scheme (\\ref{eqn:pACOK_FullDiscreteII}). For this numerical experiment, we fix $\\omega = 0.1$, and take a round disk as the initial data $\\Phi^0 = 0.5 + 0.5\\tanh(\\frac{r_0-r}{\\epsilon\/3})$ with $r_0 = \\sqrt{\\omega|\\mathbb{T}^2|\/\\pi}+0.1$. The simulation is performed until $T = 0.02$. For the rate of convergence, we take the solution generated by the scheme (\\ref{eqn:pACOK_FullDiscreteII}) with $\\tau = 10^{-6}$ and $N=2^8$ (consequently $h = h_1 = h_2 = \\frac{2X_1}{N}=\\frac{1}{128}$) as the benchmark solution. Then we take several values of step size larger than $\\tau = 10^{-6}$, each is the half of the previous one, and compute the discrete $L^2$ error between the numerical solutions with larger step sizes and the benchmark one. Table \\ref{table:convergence_rates} presents the errors and the convergence rates based on the data at $T = 0.02$ for the scheme (\\ref{eqn:pACOK_FullDiscreteII}) with time step sizes being halved from $\\tau = 10^{-4}$ to $10^{-4}\/16$. We test the convergence rates for three different values of $\\epsilon = 5h, 10h $ and $20h$. $\\gamma = 100$ is fixed. We can see from the table that the numerically computed convergence rates all tend to approach the theoretical value 1.\n\n\\label{table:convergence_rates}\n\\begin{table}[H]\n\\begin{center}\n\\begin{tabular}{ccccccc}\n\n & \\multicolumn{2}{c}{$\\epsilon = 5h$} &\\multicolumn{2}{c}{$\\epsilon = 10h$} &\\multicolumn{2}{c}{$\\epsilon = 20h$} \\\\ \\hline\n$\\tau $ & Error & Rate & Error & Rate & Error & Rate \\\\ \\hline\n1e-4\/$2^0$ & 1.936e-1 & --- & 1.555e-1 & --- & 5.902e-2 & --- \\\\\n\n1e-4\/$2^1$ & 1.542e-1& 0.33 & 9.465e-2 & 0.72 & 2.376e-2 & 1.31\n\t \t \\\\\n1e-4\/$2^2$ & 1.076e-1 & 0.52 & 4.858e-2& 0.96 & 9.233e-3 & 1.36\n\t \t \\\\\n1e-4\/$2^3$ & 6.423e-2 & 0.74 & 2.247e-2& 1.11 & 3.752e-3 & 1.29\n\t\t \\\\\n1e-4\/$2^4$ & 3.270e-2 & 0.97 & 9.787e-3 & 1.20 & 1.556e-3 & 1.27\n\t \t \\\\ \t\n1e-6 (BM) & --- & ---& --- & ---& --- & ---\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{The errors and the corresponding convergence rates at time $T=0.02$ by the scheme (\\ref{eqn:pACOK_FullDiscreteII}) for different values of $\\epsilon$. In this simulation, $\\omega = 0.1, \\gamma = 100, M = 1000, \\kappa_h = 2000, N = 256$.}\n\\end{table}\n\n\n\n\\subsection{Comparison between $f(\\phi) = 3\\phi^2 - 2\\phi^3$ and $f(\\phi) = \\phi$ regarding to MPP} \\label{subsection:pVISM}\n\nIn this section, we show an example to see the effect of $f(\\phi) = 3\\phi^2 - 2\\phi^3$ on MPP. We consider the 1D pVISM system with $f(\\phi) = 3\\phi^2 - 2\\phi^3$, which holds the MPP theoretically, and with $f(\\phi) = \\phi$, the traditional choice which might lose the MPP. The parameters are taken as $L_x = 5, N = 1024, \\epsilon = 50h, \\kappa_h = 2000$. The solute atom $X = {(0)}$ (i.e. single solute atom system), and the potential function $U(x;X)$ reads:\n\\[\nU(x;X) = \\rho_{\\text{w}}\\cdot \\frac{1}{4\\epsilon_{\\text{LJ}}}\\bigg[\\Big(\\frac{\\sigma_0}{x_{\\text{cut}}}\\Big)^{12} - \\Big(\\frac{\\sigma_0}{x_{\\text{cut}}}\\Big)^{6}\\bigg] + \\frac{Q^2}{8\\pi\\epsilon_0}\\Big( \\frac{1}{\\epsilon_{\\text{w}}} - \\frac{1}{\\epsilon_{\\text{m}}}\\Big)\\frac{1}{x_{\\text{cut}}^2}.\n\\]\nHere $\\rho_{\\text{w}} = 0.0333 \\mathring{\\text{A}}^{-3}$ is the constant solvent (water) density, $\\epsilon_{\\text{LJ}} = 0.3 k_{\\text{B}}T$ is the depth of the Lennard-Jones potential well associated with the solute atom, $\\sigma_0 = 3.5 \\mathring{A}$ is the finite distance at which the Lenard-Jones potential of the solute atom is zero, $x_{\\text{cut}} = \\max\\{|x|,2.5\\}$ is the cutoff distance of $x$ from solute atom, $Q = 1e$ is the partial charge of the solute atom, $\\epsilon_0 = 1.4321\\times 10^{-4} e^2\/(k_{\\text{B}}T\\mathring{\\text{A}})$ is the vacuum permittivity, $\\epsilon_{\\text{m}} = 1$ is the relative permittivity of the solute, and $\\epsilon_{\\text{w}} = 80$ is the relative permittivity of the solvent. See \\cite{Zhao_2018CMS} for the model details.\n\nFigure \\ref{fig:pVISM} depicts the numerical equilibrium by taking $f(\\phi) = 3\\phi^2 - 2\\phi^3$ and $f(\\phi) = \\phi$ in the pVISM system. One can see that for the model with $f(\\phi) = 3\\phi^2 - 2\\phi^3$, the numerical equilibrium remains bounded between 0 and 1, the same as the theoretical prediction. On the contrary, if $f(\\phi) = \\phi$, the numerical equilibrium becomes smaller than 0 inside the interface, and greater than 1 outside the interface. Of course, the violation of MPP can be mitigated by letting $\\epsilon \\rightarrow 0$ by the $\\Gamma$-convergence theory \\cite{LiZhao_SIAM2013}. However, in real applications, especially in the 3d simulations, $\\epsilon$ has to remain relatively large to reduce the computational cost. Therefore, the choice of $f(\\phi) = 3\\phi^2 - 2\\phi^3$ is advantageous of keeping the hyperbolic tangent profile of $\\phi$, bounding $0\\le \\phi \\le 1$ and localizing the forces only near the interfaces even for a relatively large $\\epsilon$.\n\n\\begin{figure}[!htbp]\n\\centerline{\n\\includegraphics[width=150mm]{Figures\/pVISM.eps}\n }\n\\caption{Numerical comparison between the model with $f(\\phi) = 3\\phi^2 - 2\\phi^3$ and the one with $f(\\phi) = \\phi$ for the pVISM system. The model with $f(\\phi) = 3\\phi^2 - 2\\phi^3$ holds the MPP, while the MPP is violated when taking $f(\\phi) = \\phi$.}\n\\label{fig:pVISM}\n\\end{figure}\n\n\n\n\\subsection{1D coarsening dynamics and MPP}\n\n\n\nIn this section, we verify the MPP and energy stability for the numerical scheme (\\ref{eqn:pACOK_FullDiscreteII}) for the 1D case. We take a piecewise constant function as the initial, with the constant values generated randomly between 0 and 0.8. In the simulation, the parameter values are $T = 1000, \\omega = 0.3, \\gamma = 500, M = 2000, \\tau = 10^{-3}, \\kappa = 2000$. Figure \\ref{fig:1dcoarsening_gamma500} shows the coarsening dynamics in which the system experiences phase separation from the random initial, then bumps appear from coarsening, evolve into same size, and finally are separated in equal distance. The light blue curve (values labeled on the left $y$-axis) records the discrete $L^{\\infty}$ norm for the solution $2\\Phi^n-1$ (note that $|2\\Phi^n-1|\\le 1$ is equivalent to $0\\le \\Phi^n \\le 1$), which clearly implies the boundedness of $\\Phi^n$ between 0 and 1. The red curve (values labeled on the right $y$-axis) represents the discrete energy $E_h^{\\text{pOK}}[\\Phi^n]$ in (\\ref{eqn:discreteEnergy}) which is monotonically decreasing. Indicated by different colors, the four insets correspond to the four snapshots at $t = 0, 10, 500, 1000$ of the coarsening dynamics.\n\nNow we fix all parameter values as they are in Figure \\ref{fig:1dcoarsening_gamma500} but change $\\gamma = 2000$, a larger value than it was. As $\\gamma$ represents strength of the long-range repulsive interaction, we expect that a larger $\\gamma$ generates more bumps. This is verified by Figure \\ref{fig:1dcoarsening_gamma2000} in which the system still start from a randomly generated initial, but end up with six equally-sized equally-separated bumps. Meanwhile MPP and energy stability are still held as expected.\n\n\n\n\\begin{figure}[!htbp]\n\\centerline{\n\\includegraphics[width=150mm]{Figures\/OK_1d_kappa500.eps}\n }\n\\caption{A 1D coarsening dynamics process with a small repulsive strength $\\gamma$. In this simulation, the parameter values are $T = 1000, \\omega = 0.3, \\gamma = 500, M = 2000, \\tau = 10^{-3}, \\kappa = 2000$. The light blue curve records the discrete $L^{\\infty}$ norm for the solution $\\Phi^n$, which is clearly bounded between 0 and 1. The red curve represents the discrete energy $E_h^{\\text{pOK}}[\\Phi^n]$ in (\\ref{eqn:discreteEnergy}) which is monotonically decreasing. The four insets are snapshots at different times.}\n\\label{fig:1dcoarsening_gamma500}\n\\end{figure}\n\n\n\\begin{figure}[!htbp]\n\\centerline{\n\\includegraphics[width=150mm]{Figures\/OK_1d_kappa2000.eps}\n }\n\\caption{A 1D coarsening dynamics process with a large repulsive strength $\\gamma$. In this simulation, the parameter values are $T = 1000, \\omega = 0.3, \\gamma = 2000, M = 2000, \\tau = 10^{-3}, \\kappa = 2000$.}\n\\label{fig:1dcoarsening_gamma2000}\n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{2D coarsening dynamics and MPP}\n\nIn this section, we solve the equation (\\ref{eqn:pACOK_FullDiscreteII}) in 2D and explore the corresponding discrete MPP and discrete energy stability. We take a $256\\times 256$ mesh grid and $T = 100, \\omega = 0.15, \\gamma = 2000, M = 10^{4}, \\tau = 2\\cdot 10^{-4}, \\kappa = 2000$. Similar as in the 1D case, a 2D random initial is generated on a coarse grid. The coarsening dynamics is presented in Figure \\ref{fig:2dcoarsening_gamma1000} in which the random initial is phase separated within a very short time period, resulting in a group of bubbles with different sizes, then the tiny bubbles disappear, other bubbles evolves into equal size, and eventually all the equally-sized bubbles become equally distanced, forming a hexagonal pattern in the 2D domain $\\mathbb{T}^2$. Just like the 1D case, we see that the 2D coarsening dynamics also enjoy the MPP property and energy stability in the discrete sense as the theory predicts in the previous sections. The insets are snapshots taken at $t = 0, 1, 10, 100$, each of which has a colored title indicating the corresponding colored marker on the two curves.\n\nWhen the value of $\\gamma$ become larger, say $\\gamma = 2000$, but other parameter values are fixed, the stronger long-range repulsive interaction between bubbles lead to more bubbles of equal size and equal distance. This result is depicted in Figure \\ref{fig:2dcoarsening_gamma2000} in which the MPP and energy stability are still held.\n\n\\begin{figure}[!htbp]\n\\centerline{\n\\includegraphics[width=150mm]{Figures\/OK_2d_kappa1000.eps}\n }\n\\caption{A 2D coarsening dynamics process with a small repulsive strength $\\gamma$. In this simulation, the parameter values are $T = 100, \\omega = 0.15, \\gamma = 1000, M = 10^4, \\tau = 2\\cdot10^{-4}, \\kappa = 2000$. The light blue curve is the discrete $L^{\\infty}$ norm of $2\\Phi^n-1$ which implies the bound of $\\Phi^n$ between 0 and 1, while the red curve indicates the monotonic decay of the discrete energy $E_h^{\\text{pOK}}$. The four insets are snapshots at different times. }\n\\label{fig:2dcoarsening_gamma1000}\n\\end{figure}\n\n\\begin{figure}[!htbp]\n\\centerline{\n\\includegraphics[width=150mm]{Figures\/OK_2d_kappa2000.eps}\n }\n\\caption{A 2D coarsening dynamics process with a large repulsive strength $\\gamma = 2000$. Other parameters are the same as that for Figure \\ref{fig:2dcoarsening_gamma1000}. Larger $\\gamma$ lead to more bubbles forming hexagonal pattern.}\n\\label{fig:2dcoarsening_gamma2000}\n\\end{figure}\n\n\n\n\n\\section{Summary}\n\nIn this paper, we explore the MPP property for the pACOK equation and propose a first order stabilized linear semi-implicit scheme which inherits the MPP and the energy stability in the discrete level. The third order polynomial $f(\\phi) = 3\\phi^2 - 2\\phi^3$ plays a key role in the proof of MPP for the system. We prove the MPP and energy stability in the semi-discrete and fully-discrete level in which the nonlinear terms $W$ and $f$ need not to be extended to have bounded second order derivative.\n\nIn the numerical experiments, we test the rate of convergence for the proposed scheme. We also show that in some examples, a traditional choice of $f(\\phi) = \\phi$ could violate the MPP. When $\\omega \\ll 1$, the pACOK dynamics displays pattern of hexagonal bubble assemblies. When the repulsive long-range interaction becomes stronger, there will be more bubbles appearing in the hexagonal equilibria.\n\nThis work can be extended along several directions. Firstly, we can study for higher order MPP schemes for the pACOK equation, or generally binary systems with long-range interactions. Secondly, we can further consider the MPP scheme for ternary systems, or a more general system of $N+1$ constituents in which $N$ phase field functions $\\{\\phi_j\\}_{j=1}^N$ are introduced to represent the densities of the $N$ constituents, and the $(N+1)$-th one is implicitly represented by $1- \\sum_{j=1}^N \\phi_j$.\n\nIn this paper, we mainly explore the numerical scheme for $L^2$ gradient flow dynamics based on operator splitting technique. Some other numerical methods, such as exponential time differencing based schemes, could be alternative choices for the MPP scheme, which will also be considered in the future.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Appendix}\\label{s:Appendix}\n\nIn the appendix, we briefly discuss the wellposedness of the pACOK dynamics (\\ref{eqn:pACOK}) and the $L^{\\infty}$ bound for the solution of (\\ref{eqn:pACOK}).\n\n\\begin{definition}\nLet $d = 2$ or 3. We call $\\phi(t,x)$ a global weak solution to problem (\\ref{eqn:pACOK}) if for any $T>0$, $\\phi(t,x)$ satisfies\n\\[\n\\phi \\in C([0,T]; L^p(\\mathbb{T}^d)) \\cap L^{\\infty}(0,T; H^1(\\mathbb{T}^d)) \\cap L^{2}(0,T;H^2(\\mathbb{T}^d)), \\quad p \\in [2,6)\n\\]\nand the initial condition $\\phi(0,x) = \\phi_0(x)$. Further, for any $t\\in(0,T]$, any test function $w\\in L^2(\\mathbb{T}^d)$, it holds\n\\begin{align*}\n&\\frac{\\text{d}}{\\text{d}t} \\int_{\\mathbb{T}^d} \\phi(t,x) w(x) \\ \\text{d}x \\\\\n= & \\int_{\\mathbb{T}^d} \\bigg[\\epsilon\\Delta\\phi - \\dfrac{1}{\\epsilon}W'(\\phi) - \\gamma(-\\Delta)^{-1}(f(\\phi)-\\omega)f'(\\phi) - M\\int_{\\mathbb{T}^d} (f(\\phi)-\\omega)\\ \\text{d}x\\cdot f'(\\phi) \\bigg] w(x) \\text{d}x\n\\end{align*}\nin the distributional sense in $(0,T)$.\n\\end{definition}\n\nWith the definition of the weak solution for the problem (\\ref{eqn:pACOK}), we are now ready to state the theorem for its wellposedness.\n\n\\begin{theorem}\\label{theorem:wellposedness}\nLet $d = 2$ or 3, and the initial data $\\phi_0\\in H^1(\\mathbb{T}^d)$. Then there exists a unique global weak solution $\\phi$ to the problem (\\ref{eqn:pACOK}). Further, the free energy $E^{\\emph{pOK}}$ in (\\ref{functional:pOK}) decreases as time evolves.\n\\end{theorem}\nThe proof is a standard procedure by following De Giorgi's minimizing movement scheme \\cite{Ambrosio_Rend1995,DeGiorgi_RMA1993}. We have a preprint discussing the wellposedness of a more complicated ternary system with long-range interaction, for which the proof of Theorem \\ref{theorem:wellposedness} can be viewed as a straightforward application. Therefore we will omit the proof here and recommend the readers to refer to \\cite{JooXuZhao_Preprint2020} for the details.\n\nOur next result is regarding to the $L^{\\infty}$ bound for the weak solution $\\phi(t,x)$ of the problem (\\ref{eqn:pACOK}), which can be achieved by De Giorgi's iteration \\cite{Chen_Book2003,WuYinWang_Book2006}. Note that in this case, the result holds only for $d = 2$. To begin with, we need an algebraic lemma. Without causing any confusions, we point out that the notations $M, h, k, d, \\alpha, \\beta$ picked below are exclusively for Lemma \\ref{lemma-algebra}, and might not mean the same as they are used elsewhere.\n\n\n\\begin{lemma}\\label{lemma-algebra}\nLet $\\mu(t)$ be a nonnegative, non-increasing function on $[k_0, +\\infty)$ that satisfies\n\\begin{equation}\\label{growth-condition}\n\\mu(h)\\leq\\Big(\\frac{M}{h-k}\\Big)^{\\alpha}\\mu(k)^{\\beta}, \\quad\\forall h>k\\geq k_0,\n\\end{equation}\nwhere $M>0$, $\\alpha>0$, $\\beta>1$ are all constants. Then we can find a constant $d>0$ such that\n$$\n \\mu(h)=0, \\quad\\forall h\\geq k_0+d.\n$$\n\\end{lemma}\n\\begin{proof}\nTo begin with, let\n\\[\nk_s=k_0+d-\\frac{d}{2^s}, \\quad\\forall s\\in\\mathbb{Z}^+,\n\\]\nwhere $d$ is defined to be\n\\begin{align}\\label{definition-d}\nd=M2^{\\frac{\\beta}{\\beta-1}}\\mu(k_0)^{\\frac{\\beta-1}{\\alpha}}.\n\\end{align}\nTake $h = k_{s+1}$ and $k = k_s$ in (\\ref{growth-condition}), it yields the recursive relation\n\\begin{equation}\\label{recursive}\n\\mu(k_{s+1})\\leq\\frac{M^{\\alpha}2^{(s+1)\\alpha}}{d^{\\alpha}}\\mu(k_s)^{\\beta}, \\quad\\forall s\\in\\mathbb{Z}^+.\n\\end{equation}\n\nNext we claim that\n\\begin{equation}\\label{claim}\n\\mu(k_s)\\leq\\frac{\\mu(k_0)}{r^s}, \\quad\\forall s\\in\\mathbb{Z}^+,\n\\end{equation}\nwhere $r>0$ is a constant defined as\n\\begin{equation}\\label{definition-r}\nr=2^{\\frac{\\alpha}{\\beta-1}}>1.\n\\end{equation}\nOnce \\eqref{claim} is verified, the proof of Lemma \\ref{lemma-algebra} is done by simply passing $s\\rightarrow \\infty$\nand using the assumption that $\\mu$ is non-increasing.\n\nWe finally prove (\\ref{claim}) by induction. Suppose (\\ref{claim}) is valid for $s$, then we obtain from (\\ref{definition-d}), (\\ref{recursive}) and (\\ref{definition-r}) that\n\\begin{align}\n\\mu(k_{s+1})\\leq\\frac{M^{\\alpha}2^{(s+1)\\alpha}}{d^{\\alpha}}\\frac{\\mu(k_0)^\\beta}{r^{\\beta{s}}}\n = \\frac{M^{\\alpha}2^{(s+1)\\alpha}}{M^{\\alpha}r^{\\beta}\\mu(k_0)^{\\beta-1}}\\frac{\\mu(k_0)^\\beta}{r^{\\beta{s}}}\n = \\dfrac{\\mu(k_0)}{r^{s+1}} \\dfrac{r^{s+1}2^{(s+1)\\alpha}}{r^{\\beta(s+1)}}\n = \\frac{\\mu(k_0)}{r^{s+1}}\n\\end{align}\nHence \\eqref{claim} is also valid if $s$ is replaced by $s+1$.\n\\end{proof}\n\n\n\nNow we can present our result regarding to the $L^{\\infty}$ bound for the weak solution to the problem (\\ref{eqn:pACOK}) with initial data $\\phi_0$ in 2D.\n\n\\begin{theorem}\\label{theorem: Linfity}\nFor any $\\phi_0\\in H^1(\\mathbb{T}^2)\\cap L^\\infty(\\mathbb{T}^2)$ and $T>0$, the unique weak solution\n\\[\n \\phi\\in L^\\infty(0, T; H^1(\\mathbb{T}^2))\\cap L^2(0, T; H^2(\\mathbb{T}^2))\n\\]\nto the problem (\\ref{eqn:pACOK}) satisfies\n\\begin{equation}\n\\|\\phi\\|_{L^\\infty([0,T]\\times\\mathbb{T}^2)}\\leq\\|\\phi_0\\|_{L^\\infty}+C^\\ast,\n\\end{equation}\nwhere $C^\\ast>0$ is a constant that only depends on $\\|\\phi_0\\|_{H^1}, \\epsilon^{-1}, \\omega, \\gamma$ and $M$.\n\\end{theorem}\n\\begin{proof}\nLet us denote\n\\[\n l=\\|\\phi_0\\|_{L^\\infty},\n\\]\nthe test function\n\\[\n\\xi(t,x)=(\\phi(t,x)-k)^+\\chi_{[t_1,t_2]},\\quad\\forall k>l\\ \\text{and}\\ t_2>t_1\n\\]\nand\n\\[\n\\tilde{F}(\\phi)=-\\frac{1}{\\epsilon}W'(\\phi)-\\gamma(-\\Delta)^{-1}\\big(f(\\phi)-\\omega\\big)f'(\\phi)-M\\Big(\\int_{\\mathbb{T}^2}(f(\\phi) - \\omega)\\,\\mathrm{d}{x}\\Big)f'(\\phi)\n\\]\nThen it is immediate to check for any $p>2$, there exists $M_1>0$ that only depends on $p$, $\\|\\phi_0\\|_{H^1}$, and coefficients of the equation, such that\n\\[\n\\|\\tilde{F}(\\phi(t))\\|_{L^p}\\leq M_1, \\quad\\forall t\\in [0, T].\n\\]\nConsider $\\xi$ as a test function for the weak solution $\\phi$, we obtain that\n\\begin{align}\\label{integral-equality-1}\n&\\iint_{[0, T]\\times\\mathbb{T}^2}\\partial_t(\\phi-k)^+(\\phi-k)^+\\chi_{[t_1,t_2]}\\,\\mathrm{d}{x}\\mathrm{d}{t}+\\iint_{[0, T]\\times\\mathbb{T}^2}\\big|\\nabla(\\phi-k)^+\\big|^2\\chi_{[t_1,t_2]}\\,\\mathrm{d}{x}\\mathrm{d}{t}\\nonumber\\\\\n=&\\iint_{[0, T]\\times\\mathbb{T}^2}\\tilde{F}(\\phi)(\\phi-k)^+\\chi_{[t_1,t_2]}\\,\\mathrm{d}{x}\\mathrm{d}{t}.\n\\end{align}\nIf we denote\n\\[\n\\mathrm{I}_k(t)=\\int_{\\mathbb{T}^2}|(\\phi(t,x)-k)^+|^2\\,\\mathrm{d}{x},\n\\]\nwe get from \\eqref{integral-equality-1} that\n\\begin{align}\\label{integral-inequality-1}\n\\frac12\\Big[\\mathrm{I}_k(t_2)-\\mathrm{I}_k(t_1)\\Big]+\\int_{t_1}^{t_2}\\int_{\\mathbb{T}^2}\\big|\\nabla(\\phi-k)^+\\big|^2\\,\\mathrm{d}{x}\\mathrm{d}{t}\n\\leq\\int_{t_1}^{t_2}\\int_{\\mathbb{T}^2}\\big|\\tilde{F}(\\phi)\\big|(\\phi-k)^+\\,\\mathrm{d}{x}\\mathrm{d}{t}.\n\\end{align}\n\nSuppose $\\mathrm{I}_k(t)$ attains its maximum value at $s\\in [0, T]$ (assume $s>0$ without loss of generalization). Then\n\\[\n\\mathrm{I}_k(s)-\\mathrm{I}_k(s-\\eps)\\geq 0,\n\\]\nfor any $0<\\eps0$ is a generic constant. If we further denote\n$$\n F(t,x)=|\\tilde{F}(\\phi(t,x))|+|\\varphi(t, x)|,\n$$\nthen $\\forall p>2$ we get\nafter combining \\eqref{key-inequality-1} with \\eqref{Sobolev-embedding} that\n\\begin{equation}\\label{key-inequality-2}\n\\Big(\\int_{\\mathbb{T}^2}|\\varphi(s, x)|^p\\,\\mathrm{d}{x}\\Big)^{\\frac{2}{p}}\\leq C\\int_{\\mathbb{T}^2}F(s, x)|\\varphi(s,x)|\\,\\mathrm{d}{x}.\n\\end{equation}\nNote that\n\\begin{equation}\\label{bound-F}\n\\|F(t)\\|_{L^p}\\leq M_2, \\quad\\forall t\\in [0, T].\n\\end{equation}\n\nNext, we denote\n$$\n B_k(t)=\\{x\\in\\mathbb{T}^2: \\phi(t, x)>k\\}.\n$$\nThen it follows from \\eqref{key-inequality-2} and H\\\"{o}lder's inequality that\n$$\n \\Big(\\int_{B_k(s)}|\\varphi(s, x)|^p\\,\\mathrm{d}{x}\\Big)^{\\frac{2}{p}}\\leq C\\int_{B_k(s)}F(s, x)|\\varphi(s,x)|\\,\\mathrm{d}{x}\n \\leq C\\Big(\\int_{B_k(s)}|\\varphi(s, x)|^p\\,\\mathrm{d}{x}\\Big)^{\\frac{1}{p}}\\Big(\\int_{B_k(s)}|F(s, x)|^q\\,\\mathrm{d}{x}\\Big)^{\\frac{1}{q}} ,\n$$\nwhere $q$ is the H\\\"older conjugate of $p$. It further implies\n\\begin{equation}\\label{key-inequality-3}\n\\Big(\\int_{B_k(s)}|\\varphi(s, x)|^p\\,\\mathrm{d}{x}\\Big)^{\\frac{1}{p}}\\leq C\\Big(\\int_{B_k(s)}|F(s, x)|^q\\,\\mathrm{d}{x}\\Big)^{\\frac{1}{q}},\n\\end{equation}\nwhere $C>0$ only depends on $p$, $\\|\\phi_0\\|_{H^1}$, and coefficients of equation (\\ref{eqn:pACOK}). As a consequence, for any $1k,\\,t\\in [0, T]\n\\end{align}\ndue to the fact that $\\varphi\\geq h-k$ on $B_h(t)$ and $B_h(t)\\subset B_k(t)$. Therefore, if we denote\n$$\n \\mu(k)=\\sup_{t\\in[0, T]}|B_k(t)|,\n$$\nwe get from \\eqref{direction-1} and \\eqref{direction-2} that\n\\begin{equation}\\label{iterative-inequality}\n\\mu(h)\\leq \\Big(\\frac{C}{h-k}\\Big)^2\\mu(k)^{\\frac{2p-2}{mp}+\\frac{p-2}{p}}.\n\\end{equation}\nNote that\n$$\n \\frac{2p-2}{mp}+\\frac{p-2}{p}>1\n$$\nby the choice of $m$, $p$, hence using Lemma \\ref{lemma-algebra} we know that\n$$\n \\mu(l+C^\\ast)=\\sup_{t\\in [0, T]}|B_{k+C^\\ast}(t)|=0,\n$$\nwhich indicates\n\\begin{equation}\n\\phi(t, x)\\leq l+C^\\ast, \\quad\\forall (t, x)\\in [0, T]\\times\\mathbb{T}^2.\n\\end{equation}\nHence the proof is complete.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Acknowledgements}\n\nX. Xu's work is supported by a grant from the Simons Foundation through grant No. 635288;\nY. Zhao's work is supported by a grant from the Simons Foundation through Grant No. 357963.\n\n\n\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe detection of the primordial B-mode power spectrum by the BICEP2 collaboration confirms\nthe existence of primordial gravitational wave, and the observed B-mode power\nspectrum gives the constraint on the tensor-to-scalar ratio\nwith $r=0.20^{+0.07}_{-0.05}$ at $1\\sigma$ level for the lensed-$\\Lambda$CDM model \\cite{Ade:2014xna}.\nFurthermore, $r=0$ is disfavored at $7.0\\sigma$ level. The new constraints on $r$\nand the spectral index $n_s$ exclude a wide class of inflationary models.\nFor the inflation model with non-minimal coupling with gravity \\cite{Kallosh:2013tua}, a universal\nattractor at strong coupling was found with $n_s=1-2\/N$ and $r=12\/N^2$. This model is inconsistent\nwith the BICEP2 result $r\\gtrsim 0.1$ at $2\\sigma$ level\nbecause the BICEP2 constraint on $r$ requires the number of e-folds $N=\\sqrt{12\/r}\\lesssim \\sqrt{120}\\approx 11$\nwhich is not enough to solve the horizon problem. If we require $N=50$,\nthen $r=0.0048$, so the model is excluded by the BICEP2 result. For the small-field inflation\nlike the hilltop inflation with the potential $V(\\phi)=V_0[1-(\\phi\/\\mu)^p]$ \\cite{Albrecht:1982wi,Boubekeur:2005zm},\n$r\\sim 0$, so the model is excluded by the BICEP2 result.\n\nWithout the running of the spectral index, the combination of {\\em Planck}+WP+highL data gives\n$n_s=0.9600\\pm 0.0072$ and $r_{0.002}<0.0457$ at the 68\\% confidence level for\nthe $\\Lambda$CDM model \\cite{Ade:2013zuv,Ade:2013uln} which is in tension\nwith the BICEP2 result. When the running of the spectral index is included in the data fitting,\nthe same combination gives $n_s=0.957\\pm 0.015$, $n_s'=dn_s\/d\\ln k=-0.022^{+0.020}_{-0.021}$\nand $r_{0.002}<0.263$ at the 95\\% confidence level \\cite{Ade:2013zuv,Ade:2013uln}.\nTo give a consistent constraint on $r$\nfor the combination of {\\em Planck}+WP+highL data and the BICEP2 data, we require\na running of the spectral index $n_s'<-0.002$ at the 95\\% confidence level.\nFor the single field inflation, the spectral index $n_s$ for the\nscalar perturbation deviates from the Harrison-Zel'dovich value of $1$ in the order of $10^{-2}$, so\n$n_s'$ is in the order of $10^{-3}$. The explanation of large $r$ and $n_s'$ is a challenge to\nsingle field inflation. In light of the BICEP2 data, several attempts were proposed to explain the large value\nof $r$ \\cite{Lizarraga:2014eaa,Harigaya:2014qza,Contaldi:2014zua,Collins:2014yua,Byrnes:2014xua,Anchordoqui:2014uua,Harigaya:2014sua,\nNakayama:2014koa,Zhao:2014rna,\nCook:2014dga,Kobayashi:2014jga,Miranda:2014wga,Masina:2014yga,Hamada:2014iga,Hertzberg:2014aha,\nDent:2014rga,Joergensen:2014rya,Freese:2014nla,Ashoorioon:2014nta,Ashoorioon:2013eia,\nChoudhury:2014kma,Choudhury:2013iaa,Hotchkiss:2011gz,BenDayan:2009kv}.\nIn this Letter, we use the chaotic and natural inflation models to\nexplain the challenge.\n\n\\section{Slow-roll Inflation}\n\nThe slow-roll parameters are defined as\n\\begin{gather}\n\\label{slow1}\n\\epsilon=\\frac{M_{pl}^2V_\\phi^2}{2V^2},\\\\\n\\label{slow2}\n\\eta=\\frac{M_{pl}^2V_{\\phi\\phi}}{V},\\\\\n\\label{slow3}\n\\xi=\\frac{M_{pl}^4V_\\phi V_{\\phi\\phi\\phi}}{V^2},\n\\end{gather}\nwhere $M^2_{pl}=(8\\pi G)^{-1}$, $V_\\phi=dV(\\phi)\/d\\phi$, $V_{\\phi\\phi}=d^2V(\\phi)\/d\\phi^2$\nand $V_{\\phi\\phi\\phi}=d^3V(\\phi)\/d\\phi^3$. For the single field inflation, the spectral indices,\nthe tensor-to-scalar ratio and the running are\ngiven by\n\\begin{gather}\n\\label{nsdef}\nn_s-1\\approx 2\\eta-6\\epsilon,\\\\\n\\label{rdef}\nr\\approx 16\\epsilon\\approx -8n_t,\\\\\n\\label{rundef}\nn_s'=dn_s\/d\\ln k\\approx 16\\epsilon\\eta-24\\epsilon^2-2\\xi.\n\\end{gather}\nThe number of e-folds before the end of inflation is given by\n\\begin{equation}\n\\label{efolddef}\nN(t)=\\int_t^{t_e}Hdt\\approx \\frac{1}{M_{pl}^2}\\int_{\\phi_e}^\\phi\\frac{V(\\phi)}{V_\\phi(\\phi)}d\\phi,\n\\end{equation}\nwhere the value $\\phi_e$ of the inflaton field at the end of inflation is defined by $\\epsilon(\\phi_e)=1$.\nThe scalar power spectrum is\n\\begin{equation}\n\\label{power}\n\\mathcal{P}_{\\mathcal{R}}=A_s\\left(\\frac{k}{k_*}\\right)^{n_s-1+n_s'\\ln(k\/k_*)\/2},\n\\end{equation}\nwhere the subscript ``*\" means the value at the horizon crossing, the scalar amplitude\n\\begin{equation}\n\\label{power1}\nA_s\\approx \\frac{1}{24\\pi^2M^4_{pl}}\\frac{\\Lambda^4}{\\epsilon}.\n\\end{equation}\nWith the BICEP2 result $r=0.2$, the energy scale of inflation is $\\Lambda\\sim 2.2\\times 10^{16}$GeV.\n\nFor the chaotic inflation with the power-law potential $V(\\phi)=\\Lambda^4(\\phi\/M_{pl})^p$ \\cite{linde83}, the slow-roll parameters\nare $\\epsilon=p\/(4N_*)$, $\\eta=(p-1)\/(2N_*)$ and $\\xi=(p-1)(p-2)\/(4N^2_*)$. The spectral index $n_s=1-(p+2)\/(2N_*)$,\nthe running of the spectral index $n_s'=-(2+p)\/(2N_*^2)=-2(1-n_s)^2\/(p+2)<0$ and the tensor-to-scalar ratio $r=4p\/N_*=8p(1-n_s)\/(p+2)$.\nWe plot the $n_s-r$ and $n_s-n_s'$ relations in Figs.\n\\ref{pwrnsr} and \\ref{pwrrun} for $p=1$, $p=2$, $p=3$ and $p=4$. In Fig. \\ref{pwrnsr}, we also show the points with\n$N_*=50$ and $N_*=60$. From Figs. \\ref{pwrnsr} and \\ref{pwrrun}, we see that $r$ increases with the power $p$, but\n$|n_s'|$ decreases with the power $p$. Therefore, it is not easy to satisfy both the requirements $r\\gtrsim 0.1$ and\n$n_s'<-0.002$. The chaotic inflation with $20$, $i = 1, \\dots, N$, that determines the direction ${\\bf e}(\\Theta_i) = (\\cos \\Theta_i, \\sin \\Theta_i)$ of self-propulsion at constant speed $v_0$. The orientation $\\Theta_i$ also undergoes free rotational diffusion with diffusion coefficient $D_R$. \nIn its more general form, particles interact through a pair potential $u(r, \\varphi)$ with total potential energy \n\\begin{align}\\label{e:U}\n U = \\chi\\sum_{1\\le i \\epsilon, \\forall j\\ne i, \\\\\n\t\\label{sde_angle_hc}\n\t\\mathrm{d} \\Theta_i &= \\sqrt{2 D_R} \\mathrm{d} W_i.\n\\end{align}\n\\end{subequations}\nThis represents particles as hard disks of diameter $\\epsilon$: particles only sense each other when they come into contact, and they are not allowed to get closer than $\\epsilon$ to each other (mutual impenetrability condition). In comparison with the mean-field scaling, here instead the scaling is $\\chi = 1, \\ell = \\epsilon \\ll 1$ so that each particle only interacts with the few particles that are within a distance $O(\\epsilon)$, the interaction is very strong. Using the method of matched asymptotics, from \\eqref{sde_model_hc} one obtains to order $\\phi$ the following model:\n\\begin{align}\\label{model3}\n\\partial_t f + v_0\\nabla \\cdot \\left[ f (1-\\phi \\rho) {\\bf e}(\\theta) + \\phi {\\bf p} f\\right] &= D_T \\nabla \\cdot \\left[ (1- \\phi \\rho) \\nabla f + 3 \\phi f \\nabla \\rho \\right] + D_R \\partial_{\\theta}^2 f.\n\\end{align}\nHere $\\phi$ is the effective occupied area $\\phi = (N-1)\\epsilon^2 \\pi\/2$. Model \\eqref{model3} is obtained formally in the limit of $\\epsilon$ and $\\phi$ small. \nNote that this equation is consistent with the case $N=1$: if there is only one particle, then $\\phi = 1$ and we recover a linear PDE (no interactions). The equation for the spatial density is\n\\begin{equation}\n\\partial_t \\rho + v_0 \\nabla \\cdot {\\bf p} = D_T \\nabla \\cdot \\left[ (1+ 2\\phi \\rho) \\nabla \\rho \\right],\n\\end{equation}\nwhich indicates the collective diffusion effect: the higher the occupied fraction $\\phi$, the higher the effective diffusion coefficient. \nWe note that, due to the nature of the excluded-volume interactions, models \\eqref{model2} and \\eqref{model3} are obtained via approximations (closure at the pair correlation function and matched asymptotic expansions, respectively) and no rigorous results are available. A nice exposition of the difficulties of going from micro to macro in the presence of hard-core non-overlapping constraints is given in \\cite{Maury:2011eu}. In particular, they consider hard-core interacting particles in the context of congestion handling in crowd motion. In contrast to \\eqref{sde_model_hc}, the dynamics involve only position and are deterministic. Collisions can then be handled via the projection of velocities onto the set of feasible velocities. In \\cite{Maury:2011eu} they do not attempt to derive a macroscopic model from the microscopic dynamics but instead propose a PDE model for the population density $\\rho({\\bf x},t)$ that expresses the congestion assumption by setting the velocity to zero whenever $\\rho$ attains a saturation value (which they set to one). \n\n\\subsection{Discrete random walks}\\label{sec:discrete}\n\nNext we discuss fully discrete models for active particles with size exclusion effects. We start by considering a \nsimple exclusion model for active particles on a one-dimensional lattice, which has been investigated in \\cite{kourbane2018exact}. The brief description of the microscopic lattice model is as follows: $N$ particles of size $\\epsilon$ evolve on a discrete ring of $1\/\\epsilon$ sites, with occupancy $\\phi = \\epsilon N \\le 1$. Each lattice is occupied by at most one particle (thus modelling a size exclusion), and particles can either be moving left ($-$ particles) or right ($+$ particles). The configuration can be represented using occupation numbers $\\sigma_i$ at site $i$ with values in $\\{-1, 0, 1\\}$. \nThe dynamics combine three mechanisms: \n\\begin{enumerate}[label=(\\alph*)]\n\t\\item Diffusive motion: for each bond $(i, i+1)$, $\\sigma_i$ and $\\sigma_{i+1}$ are exchanged at rate $D_T$.\n\t\\item Self-propulsion and size exclusion: for each bond $(i, i+1)$, a $+$ particle in $i$ jumps to $i+1$ if $\\sigma_{i+1} = 0$; or a $-$ particle in $i+1$ jumps to $i$ if $\\sigma_i = 0$ at rate $\\epsilon v_0$.\n\t\\item Tumbling: particles switch direction $\\sigma_i \\to - \\sigma_i$ at rate $\\epsilon^2 \\lambda$. \n\\end{enumerate}\nRescaling space and time as $\\epsilon i$ and $\\epsilon^2 t$ respectively, and starting from a smooth initial conditions, the macroscopic equations can be derived exactly as \\cite{kourbane2018exact}\n\\begin{align} \\label{model_lattice1D}\n\\begin{aligned}\n\\partial_t f_+ + v_0\\partial_x [f_+ (1-\\phi \\rho)] &= D_T\\partial_{xx} f_+ + \\lambda (f_- - f_+), \\\\\n\\partial_t f_- - v_0\\partial_x [f_- (1-\\phi \\rho)] &= D_T\\partial_{xx} f_- + \\lambda (f_+ - f_-),\n\\end{aligned}\n\\end{align}\nwhere $f_+$ and $f_-$ are the probability densities corresponding to the $+$ and $-$ particles, respectively, and $\\rho = f_+ + f_-$. \nIntroducing the number densities\n\\begin{align}\\label{number_densities}\nr({\\bf x},t) = N f_+({\\bf x},t), \\qquad b({\\bf x},t) = N f_-({\\bf x},t),\n\\end{align}\nwhich integrate to $N_1$ and $N_2$ respectively, we can rewrite \\eqref{model_lattice1D} as \n\\begin{align}\\label{model_lattice1D_number_densities}\n\\begin{aligned}\n\\partial_t r + v_0\\partial_x [r (1-\\bar \\rho)] &= D_T\\partial_{xx} r + \\lambda (b - r), \\\\\n\\partial_t b - v_0\\partial_x [b (1- \\bar\\rho)] &= D_T\\partial_{xx} b + \\lambda (r - b),\n\\end{aligned}\n\\end{align}\nwith $\\bar \\rho = \\epsilon (r + b)$.\nOne can also consider the same process in higher dimensions with a finite set of orientations ${\\bf e}_k, k = 1, \\dots, m$. The most straightforward generalisation of \\eqref{model_lattice1D} is to consider a two-dimensional square lattice with $m=4$ directions, namely ${\\bf e}_1 = (1,0), {\\bf e}_2 = (0,1), {\\bf e}_3 = (-1,0), {\\bf e}_4 = (0,-1)$ (see Fig.~1 in \\cite{kourbane2018exact}). In this case, the configuration would take five possible values, $\\sigma_i= \\{-1, -i, 0, i, 1\\}$ and the resulting macroscopic model would consist of a system of four equations for the densities of each subpopulations\n\\begin{equation} \\label{lattice2D}\n\t\\partial_t f_k + v_0\\nabla \\cdot [f_k (1-\\phi \\rho) {\\bf e}_k] = D_T \\Delta f_k + \\lambda (f_{k+1} + f_{k-1} - 2 f_k), \\qquad k = 1, \\dots,4\n\\end{equation}\nwhere now $\\phi = \\epsilon^2 N$, $f_k({\\bf x},t)$ stands for the probability density of particles going in the ${\\bf e}_k$ direction, and $\\rho = \\sum_k f_k$. Periodicity in angle implies that $f_5=f_1, f_{-1} = f_4$. \n\nNote how the model in \\cite{kourbane2018exact} differs from an asymmetric simple exclusion processes (ASEP) in that particles are allowed to swap places in the diffusive step (see (a) above). As a result, the macroscopic models \\eqref{model_lattice1D} and \\eqref{lattice2D} lack any cross-diffusion terms. We can also consider an actual ASEP process, in which simple exclusion is also added to the diffusive step, that is, point (a) above is replaced by\n\\begin{enumerate}[label=(\\alph*')]\n\t\\item Diffusive motion: a particle in $i$ jumps to $i+ 1$ at rate $D_T$ if $\\sigma_{i+ 1} = 0$ (and similarly to $i-1$).\n\\end{enumerate}\nIn this case, the resulting macroscopic model is\n\\begin{equation} \\label{ASEP_2D}\n\t\\partial_t f_k + v_0\\nabla \\cdot [ f_k (1- \\phi \\rho) {\\bf e}_k] = D_T \\nabla \\cdot [(1- \\phi \\rho) \\nabla f_k + \\phi f_k \\nabla \\rho] + \\lambda (f_{k+1} + f_{k-1} - 2 f_k), \\qquad k = 1, \\dots,4.\n\\end{equation}\n\n\n\n\\subsection{Hybrid random walks}\\label{sec:hybrid}\n\nIn the previous two subsections we have discussed models that consider both the position and the orientation as continuous, or discrete. Here we discuss hybrid random walks, that is, when positions are continuous and orientations finite, or vice-versa.\n\nThe first hybrid model we consider is an active exclusion process whereby the orientation is a continuous process in $[0, 2 \\pi)$ evolving according to a Brownian motion with diffusion $D_R$, \\eqref{sde_angle}, while keeping the position evolving according to a discrete asymmetric exclusion process (ASEP) \\cite{Bruna:2021tb}. The advantage of this approach is to avoid the anisotropy imposed by the underlying lattice. \nHere we present the model in two-dimensions so that we can compare it to the models presented above. \n\nWe consider a square lattice with spacing $\\epsilon$ and orientations ${\\bf e}_k, k=1, \\dots, 4$ as given above. A particle at lattice site ${\\bf x}$ can jump to neighbouring sites ${\\bf x}+\\epsilon {\\bf e}_k$ if the latter is empty at a rate $\\pi_k(\\theta)$ that depends on its orientation $\\theta$, namely \n\\begin{equation*}\n\t\\pi_k(\\theta) = \\alpha_\\epsilon \\exp(\\beta_\\epsilon {\\bf e}(\\theta) \\cdot {\\bf e}_k),\n\\end{equation*}\nwhere $\\alpha_\\epsilon = D_T \/\\epsilon^2$ and $\\beta_\\epsilon = v_0 \\epsilon\/(2 D_T)$. Therefore, the diffusive and self-propulsion mechanisms in \\eqref{model_lattice1D} are now accounted for together: jumping in the direction opposite to your orientation reduces the rate to $\\sim \\alpha_\\epsilon(1-\\beta_\\epsilon)$, whereas the there is a positive bias $\\sim \\alpha_\\epsilon(1+\\beta_\\epsilon)$ towards jumps in the direction pointed to by ${\\bf e}(\\theta)$. The tumbling (point 3 above) is replaced by a rotational Brownian motion. \nTaking the limit $\\epsilon\\to 0$ while keeping the occupied fraction $\\phi =N \\epsilon^2$ finite one obtains the following macroscopic model for $f = f({\\bf x},\\theta,t)$\n\\begin{align} \\label{model_hy}\n\\partial_t f + v_0\\nabla \\cdot [ f (1- \\phi \\rho) {\\bf e}(\\theta)] &= D_T \\nabla \\cdot ( (1- \\phi \\rho) \\nabla f + \\phi f \\nabla \\rho) + D_R \\partial_{\\theta}^2 f.\n\\end{align}\nThis model can be directly related to the fully discrete model \\eqref{ASEP_2D}: they are exactly the same if one considers \\eqref{ASEP_2D} as the discretised-in-angle version of \\eqref{model_hy} by identifying \n$$\nD_R \\partial_\\theta^2 f_k \\approx D_R \\frac{f_{k+1} + f_{k-1} - 2 f_k}{(2\\pi\/m)^2}, \n$$\nthat is, $\\lambda = D_R m^2\/(2 \\pi^2)$, where $m$ is the number of orientations in the fully discrete model.\n\n\n\nThe other possible hybrid model is to consider a continuous random walk with interactions in space \\eqref{sde_x}, while only allowing a finite number of orientations, $\\Theta_i \\in \\{\\theta_1, \\dots, \\theta_m\\}$. In its simplest setting, we can consider that $\\theta_k$ are equally spaced in $[0, 2\\pi)$ and a constant switching rate $\\lambda$ between the neighbouring angles. The $N$ particles evolve according to the stochastic model:\n\\begin{subequations} \\label{hybrid2}\n\\begin{align}\n\\label{hybrid_x}\n\\mathrm{d} {\\bf X}_i &= \\sqrt{2 D_T} \\mathrm{d} {\\bf W}_i - \\nabla_{{\\bf x}_i} U\\mathrm{d} t + v_0 {\\bf e}(\\Theta_i) \\mathrm{d} t,\\\\\n\\label{hybrid_angle}\n\\Theta_i &= \\{\\theta_k\\}_{k=1}^m, \\quad \\theta_k \\xrightarrow{\\ \\lambda \\ } \\theta_{k+ 1}\\pmod{2\\pi}, \\quad \\theta_k \\xrightarrow{\\ \\lambda \\ } \\theta_{k- 1}\\pmod{2\\pi}.\n\\end{align}\n\\end{subequations}\nIf we assume exclude-volume interactions through a hard-core potential, the resulting model is \\cite{Wilson:2018fg}\n\\begin{equation}\\label{model_wilson}\n\t\\partial_t f_k + v_0\\nabla \\cdot \\left[ f_k (1-\\phi \\rho) {\\bf e}_k + \\phi {\\bf p} f_k\\right] = D_T \\nabla \\cdot \\left[ (1- \\phi \\rho) \\nabla f_k + 3 \\phi f_k \\nabla \\rho \\right] + \\lambda \\left ( f_{k+1} + f_{k-1} -2 f_k \\right),\n\\end{equation}\nwhere $\\rho = \\sum_{k=1}^m f_k$, ${\\bf p} = \\sum_{k=1}^m f_k {\\bf e}(\\theta_k)$, and ${\\bf e}_k = {\\bf e}(\\theta_k)$. \nThe density $f_k({\\bf x},t)$ represents the probability of finding a particle at position ${\\bf x}$ at time $t$ with orientation $\\theta_k$ (naturally, we identify $f_{m+1} = f_1$ and $f_{-1} = f_m$). Here $\\phi = (N-1) \\epsilon^2 \\pi\/2$ represents the effective excluded region as in \\eqref{model3}. We note how this model is consistent with the continuous model \\eqref{model3}, in that if we had discretised angle in \\eqref{model3} we would arrive at the cross-diffusion reaction model \\eqref{model_wilson}. \n\nA variant of the hybrid model \\eqref{hybrid2} is to allow for jumps to arbitrary orientations instead of rotations of $2\\pi\/m$, namely, from $\\theta_k$ to $\\theta_{j}\\pmod{2\\pi}$, $j\\ne k$, at a constant rate $\\lambda$ independent of the rotation. This is a convenient way to model the tumbles of a run-and-tumble process, such as the one used to describe the motion of \\emph{E. Coli} \\cite{Berg:1993ug}, see also \\S \\ref{sec:biological_transport}. In this case, the reaction term in \\eqref{model_wilson} changes to \n\\begin{equation}\\label{hybrid3}\n\t\\partial_t f_k + v_0\\nabla \\cdot \\left[ f_k (1-\\phi \\rho) {\\bf e}(\\theta_k) + \\phi {\\bf p} f_k\\right] = D_T \\nabla \\cdot \\left[ (1- \\phi \\rho) \\nabla f_k + 3 \\phi f_k \\nabla \\rho \\right] + \\lambda \\sum_{j \\ne k } \\left( f_{j} - f_k \\right).\n\\end{equation}\n\nWe may generalise the jumps in orientation by introducing a turning kernel $T(\\theta, \\theta')$ as the probability density function for a rotation from $\\theta'$ to $\\theta$. That is, if $\\Theta_i(t)$ is the orientation of the $i$th particle at time $t$ and the jump occurs at $t^*$, \n$$\nT(\\theta, \\theta') \\mathrm{d} \\theta = \\mathbb P\\left \\{ \\theta \\le \\Theta_i(t^*_+) \\le \\theta + \\mathrm{d} \\theta \\ | \\ \\Theta_i(t^*_-) = \\theta' \\right \\}.\n$$\nClearly for mass conservation we require that $\\int T(\\theta, \\theta') \\mathrm{d} \\theta = 1$. \nThe jumps may only depend on the relative orientation $\\theta - \\theta'$ in the case of a homogeneous and isotropic medium, in which case $T(\\theta, \\theta') \\equiv T(\\theta-\\theta')$. This is the case of the two particular examples above: in \\eqref{model_wilson}, the kernel is \n$$\nT(\\theta, \\theta') = \\frac{1}{2} \\left [ \\delta(\\theta -\\theta'- \\Delta) + \\delta(\\theta-\\theta' + \\Delta) \\right], \\qquad \\Delta = \\frac{2\\pi}{m},\n$$\nwhereas the rotation kernel in \\eqref{hybrid3} is\n$$\nT(\\theta, \\theta') = \\frac{1}{m-1} \\sum_{k = 1}^{m-1} \\delta(\\theta -\\theta' + k\\Delta), \\qquad \\Delta = \\frac{2\\pi}{m},\n$$\nwhere the argument of the delta function is taken to be $2\\pi$-periodic. If the turning times $t^*$ are distributed according to a Poisson process with intensity $\\lambda$, the resulting macroscopic model for the phase density $f = f({\\bf x},\\theta,t)$ with a general turning kernel $T$ becomes\n\\begin{equation}\\label{run-tumble-cont}\n\t\\partial_t f + v_0\\nabla \\cdot \\left[ f (1-\\phi \\rho) {\\bf e}(\\theta) + \\phi {\\bf p} f\\right] = D_T \\nabla \\cdot \\left[ (1- \\phi \\rho) \\nabla f + 3 \\phi f \\nabla \\rho \\right] -\\lambda f + \\lambda \\int_0^{2\\pi} T(\\theta,\\theta') f({\\bf x},\\theta',t) \\mathrm{d} \\theta '.\n\\end{equation}\nWe note that the microscopic process associated with \\eqref{run-tumble-cont} is continuous (and not hybrid) if the support of $T$ has positive measure.\n\n\n\n\\section{Models for externally activated particles}\\label{s:nonact}\n\nIn this section we go from active to passive particles and consider models with time reversal at the microscopic level. As mentioned in the introduction, the defining factor of active matter models is the self-propulsion term, which makes them out-of-equilibrium. Mathematically, this can be expressed by saying that even the microscopic model lacks a gradient-flow structure (either due to the term ${\\bf e}(\\theta)$ in the transport term, see \\eqref{model3}, or the reaction terms in \\eqref{model_lattice1D}, \\eqref{lattice2D}, see section \\ref{sec:gen_structure}).\n \nIn the previous section we have seen the role the orientation $\\theta$ plays. If it is kept continuous, the resulting macroscopic model is of kinetic type for the density $f({\\bf x},\\theta,t)$. If instead only a fixed number $m$ of orientations are allowed, then these define a set of $m$ species, whereby all the particles in the same species have the same drift term. This motivates the connection to cross-diffusion systems for passive particles, which are obtained by turning off the active change in directions in the models of section \\ref{sec:active} and look at the resulting special cases. This is a relevant limit in many applications, such as in pedestrian dynamics (see section \\ref{sec:pedestrian}).\nOnce the orientations are fixed, we are left with two possible passive systems: either originating from a spatial Brownian motion or a spatial ASEP discrete process. \n\n\\subsection{Continuous models}\nThe starting point is the microscopic model \\eqref{sde_model} taking the limit $D_R \\to 0$. We could still keep the interaction potential as depending on the relative orientations, which would lead to different self- and cross-interactions (which might be useful in certain applications). Here for simplicity we assume interactions are all the same regardless of the orientations:\n\\begin{subequations}\n\t\\label{sde_model_passive}\n\\begin{align}\n\\mathrm{d} {\\bf X}_i &= \\sqrt{2 D_T} \\mathrm{d} {\\bf W}_i - \\nabla_{{\\bf x}_i} U\\mathrm{d} t + v_0 {\\bf e}(\\Theta_i) \\mathrm{d} t,\\\\\n\\Theta_i(t) &= \\theta_k, \\qquad \\text{if } i \\in \\mathcal I_k, \\qquad k = 1,\\dots, m,\n\\end{align}\n\\end{subequations}\nwhere $\\mathcal I_k$ is the set of particles belonging to species $k$. The number of particles in each species is $|\\mathcal I_k| = N_k$.\n\nThe mean-field limit of \\eqref{sde_model_passive} is given by (taking $N = \\sum_k N_k \\to \\infty$ as in \\eqref{1_mfa})\n\\begin{equation}\n\t\\label{mfa_passive}\n\t\\partial_t f_k({\\bf x}, t) = \\nabla_{{\\bf x}} \\cdot \\left[ D_T \\nabla_{\\bf x} f_k -v_0 {\\bf e}(\\theta_k) f_k + f_k \\nabla_{{\\bf x}} (u \\ast \\rho) \\right],\n\\end{equation}\nand $\\rho({\\bf x},t) = \\sum_k f_k$. For consistency with the active models, here we do not take $f_k$ to be probability densities but to integrate to the relative species fraction, whereas as before the total density $\\rho$ has unit mass:\n\\begin{equation} \\label{normalisation_passive}\n\t\\int_\\Omega f_k({\\bf x},t) \\mathrm{d} {\\bf x} = \\frac{N_k}{N},\\qquad \\int_\\Omega \\rho({\\bf x},t) \\mathrm{d} {\\bf x} = 1.\n\\end{equation}\nThus $f_k = f_k({\\bf x},t)$ describes the probability that a particle is at position ${\\bf x}$ at time $t$, \\emph{and} is in the $\\mathcal I_k$ set.\n\nThe microscopic model \\eqref{sde_model_passive} with the interaction term $U$ replaced by a hard-core potential for particles with diameter $\\epsilon$ can be dealt with via the method of matched asymptotics. In this case, the resulting cross-diffusion model is\n\\begin{align}\\label{eq:MF_cross_diff_sys}\n\t\\partial_t f_k + v_0\\nabla \\cdot \\left[ f_k {\\bf e}_k + \\phi_{kl}({\\bf e}_l - {\\bf e}_k) f_k f_l \\right] &= D_T \\nabla \\cdot \\left[ (1 + \\phi_{kk} f_k) \\nabla f_k + \\phi_{kl} (3 f_k \\nabla f_l - f_l \\nabla f_k) \\right], \\qquad l \\ne k,\n\\end{align}\nwhere $\\phi_{kk} = (N_k -1) N\/N_k \\epsilon^2 \\pi$, $\\phi_{kl} = N\\epsilon ^2 \\pi \/2$ for $l\\ne k$, and $f_k({\\bf x},t)$ are defined as above. \nThis model was first derived in \\cite{Bruna:2012wu} for just two species but in a slightly more general context, whereby particles many have different sizes and diffusion coefficients\n(also, note that in \\cite{Bruna:2012wu}, \\eqref{eq:MF_cross_diff_sys} appears written in terms of probability densities). \nEquation \\eqref{eq:MF_cross_diff_sys} can be directly related to model \\eqref{model_wilson} with $\\lambda = 0$ if in both models we assume $N_k$ large enough such that $N_k - 1 \\approx N_k, N-1 \\approx N$: \n\\begin{equation}\\label{cross-diff_num_den}\n\t\\partial_t f_k + v_0\\nabla \\cdot \\left[ f_k (1-\\phi \\rho) {\\bf e}(\\theta_k) + \\phi {\\bf p} f_k\\right] = D_T \\nabla \\cdot \\left[ (1- \\phi \\rho) \\nabla f_k + 3 \\phi f_k \\nabla \\rho\t \\right],\n\\end{equation}\nwhere $\\phi = N \\epsilon^2 \\pi\/2$, $\\rho = \\sum_k f_k$, and ${\\bf p} =\\sum_k f_k {\\bf e}(\\theta_k)$. Model \\eqref{cross-diff_num_den} is the cross-diffusion system for red and blue particles studied in \\cite{BBRW2017} in disguise. First, set the number of species to $m=2$ and define the number densities\n\\begin{align}\\label{number_densitiesb}\nr({\\bf x},t) = N f_1({\\bf x},t), \\qquad b({\\bf x},t) = N f_2({\\bf x},t),\n\\end{align}\nwhich integrate to $N_1$ and $N_2$ respectively. Then define the potentials $V_r = - (v_0\/D_T) {\\bf e}(\\theta_1) \\cdot {\\bf x}$ and $V_b = - (v_0\/D_T) {\\bf e}(\\theta_2) \\cdot {\\bf x}$. In terms of these new quantities, system \\eqref{cross-diff_num_den} becomes\n\\begin{subequations}\\label{e:aa_cross_sys}\n\\begin{align}\n\\partial_t r &= D_T \\nabla \\cdot \\left[ (1+ 2\\varphi r - \\varphi b) \\nabla r + 3\\varphi r \\nabla b + r \\nabla V_r + \\varphi r b \\nabla (V_b - V_r) \\right],\\\\\n\\partial_t b &= D_T \\nabla \\cdot \\left[ (1+2\\varphi b - \\varphi r) \\nabla b + 3\\varphi b \\nabla r + b \\nabla V_b + \\varphi r b \\nabla (V_r - V_b) \\right],\n\\end{align}\n\\end{subequations}\nwhere $\\varphi = \\epsilon^2 \\pi\/2$.\nThis is exactly the cross-diffusion system for particles of the same size and diffusivity studied in \\cite{BBRW2017} for $d=2$ (see Eqs. (11) in \\cite{BBRW2017}).\\footnote{We note a typo in \\cite{BBRW2017}: the coefficient $\\beta$ below system (11) should have read $\\beta = (2d-1)\\gamma$.} \n\n\\subsection{Discrete models}\\label{sec:discrete_passive}\n\n\nIn this category there are discrete processes in space without changes in orientations. The most well-known model in the context of excluded-volume interactions is ASEP, which was used above in combination of either continuous change in angle, see \\eqref{model_hy}, or discrete jumps, see \\eqref{ASEP_2D}. We obtain the corresponding passive process by either setting $D_R$ or $\\lambda$ to zero, respectively. The resulting model in either case is\n\\begin{align} \\label{model3_passive}\n\\partial_t f_k + v_0\\nabla \\cdot [ f_k (1- \\phi \\rho) {\\bf e}(\\theta_k)] &= D_T \\nabla \\cdot [(1- \\phi \\rho) \\nabla f_k + \\phi f_k \\nabla \\rho], \\qquad k = 1, \\dots, m,\n\\end{align}\nwhere $f_k$ satisfy \\eqref{normalisation_passive} as before, and $\\phi = N \\epsilon^2$. We notice three differences with its continuous passive counterpart \\eqref{cross-diff_num_den}: in the latter, the effective occupied fraction $\\phi$ has a factor of $\\pi\/2$, the coefficient in the cross-diffusion term $f_k \\rho$ has a factor in three, and the transport term has an additional nonlinearity that depends on the polarisation.\nThe cross-diffusion system \\eqref{model3_passive} was derived in \\cite{Simpson:2009gi} and analysed in \\cite{Burger:2010gb} for two species ($m=2$). Specifically, if we introduce the number densities $r, b$ and general potentials $V_r, V_b$ as above, it reads\n\\begin{subequations}\n\\label{eq:MF_cross_diff}\n\\begin{align}\n\\partial_t r &= D_T \\nabla \\cdot \\left[(1-\\bar \\rho) \\nabla r + r \\nabla \\bar \\rho + r (1- \\bar \\rho) \\nabla V_r \\right]\\\\\n\\partial_t b &= D_T \\nabla \\cdot \\left[(1-\\bar \\rho) \\nabla b + b \\nabla \\bar \\rho + b (1-\\bar \\rho) \\nabla V_b \\right],\n\\end{align}\n\\end{subequations}\nwhere $\\bar \\rho = \\epsilon^2 (r+b) = \\epsilon^2 (N_1 f_1 + N_2 f_2)$ (compare with (3.7)-(3.8) in \\cite{Burger:2010gb}).\\footnote{In the system (3.7)-(3.8) of \\cite{Burger:2010gb}, $r$ and $b$ are volume concentrations, thus having a factor of $\\epsilon^2$ compared to those used in \\eqref{eq:MF_cross_diff}, and the diffusivities of the two species are $1$ and $D$ instead of $D_T$ for both.} \n\n\\section{General model structure}\\label{sec:gen_structure}\n\nWe now put the models presented in the previous sections into a more general picture. We assume that $f = f({\\bf x}, \\theta, t)$, where $\\theta$ is a continuous variable taking values in $[0, 2\\pi)$ or a discrete variable taking values $\\theta_k$ for $k = 1, \\dots, m$ (ordered increasingly on $[0,2\\pi)$). In the latter case we shall also use the notation $f_k({\\bf x},t) = f({\\bf x},\\theta_k,t).$ We also recall the definition of the space density $\\rho$ and the polarisation $\\mathbf{p}$:\n \\begin{align*}\n \\rho({\\bf x},t) = \\int_0^{2\\pi} f({\\bf x}, \\theta, t ) \\, \\mathrm{d} \\mu(\\theta) \\quad \\text{and} \\quad \\mathbf{p}({\\bf x},t) = \\int_0^{2\\pi} {\\bf e}(\\theta) f({\\bf x}, \\theta, t) \\, \\mathrm{d} \\mu(\\theta),\n \\end{align*}\n where the integral in $\\theta$ is either with respect to the Lebesgue measure for continuum angles or with respect to a discrete measure (a finite sum) for discrete angles.\n \nThe models presented have the following general model structure:\n\\begin{equation}\n \\label{eq:genform}\n \\partial_t f + v_0 \\nabla \\cdot ( f ((1-\\phi \\rho) {\\bf e}(\\theta) + a\\phi \\mathbf{p} f)) = \n D_T \\nabla \\cdot \\left( \\mathcal{B}_1(\\rho ) \\nabla f + \\mathcal{B}_2(f) \\nabla \\rho \\right) + D_R \\Delta_\\theta f.\n\\end{equation}\nwith $a \\in \\lbrace 0,1 \\rbrace$. In \\eqref{eq:genform} the derivative operator $\\nabla$ is the standard gradient with respect to the spatial variable $x$, while the Laplacian $\\Delta_\\theta$ is either the second derivative $\\partial_{\\theta \\theta}$ with respect to $\\theta$ in the continuous case, the second order difference\n$$ \\Delta_\\phi f = {\\cal D}^2f = (f_{k+1} + f_{k-1} - 2f_k) $$\nwith cyclic extension of the index $k$, or the graph Laplacian with uniform weights\n$$ \\Delta_\\phi f = {\\cal D}_G f = \\sum_{j \\neq k} (f_j - f_k). $$ \nLet us mention that similar structures and results hold true for graph Laplacians with other non-negative weights.\nWe provide an overview of the respective differential operators and constants for most of the presented models in Table \\ref{tab:my_label}.\n\n \\setlength{\\tabcolsep}{12pt}\n\\renewcommand{\\arraystretch}{1.2}\n\n \\begin{table}[ht]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n Eq. Nr. & $\\Delta_{\\theta}$ &$a$ & $\\mathcal{B}_1$ & $\\mathcal{B}_2$ & $D_R$ \\\\\n \\hline\n \\eqref{model2} & $\\partial_{\\theta \\theta}$ & $0 $ & 1 & 0 & $> 0$ \\\\\n \\hline\n \\eqref{model3} & $\\partial_{\\theta \\theta}$ & $1$ &$(1-\\phi \\rho)$ & $3 \\phi f$ & $> 0$ \\\\\n \\hline \n \\eqref{lattice2D} & ${\\cal D}^2$ & $0$ & 1 & 0 & $=\\lambda>0$ \\\\\n \\hline \n \\eqref{ASEP_2D} & ${\\cal D}^2$ & $0$ & $(1-\\phi\\rho)$ & $\\phi f$ & $=\\lambda>0$\\\\\n \\hline\n \\eqref{model_hy} & $\\partial_{\\theta \\theta}$ & $0$& $(1-\\phi \\rho)$& $\\phi f$ & $ > 0$ \\\\\n \\hline \n \\eqref{model_wilson} & ${\\cal D}^2$ & $1$& $(1-\\phi \\rho)$ & $3 \\phi f$ & $> 0$\\\\\n \\hline \n \\eqref{hybrid3} & ${\\cal D}_G$ & $1$& $(1-\\phi \\rho)$ & $3 \\phi f$ & $> 0$\\\\\n \\hline\n \\eqref{eq:MF_cross_diff_sys} & & $0$ & $(1-\\phi \\rho)$ & $3\\phi f$ & 0\\\\\n \\hline\n \\eqref{model3_passive} & & $0$ & $(1-\\phi \\rho)$ & $\\phi f$ & 0 \\\\\n \\hline\n \\end{tabular}\n \\caption{Table recasting most models in the general form of \\eqref{eq:genform}.}\n \\label{tab:my_label}\n \\end{table}\n \n\\paragraph{Small and large speed}\n \nNatural scaling limits for the general system \\eqref{eq:genform} are the ones for large and small speed, i.e. $v_0 \\rightarrow 0$ and $v_0 \\rightarrow \\infty$, respectively. The first case is rather obvious, since at $v_0=0$ the model is purely diffusive, i.e. \n$$ \\partial_t f = \n D_T \\nabla \\cdot ( \\mathcal{B}_1(f,\\rho ) \\nabla f + \\mathcal{B}_2(f,\\rho) \\nabla \\rho) + D_R \\Delta_\\theta f.$$\nThe model can then be written as a gradient flow structure (respectively a generalised gradient structure in the case of discrete angles, see for example \\cite{M2011,PRST2020}) for an entropy of the form \n \\begin{equation}\n{\\cal E}(f) = \\int \\int f \\log f ~\\mathrm{d} {\\bf x} ~\\mathrm{d} \\theta + \nc \\int (1-\\rho) \\log (1 -\\rho) ~\\mathrm{d} {\\bf x},\n\\end{equation}\nwith $c \\in \\{0,1,3\\}$ corresponding to the coefficients of ${\\cal B}_2$.\nIn the case of small $v_0$ the gradient flow structure is broken, but we still expect the diffusive part to dominate. In particular we expect long-time convergence to a unique stationary solution. \n\nIn the case $v_0 \\rightarrow \\infty$ there are two relevant time scales. At a small time scale, i.e. $\\tau = \\frac{L}{v_0}$, where $L$ is a typical length scale, the evolution is governed by the first order equation\n$$\\partial_t f + \\nabla \\cdot ( f ((1-\\phi \\rho) {\\bf e}(\\theta) + a\\phi \\mathbf{p} f)) = 0 .$$\nThe divergence of the corresponding velocity field \n$u = ((1-\\phi \\rho) {\\bf e}(\\theta) + a\\phi \\mathbf{p}$ is given by\n$$ \\nabla \\cdot u = - \\phi \\nabla \\rho \\cdot {\\bf e}(\\theta) + a \\phi \\nabla \\cdot \\mathbf{p}. $$\nIn particular in the case of $a=0$ we see that the question of expansion or compression of the velocity field is determined by the angle between $\\nabla \\rho$ and the unit vector ${\\bf e}(\\theta)$. Unless $\\nabla \\rho = 0$, the velocity field is compressible for a part of the directions and expansive for the opposite directions. A consequence to be expected is the appearance of patterns with almost piecewise constant densities. Inside the structures with constant densities ($\\nabla \\rho = 0$) the velocity field is incompressible, while the compression or expansion arises at the boundaries of such regions. This is rather described by a large time scale, i.e. the equation without time rescaling. Then one expects a slow interface motion, which is also observed in numerical simulations. In a simple case with only one direction this has been made precise in \\cite{burger2008asymptotic}.\n \n\\paragraph{Small and large rotational diffusion}\n\nThe limit of small rotational diffusion $D_R=0$ corresponds to a more standard nonlinear Fokker-Planck system with a given linear potential. \n$$ \\partial_t f + v_0 \\nabla \\cdot ( f ((1-\\phi \\rho) {\\bf e}(\\theta) + a\\phi \\mathbf{p})) = \n D_T \\nabla \\cdot ( \\mathcal{B}_1(f,\\rho ) \\nabla f + \\mathcal{B}_2(f,\\rho) \\nabla \\rho). $$\nModels of this kind have been investigated previously, see for example \\cite{Burger:2010gb, BHRW2016}. They tend to develop patterns such as jams or lanes, depending on the initial condition. This happens in particular for large speeds $v_0$.\n\nThe case of large rotational diffusion $D_R \\rightarrow \\infty$ will formally lead to $f$ being constant with respect to $\\theta$ at leading order. The corresponding equation at leading order can thus be obtained by averaging \\eqref{eq:genform} in $\\theta$. Since $f$ does not depend on $\\theta$ the polarisation is zero, that is\n$$ \\mathbf{p} = \\int_0^{2\\pi} f {\\bf e}(\\theta) ~ \\mathrm{d} \\theta = 0, $$\nand the transport term drops out in all the models. Indeed the nonlinear diffusion terms in any case average to linear diffusion with respect to $x$. Hence, the evolution of $f$ at leading order is governed by a linear diffusion equation. \n\n\n\n\\subsection{Wasserstein gradient flows}\n\nWe have seen in the previous subsection that several models for externally activated particle have an underlying gradient flow structure, which should ideally be maintained in the mean field limit. Adams et al. \\cite{ADPZ2011} showed in their seminal work that then the Wasserstein metric arises naturally in the mean-field limit (under suitable scaling assumptions). However, this limit is only well understood in a few cases (for example for point particles) and rigorous results are often missing. In case of excluded-volume effects, as discussed in subsections \\ref{sec:evi} and \\ref{sec:hybrid}, the only known rigorous continuum models are derived in 1D \\cite{Rost1984, BV2005, Gavish:2019tu}, with only approximate models for higher space dimension. We will see that these approximate mean-field limits often lack a full gradient flow structure, but are sufficiently close to it. In the following we give a brief overview on how Wasserstein gradient flows and energy dissipation provides useful a-priori estimates that can be used in existence proofs or when studying the long time behaviour of solutions. These techniques are particularly useful for systems with cross diffusion terms, for which standard existence results do not necessarily hold.\n\nWe will briefly outline the main ideas for functions $f = f({\\bf x}, \\theta, t)$ where $\\theta$ is either continuous or taking discrete values $\\theta_k$ with $k = 1, \\ldots m$. We say that a mean-field model has a Wasserstein gradient flow structure if it can be written as\n \\begin{align}\n \\label{e:w2sys}\n \\partial_t f({\\bf x}, \\theta, t) &=\n \\nabla \\cdot \\left( \\mathcal{M}(f) \\nabla_{{\\bf x}, \\theta}\n \\partial_f \\mathcal{E}\n \\right),\n \\end{align}\n where $\\mathcal{M}$ is the mobility operator and $w = \\partial_f \\mathcal{E}$ the variational derivatives of an entropy\/energy functional $\\mathcal{E}$ with respect to $f$. Note that for discrete $\\theta_k$, $k=1, \\ldots m$ the mobility $\\mathcal{M}$ is a positive definite matrix in $\\mathbb{R}^{m \\times m}$ and $\\partial_f \\mathcal{E}$ is replaced by the vector $\\partial_{f_k} \\mathcal{E}$. We have seen possible candidates for energies in the previous subsection - the usually comprise negative log entropy terms of the particle distribution and the total density (corresponding to linear and non-linear diffusion) as well as potentials (relating to the operators $\\mathcal{B}_1$ and $\\mathcal{B}_2$).\n \nIf the system has a Wasserstein gradient flow structure \\eqref{e:w2sys} then the entropy $\\mathcal{E}$ changes in time as\n\\begin{align}\\label{e:entropydiss}\n \\frac{d \\mathcal{E}}{dt} = \\int_{\\Omega} \\partial_t f \\cdot w \\, \\mathrm{d} {\\bf x} = -\\int_{\\Omega} \\bar{\\mathcal{M}}(w) \\lvert \\nabla w \\rvert^2 \\, \\mathrm{d} {\\bf x},\n \\end{align}\n where $\\bar{\\mathcal{M}}$ is the transformed mobility matrix. If $\\bar{\\mathcal{M}}$ is positive definite, then the energy is dissipated. In the next subsection we will define an entropy for the general model \\eqref{eq:genform} and show that the system is dissipative for several of the operator choices listed in Table \\ref{tab:my_label}. \n\nNote that these entropy dissipation arguments are mostly restricted to unbounded domains and bounded domains with no-flux or Dirichlet boundary conditions. It is possible to generalise them in the case of non-equilibrium boundary conditions, as such discussed in Section \\ref{s:bc}, but a general theory is not available yet. We will see in the next subsection that entropy dissipation may also hold for systems, which do not have a full gradient flow structure. \n \nSince system \\eqref{e:w2sys} is dissipative, we expect long time convergence to an equilibrium solution. The respective equilibrium solutions $f_\\infty$ to \\eqref{e:w2sys} then correspond to minimisers of the entropy $\\mathcal{E}$. To show exponential convergence towards equilibrium it is often helpful to study the evolution of the so-called relative entropy, that is\n\\begin{align*}\n \\mathcal{E}(f, f_{\\infty}) := \\mathcal{E}(f) - \\mathcal{E}(f_{\\infty}).\n\\end{align*}\nIn general one wishes the establish so-called entropy-entropy dissipation inequalities for the relative entropy\n\\begin{align*}\n \\frac{d\\mathcal{E}}{dt} \\leq -c \\mathcal{E},\n\\end{align*}\nwith $c>0$. Then Gronwall's lemma gives desired exponential convergence. This approach is also known as the Bakry-Emery method, see \\cite{BE1985}.\n\n We discussed the challenges in the rigorous derivation of mean-field models in the previous sections and how often only formal or approximate limiting results are available. These approximate mean-field models are often 'close' to a full gradient flow, meaning that they only differ by higher order terms (which were neglected in the approximation). This closeness motivated the definition of so-called asymptotic gradient flow, see \\cite{BBRW2017}. A dynamical system of the form \n \\begin{align}\\label{e:agf}\n \\partial_t z = \\mathcal{F}(z; \\epsilon)\n \\end{align}\n has a an asymptotic gradient flow structure of order $k$ if \n \\begin{align*}\n \\mathcal{F}(z; \\epsilon) + \\sum_{j=k+1}^{2k} \\epsilon^j \\mathcal{G}_j(z) = -\\mathcal{M}(z; \\epsilon) \\mathcal{E}'(z, \\epsilon),\n \\end{align*}\n for some parametric energy functional $\\mathcal{E}.$ For example, \\eqref{eq:MF_cross_diff_sys} exhibits a GF structure if the red and blue particles have the same size and diffusivity, but lacks it for differently sized particles (a variation of the model not discussed here). The closeness of AGF to GF can be used to study for example its stationary solutions and the behaviour of solution close to equilibrium, see \\cite{ABC2018, ARSW2020, BBRW2017}.\n \n\\subsection{Entropy dissipation}\\label{sec:entropydiss}\n\nNext we investigate the (approximate) dissipation of an appropriate energy for the general formulation \\eqref{eq:genform}. The considered energy functional is motivated by the entropies of the scaling limits considered before. In particular we consider\n \\begin{equation}\n{\\cal E}(f) = \\int \\int f \\log f + V({\\bf x},\\theta) f ~\\mathrm{d} {\\bf x}~\\mathrm{d} \\mu(\\theta) + \nc \\int (1-\\rho) \\log (1 -\\rho) ~\\mathrm{d} {\\bf x},\n\\end{equation}\nfor which the models can be formulated as gradient flows in the case $D_R=0$ with $c \\in \\{0,1,3\\}$ chosen appropriately. For simplicity we set $\\phi = 1$ as well as $D_T=1$ in the following in order to shorten the computations. As before, we interpret integrals in $\\theta$ with respect to the Lebesgue measure for continuum angles and with respect to the discrete measure (sum) in case of a finite number of directions. We recall that the potential $V$ is given by \n$$\nV({\\bf x}, \\theta) = - v_0 \\, {\\bf e}(\\theta) \\cdot {\\bf x} = - v_0\\, (\\cos \\theta_k x + \\sin \\theta_k y).\n$$\nIn the following we provide a formal computation assuming sufficient regularity of all solutions. We have\n\\begin{align*}\n \\frac{d \\mathcal{E}}{dt} &= \\int \\int \\partial_t f ( \\log f + V - c \\log(1-\\rho)) ~\\mathrm{d} {\\bf x}~\\mathrm{d} \\theta \\\\\n &= - \\int \\int \\nabla ( \\log f + V - c\\log(1-\\rho)) \n (- v_0 ( f ((1-\\rho) {\\bf e}(\\theta) + a \\mathbf{p} f)) + \n ( \\mathcal{B}_1(f,\\rho ) \\nabla f + \\mathcal{B}_2(f,\\rho) \\nabla \\rho)) \n ~\\mathrm{d} {\\bf x}~\\mathrm{d} \\theta\\\\\n & \\phantom{=} + D_R \\int \\int ( \\log f + V -c \\log(1-\\rho)) \\Delta_\\theta f ~\\mathrm{d} {\\bf x}~\\mathrm{d} \\theta\n\\end{align*}\n\nLet us first investigate the last term.\nSince $\\rho$ is independent of $\\theta$, we have using the properties of the generalised Laplacian $\\Delta_\\theta$ with periodic boundary conditions\n$$ \\int \\log(1-\\rho) \\Delta_\\theta f~d\\theta = \n\\log(1-\\rho) \\int \\Delta_\\theta f~d\\theta = 0.$$\nUsing the fact that $\\Delta_\\theta {\\bf e}(\\theta)$ is uniformly bounded in all cases, we find\n\\begin{align*}\n \\int \\int ( \\log f + V - c\\log(1-\\rho)) \\Delta_\\theta f ~d{\\bf x}~d\\theta &= \n - \\int \\int {\\cal F}_\\theta(f) - v_0 \\Delta_\\theta {\\bf e}(\\theta) {\\bf x} f ~d{\\bf x}~d\\theta, \\\\\n &\\leq C |v_0| \\int |x| f~d{\\bf x}~d\\theta = C |v_0| \\int |x| \\rho~d{\\bf x},\n\\end{align*}\nwhere ${\\cal F}_\\theta(f) \\geq 0$ is the Fisher information with respect to the generalised Laplacian $\\Delta_\\theta$ \n$$ {\\cal F}_\\theta(f) = \\left\\{ \\begin{array}{ll} \\frac{|\\partial_\\theta f|^2}f & \\text{for }\\partial_{\\theta \\theta} \\\\\n\\frac{|f_{k+1}-f_k|^2}{M(f_k,f_{k+1})}& \\text{for } {\\cal D}^2 \\\\\n\\sum_j \\frac{|f_{j}-f_k|^2}{M(f_j,f_k)} & \\text{for } {\\cal D}_G\n\\end{array} \\right. , $$\nwhere \n$$M(f,g) = \\frac{f-g}{\\log(f) - \\log(g)}$$\nis the logarithmic mean. \n\nNow we further investigate the first term for the models with $a =0$, where for appropriate choice of $c$ we can achieve\n\\begin{align*}\n\\int \\int \\nabla ( \\log f + V - c\\log(1-\\rho)) (\n v_0 f (1- \\rho) {\\bf e}(\\theta) - & ( \\mathcal{B}_1(f,\\rho ) \\nabla f + \\mathcal{B}_2(f,\\rho) \\nabla \\rho) )~d{\\bf x}~d\\theta \\\\\n &= - \\int \\int f(1-\\rho) |\\nabla ( \\log f + V - c\\log(1-\\rho)) |^2~d{\\bf x}~d\\theta \\leq 0.\n\\end{align*}\nOverall we finally find\n$$ \\frac{d \\mathcal{E}}{dt} \\leq C ~|v_0|~\\int |{\\bf x}| \\rho~d{\\bf x} \\leq C ~|v_0|~ \\sqrt{\\int |{\\bf x}|^2 \\rho~d{\\bf x}} . $$\nThus, the growth of the entropy in time is limited by the second moment. Note that for $a=1$ one can employ analogous reasoning to obtain the above negative term. However it is unclear how to control the additional term \n$\\int \\int \\nabla ( \\log f + V - c\\log(1-\\rho)) \n v_0 {\\bf p} f ~d{\\bf x}~d\\theta$.\\\\\n The obtained bounds provide useful a-priori estimates, which can be used in existence results and to study the long-time behaviour, see for example \\cite{BSW2012, J2015}.\n\n\n\n\\section{Boundary effects}\\label{s:bc}\n\n\n\nSo far, we have concentrated on the dynamics on domains with periodic boundary conditions, thus neglecting its effects entirely. In the following we will outline how boundaries as well as inflow and outflow conditions can be included in all models on the micro- as well as macroscopic level. We remark that the continuous random walks models are difficult to treat on bounded domains. Thus we only mention a few aspects and comment in more detail on the time-discrete situation which is easier to tackle, see remark \\ref{rem:time_discrete}.\n\n\n\\subsubsection*{Mass conserving boundary conditions} We first discuss conditions that conserve the total mass, i.e. the number of particles for discrete models, or the integral of the density for continuous ones, in a given domain.\nIn case of the coupled SDE model \\eqref{sde_model}, we are interested in conditions that ensure that particles remain inside the domain. Intuitively, particles need to be reflected whenever they hit the boundary. However, as we are dealing with a problem that is continuous in time, we have to ensure that the particle path remains continuous. In his seminal paper \\cite{Skorokhod1961_bounded} Skorokhod solved this problem by introducing an additional process that increases whenever the original process hits the boundary, see \\cite{Pilipenko2014_reflection} for a detailed discussion. \nFor the microscopic models on a lattice, such boundary conditions \ncorrespond to allowing only jumps into domain whenever a particle is located at the boundary. \nFor the macroscopic models, mass conservation corresponds to no-flux boundary conditions that are implemented by setting the normal flux over the boundary to zero, i.e. \n\\begin{align}\\label{eq:noflux}\n{\\bf J} \\cdot {\\bf n} = 0 \\text{ a.e. in } \\Upsilon \\times (0,T),\n\\end{align}\nwhere, using the general form \\eqref{eq:genform}, the flux density is given as \n\\begin{align}\\label{eq:J_general}\n {\\bf J} =v_0 ( f ((1-\\phi \\rho) {\\bf e}(\\theta) + a\\phi \\mathbf{p})) - D_T ( \\mathcal{B}_1(\\rho ) \\nabla f + \\mathcal{B}_2(f) \\nabla \\rho).\n\\end{align}\n\n\n\\subsubsection*{Flux boundary conditions}\nApart from periodic or no-flux boundary conditions, there is also the possibility for boundary conditions that allow for the in- or outflow of particles (mass) via the boundary. Such effects are of particular interest in the context of this chapter, since they yield an active system even if the motion of the particles within the domain is purely passive (i.e. due to diffusion). \n\nFor the SDE model \\eqref{sde_model}, such boundary conditions correspond to partially reflecting or radiation conditions.\nIntuitively, once a particle reaches the boundary it is, with a certain probability, either removed or otherwise reflected, see \\cite{grebenkov2006partially} and \\cite[Section 4]{Lions1984}.\nFor the discrete models of section \\ref{sec:discrete} let us consider first the special case of a single species which yields the setting of a asymmetric simple exclusion process (ASEP) with open boundary conditions, the paradigmatic models in non-equilibrium thermodynamics, \\cite{chou2011_nonequilibrium}. \nThe dynamics of such a process is well understood and can be solved explicitly, \\cite{Derrida1993:TASEP,derrida1998_exactly} (see also \\cite{Wood2009:TASEP_boundary}). \nWe denote by $\\alpha$ and $\\beta$ the rates by which particles enter (at the left boundary) or exit (at the right boundary) the lattice. Then, the key observation here is that in the steady state, system can be in one of three distinct states, characterised by the value of the one-dimensional current and the density as follows\n\\begin{itemize}\n \\item \\emph{low density} or \\emph{influx limited} meaning the density takes the value $\\alpha$ and the flux $\\alpha(1-\\alpha)$; occurs whenever $\\alpha < \\min\\{\\beta, 1\/2\\}$\n \\item \\emph{high density} or \\emph{outflux limited} meaning density $1-\\beta$ and flux $\\beta(1-\\beta)$; occurs whenever $\\beta < \\min\\{\\alpha, 1\/2\\}$\n \\item \\emph{maximal density} or \\emph{maximal current} if the density is $1\/2$ and the flux $1\/4$; occurs whenever $\\alpha > 1\/2$ and $\\beta > 1\/2$.\n \\end{itemize}\n \nA similar behaviour can be verified for the macroscopic, on-lattice model \\eqref{eq:MF_cross_diff} (or also \\eqref{model_lattice1D_number_densities} with $\\lambda =0$) for a single species on the domain $\\Omega = [0,L]$, which reduces to a single equation for the unknown density $r$, i.e.\n\\begin{align*}\n \\partial_t r + \\partial_x j = 0 \\text{ with } j = -D_T \\partial_x r + r(1-r) \\partial_x V.\n\\end{align*}\nWe supplement the equation with the flux boundary conditions \n\\begin{align}\\label{eq:inoutflux}\n -j \\cdot n = \\alpha (1- r) \\text{ at } x = 0 \\text{ and } j \\cdot n = \\beta r \\text{ at } x = L,\n\\end{align}\nsee \\cite{Burger2016}. Indeed, one can show that for positive $D_T> 0$, stationary solutions are close to one of the regimes and as $D_T \\to 0$, they obtain the exact values for flux and density. Interestingly, for positive $D_T$ it is possible to enter the maximal current regime for values of $\\alpha$ and $\\beta$ strictly less than $1\/2$. The long time behaviour of these equations, using entropy--entropy-dissipation inequalities, has been studied in \\cite{Burger2016}.\n\nFor the kinetic models \\eqref{model3} and \\eqref{model_hy}, a similar condition can be formulated for the unknown quantity $f$. However, as $f$ depends not only on $\\bf{x}$ and $t$ but also on the angle $\\theta$, the coefficients may also depend on it. In the most general situation we obtain\n\\begin{align}\\label{eq:flux_bc}\n {\\bf J} \\cdot {\\bf n} =- \\alpha(\\theta, {\\bf n}) {(1- \\phi\\rho)} + \\beta(\\theta, {\\bf n})f,\n\\end{align}\nwith ${\\bf J}$ defined in \\eqref{eq:J_general}.\nHere, the choice of the functions $\\alpha$ and $\\beta$ is subject to modelling assumptions or properties of microscopic stochastic models for the in- and outflow. Typically one has a separation into inflow- and outflow regions, which means that $\\alpha$ is supported on inward pointing directions ${\\bf e}(\\theta \\cdot n) > 0$, while $\\beta$ is supported outward pointing directions ${\\bf e}(\\theta \\cdot n) > 0$.\n\n\n\n\\subsubsection*{Other boundary conditions}\nLet us also discuss other types of boundary conditions. Homogeneous Dirichlet boundary conditions can be applied to all types of models: for the SDE \\eqref{sde_model}, one has to remove a particle once it reaches the boundary. The same holds for the discrete random walk models. For the macroscopic models, one sets the trace at the boundary to zero. Finally, also mixed boundary conditions are possible, combining the effects described above on different parts of the boundary.\n\n\\begin{remark}[Boundary conditions for discrete time random walks] \\label{rem:time_discrete}\nWe briefly comment on the situation for time-discrete random walks, that is when the SDE \\eqref{sde_model} is replaced by the time-discrete system\n\\begin{subequations}\n\t\\label{sde_model_discrete}\n\\begin{align}\n\\label{sde_x_discrete}\n\t{\\bf X}_i(t+\\Delta t) &= {\\bf X}_i(t) + \\Delta t\\sqrt{2 D_T} \\zeta_i - \\Delta t \\nabla_{{\\bf x}_i} U + \\Delta t v_0 {\\bf e}(\\Theta_i),\\\\\n\t\\label{sde_angle_discrete}\n\t\\Theta_i(t + \\Delta t) &= \\Theta_i(t) + \\Delta t\\sqrt{2 D_R} \\bar \\zeta_i - \\Delta t\\partial_{\\theta_i} U,\n\\end{align}\n\\end{subequations}\nfor some time step size $\\Delta t > 0$ and where $\\zeta_i, \\, \\bar \\zeta_i$ are normally distributed random variables with zero mean and unit variance. To implement boundary conditions, one has to calculate the probability that ${\\bf{X}}_i(t + \\Delta t) \\notin \\Omega$ (considering also the case that the particles leaves the domain but moves back into it within the time interval $[t, t+\\Delta t]$), see \\cite{Andrews2004_time_discrete} for detailed calculations in the case of pure diffusion. If a particle is found to have left the domain, it can either be removed with probability one (corresponding to homogeneous Dirichlet boundary conditions) or less than one, called a partially reflective boundary condition (corresponding to Robin boundary conditions). In our setting, this probability can depend on the current angle of the particle, $\\Theta_i(t)$, allowing for additional modelling. \nIt is also possible to add a reservoir of particles at the boundary to implement flux boundary conditions in the spirit of \\eqref{eq:flux_bc} by prescribing a probability to enter the domain. In the case of excluded volume, the probability to enter will depend on the number of particles close to the entrance.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\\section{Active crowds in the life and social science}\\label{sec:applications}\n\n\\subsection{Pedestrian dynamics} \\label{sec:pedestrian}\n\nA prominent example of active and externally activated dynamics in the context of socio-economic applications is the motion of large pedestrian crowds. There is an extensive literature on mathematical modelling for pedestrians in the physics and the transportation community, which is beyond the scope of this paper. We will therefore review the relevant models in the context of active crowds only and refer for a more comprehensive overview to \\cite{CPT2014, MF2018}. \n\n\\paragraph{Microscopic models for pedestrian flows}\nMicroscopic off-lattice models are the most popular approach in the engineering and transportation research literature. Most software packages and simulations are based on the so called social force model by Helbing \\cite{Helbing1995:social,Helbing2000:social}. The social force model is a second order SDE model, which does not take the form of active models considered here. However, it is easy to formulate models for pedestrians in the context of active particles satisfying \\eqref{sde_model}. For example, assume that all pedestrians move with the same constant speed in a desired direction $\\Theta_d$ avoiding collisions with others. Then their dynamics can be described by the following second order system:\n\\begin{subequations} \n\\label{e:activepedestrians}\n\\begin{align}\n \\mathrm{d} {\\bf X}_i &= -\\nabla_{{\\bf X}_i} U \\mathrm{d} t + v_0 \\frac{e(\\Theta_i)- \\Theta_d}{\\tau}dt + \\sqrt{2 D_T}\\, \\mathrm{d} {\\bf W}_i\\\\\n \\mathrm{d} \\Theta_i &= -\\partial_{\\Theta_i} U \\mathrm{d} t + \\sqrt{2D_R}\\,\\mathrm{d}{\\bf W}_i.\n\\end{align}\n\\end{subequations}\nThe potential $U$ takes the form \\eqref{e:U}, where the pairwise interactions $u$ should be related to the likelihood of a collision. One could for example consider\n\\begin{align*}\n u(\\lvert {\\bf X}_i - {\\bf X}_j \\rvert\/\\ell, \\Theta_i - \\Theta_j) = C \\frac{\\Theta_i - \\Theta_j}{\\lvert {\\bf X}_i - {\\bf X}_j\\rvert},\n\\end{align*}\nwhere $C \\in \\mathbb{R}^+$ and $\\ell$ relates to the personal comfort zone. Another possibility corresponds to a Lennard Jones type potential to model short range repulsion and long range attraction. \nAnother popular microscopic approach are so-called cellular automata, which correspond to the discrete active and externally activated models discussed before. In cellular automata a certain number of pedestrians can occupy discrete lattice sites and individuals move to available (not fully occupied) neighbouring sites, with transition rates. These transition rates may depend on given potentials, as discussed in the previous sections, which relate to the preferred direction.\n\nThere is also a large class of microscopic on-lattice models, so called cellular automata, see \\cite{Kirchner2002:Cellular}, which relate to the microscopic discussed in Section \\ref{sec:discrete_passive}. In cellular automata pedestrians move to neighbouring sites at given rates, if these sites are not already occupied. Their rates often depend on an external given potential, which relates to the desired direction $\\Theta_d$. Cellular automata often serve as the basis for the macroscopic pedestrian models, which will be discussed in the next paragraph, see for example \\cite{BMP2011, Burger2016:sidestepping}. \n\n\\paragraph{Macroscopic models for pedestrian flows}\nMean field models derived from microscopic off-lattice approaches have been used successfully to analyse the formation of directional lanes or aggregates in bi-directional pedestrian flows. This segregation behaviour has been observed in many experimental and real-life situations. Several models, which fall into the category of externally activated particles introduced in section \\ref{sec:discrete_passive}, were proposed and investigated in this context. These models take the form \\eqref{eq:MF_cross_diff}, in which the densities $r$ and $b$ relate to different directions of motion. For example in the case of bi-directional flows in a straight corridor 'red particles' correspond to individuals moving to the right, while blue ones move to the left. We will see in section \\ref{sec:numerics} that we can observe temporal as well as stationary segregated states. Depending on the initial and inflow conditions directional lanes or jams occur. Then the gradient flow structure can then be used to investigate the stability of stationary states using for example the respective entropy functionals. Due to the segregated structure of stationary solutions, one can also use linear stability analysis around constant steady states to understand for example the formation of lanes, see \\cite{Pietschmann2011:lane}.\n\nMore pronounced segregated states and lanes can be observed when allowing for side-stepping. In the respective microscopic on lattice models, individuals step aside when approached by a member of the other species. The respective formally derived mean-field model has a perturbed gradient flow structure, which can be used to show existence of solutions, see \\cite{Burger2016:sidestepping}. \nMore recently, a model containing both and active and a passive species has been introduced, \\cite{Cirillo2020:ActivePassive}\n\n\n\\subsection{Transport in biological systems}\\label{sec:biological_transport}\nAnother example where active particles play an important role are transport process in biological systems. We will discuss two important types of such processes in the following - chemotaxis and transport in neurons. \n\n\\subsubsection*{Chemotaxis}\n\nWe consider bacteria in a given domain that aim to move along the gradient of a given chemical substance, called chemo-attractant and modelled by a function $c:\\Omega \\to \\mathbb{R}_+$. Due to their size, bacteria cannot sense a gradient by, say, comparing the value of $c$ at their head with that at their tail. \nThus, they use a different mechanism based on comparing values of $c$ at different time instances and different points in space, called run-and-tumble. In a first step, they perform a directed motion into a fixed direction (run), then rotate randomly (tumble). These two steps are repeated, however, the probability of tumbling depends on $c$ as follows: If the value of $c$ is decreasing in time, bacteria tumble more frequently as they are not moving up the gradient. If the value of $c$ increased, they turn less often. \nRoughly speaking, this mechanism reduces the amount of diffusion depending on the gradient of $c$. Here, we consider a slightly different idea that fits into the hybrid random walk model introduced in \\eqref{hybrid2}, assuming $D_T$ to be small (run) and the rate of change for the angle depends on $c$. To this end $\\lambda$ is taken different for each angle (thus denoted by $\\lambda_k$) and is assumed to depend on the difference of the external signal $c$ at the current and past positions, only. Denoting by $t_k$, $k=1,2, \\ldots$ the times at which the angle changes, at time $t_n$ we have $\\lambda_k = \\lambda_k((c(\\mathbf{X}_i(t_n))-c(\\mathbf{X}_i(t_{n-1}))$.\nAdditionally, we introduce a fixed base-line turning frequency $\\bar\\lambda$, and consider\n$$\n\\lambda_k = \\bar \\lambda + (c(\\mathbf{X}_k(t_{n-1})) - c(\\mathbf{X}_k(t_n))),\n$$ \nNow going from discrete to time-continuous jumps, i.e. $t_{n} - t_{n-1} \\to 0$, and appropriate rescaling, we obtain via the chain rule\n$$\n\\lambda_k = \\bar \\lambda - \\dot{\\mathbf{X}}_k\\cdot \\nabla c(\\mathbf{X}_i).\n$$\nHowever, due to the stochastic nature of the equation governing the evolution $\\mathbf{X}_k$, its time derivative is not defined. Thus, as a modelling choice, we replace this velocity vector by $v_0 {\\bf e}(\\theta_k)$, i.e. the direction of the active motion of the respective particle. This is also motivated by the fact that for $D_T=0$ and $U=0$ in \\eqref{hybrid_x}, this is exact. We obtain\n$$\n\\lambda_k = \\bar \\lambda - v_0{\\bf e}(\\theta_k)\\cdot \\nabla c(\\mathbf{X}_i).\n$$\nIn the particular case on one spatial dimension with only two possible angles (denoted by $+$ and $-$) and for $v_0=1$ this reduces to\n$$\n\\lambda_\\pm = \\bar\\lambda \\mp \\partial_x c,\n$$\nwhich is exactly the model analysed in \\cite{Ralph:2020cj}. There, it was also shown that using an appropriate parabolic scaling, one can obtain a Chemotaxis-like model with linear transport but non-linear diffusion in the diffusive limit.\n\n\\subsubsection*{Transport in neurons}\nAnother interesting example are transport processes within cells and we focus on the example of vesicles in neurons. Vesicles are small bubbles of cell membrane that are produced in the cell body (soma) and are then transported along extensions of the cell called axons. The transport itself is carried out by motor proteins that move along microtubules and are allowed to change their direction of motion. \nThis situation can be modelled using the discrete random walks from section \\ref{sec:discrete} by considering the one-dimensional case which, in the macroscopic limit, yields equations \\eqref{model_lattice1D}. \nSince we are now dealing with two species $f_-$ and $f_+$, denoting left- and right-moving complexes, we also have to adopt our boundary conditions as follows: Denoting by $j_+$ and $j_-$ the respective fluxes,\n\\begin{align*}\n -j_+ &= \\alpha_+ (1-\\phi\\rho), \\quad j_- = \\beta_- f_- \\qquad\\text{ at } x = 0,\\\\\n -j_- &= \\alpha_- (1-\\phi\\rho), \\quad j_+ = \\beta_+ f_+ \\qquad\\text{ at } x = 1.\n\\end{align*}\nSystem \\eqref{model_lattice1D} has, to the best of our knowledge, not yet been considered with these boundary conditions. From an application point of view, it is relevant to study whether these models are able to reproduce the almost uniform distribution of motor complexes observed in experiments, see \\cite{Bressloff2015_democracy, Bressloff2016_exclusion} for an analysis.\n\nMore recently, the influence of transport in developing neurites has been studied in \\cite{humpert_role_2019} with an emphasis on the mechanism that decides which of the growing neurites becomes an axon. To model this situation, the concentration of vesicles at some and growth cones is modelled separately by ordinary differential equations which are connected to to instances of \\eqref{eq:MF_cross_diff} via flux boundary conditions. \n\n\n\n\n\n\\section{Numerical simulations}\\label{sec:numerics}\n\n\nIn the following, we present numerical examples in one spatial dimension comparing a subset of models presented above. All simulations are based on a finite element discretisation in space (using P$1$ elements). The time discretisation is based on the following implicit-explicit (IMEX) scheme \n\\begin{equation*}\n \\frac{f^{n+1} - f^n}{\\tau} + v_0 \\nabla \\cdot ( f ((1-\\phi \\rho^n) {\\bf e}(\\theta) + a\\phi \\mathbf{p} f)) = \n D_T \\nabla \\cdot ( \\mathcal{B}_1(\\rho^n ) \\nabla f^{n+1} + \\mathcal{B}_2(f^n) \\nabla \\rho^{n+1}) + D_R \\Delta_\\theta f^n,\n\\end{equation*}\nin which the superscript index $n$ refers to the $n$th time step, that is $t^n = n \\tau$, $\\tau>0.$\nHere transport and rotational diffusion are taken explicitly, while in the diffusive part terms of second order are treated implicitly. Thus, in every time step, a linear system has to be solved. All schemes were implemented using the finite element library NgSolve, see \\cite{Sch\u00f6berl1997}. \n\nWe will illustrate the behaviour of solutions for models \\eqref{model_lattice1D_number_densities}, \\eqref{e:aa_cross_sys}, \\eqref{eq:MF_cross_diff}, in case of in- and outflux \\eqref{eq:inoutflux}, no-flux \\eqref{eq:noflux} or periodic boundary conditions in case of two species, referred to as red $r$ and blue $b$ particles. We use subscript $r$ and $b$, when referring to their respective in- and outflow rates as well as diffusion coefficients. Note that while for the models \\eqref{ASEP_2D}, \\eqref{eq:MF_cross_diff}, the one-dimensional setting is meaningful, for model \\eqref{e:aa_cross_sys}, the simulations are to be understood as two-dimensional but with a potential that is constant in the second dimension.\nFor all simulations, we discretised the unit interval into $150$ elements and chose time steps of size $\\tau = 0.01$.\n\\subsubsection*{Flux boundary conditions} Figures \\ref{fig:Regime2} and \\ref{fig:Regime3} show density profiles for the respective models at time $t=0.5,\\, 2,\\, 3, \\, 30$. In figure \\ref{fig:Regime2}, we chose rather low rates (in particular below $1\/2$) and with $\\alpha_r > \\beta_r$ as well as $\\alpha_b < \\beta_b$ which resulted in species $r$ being in a outflux limited and species $b$ an influx limited phase. We observe that for these low rates, all models are quite close to one another, yet with different shapes of the boundary layers. Model \\eqref{model_lattice1D_number_densities}, having a linear diffusion term, showing a different slope as \\eqref{eq:MF_cross_diff} where cross-diffusion seems to play a role an \\eqref{e:aa_cross_sys} being in between.\n\nIn figure \\ref{fig:Regime3} we chose rates above $1\/2$ to obtain the maximal-current phase. There, interestingly, it turns our that the dynamics of model \\eqref{e:aa_cross_sys} shows a completely different behaviour. This constitutes an interesting starting point for further analytical considerations on the phase behaviour. Figure \\ref{fig:mass} displays the evolution of the total mass of the respective species for different in- and outflow rates. We observe that the reaction-diffusion \\eqref{model_lattice1D_number_densities} and the lattice based cross diffusion system \\eqref{eq:MF_cross_diff} show a similar quantitative behaviour in several in- and outflow regimes, while the cross-diffusion system obtained via asymptotic expansion \\eqref{e:aa_cross_sys} behaves only qualitatively similar.\n\\subsubsection*{Periodic boundary conditions}\nFor periodic boundary conditions, noting that the velocity is constant, thus periodic, we expect constant stationary solutions whose value is determined by the initial mass. This is indeed observed in figure \\ref{fig:periodic}. However, for earlier times, their dynamics differs substantially, in particular for \\eqref{eq:MF_cross_diff}, the influence of cross-diffusion (\"jams\") is most pronounced.\n\n\\subsubsection*{Confining potential}\nFinally in figure \\ref{fig:confined}, we consider the situation of no-flux conditions together with a confining potential $V(x) = (x-\\frac{1}{2})^2$. Here we observe very similar behaviour for all models, probably due to the fact that the transport term dominates the dynamics. \n\n\n\n\n\\begin{figure}\n \\centering\n \\subfigure[$t=0.5$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime2DensitiesAtTime_t=0.50}.pdf}\n }\n \\subfigure[$t=2$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime2DensitiesAtTime_t=2.00}.pdf}\n }\n \\subfigure[$t=3$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime2DensitiesAtTime_t=3.00}.pdf}\n }\n \\subfigure[$t=200$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime2DensitiesAtTime_t=200.00}.pdf}\n }\n \\caption{Flux boundary conditions with: $\\lambda = 0.01$, $D_r = 0.1,\\, D_b = 0.1,\\, \\alpha_r = 0.02, \\, \\beta_r = 0.01, \\, \\alpha_b = 0.01, \\, \\beta_b = 0.02$ which yields the influx-limited phase for species $r$ and outflux-limited for $b$.}\n \\label{fig:Regime2}\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\subfigure[$t=0.5$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime3DensitiesAtTime_t=0.50}.pdf}\n }\n \\subfigure[$t=2$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime3DensitiesAtTime_t=2.00}.pdf}\n }\n \\subfigure[$t=3$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime3DensitiesAtTime_t=3.00}.pdf}\n }\n \\subfigure[$t=200$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime3DensitiesAtTime_t=200.00}.pdf}\n }\n \\caption{Flux boundary conditions with: $\\lambda = 0.01$, $D_r = 0.1,\\, D_b = 0.1,\\, \\alpha_r = 0.6, \\, \\beta_r = 0.8, \\, \\alpha_b = 0.7, \\, \\beta_b = 0.9$ which yields the maximal current phase.}\n \\label{fig:Regime3}\n\\end{figure}\n\n\n\n\n\n\n\\begin{figure}\n \\centering\n \n \n \n \\subfigure[$\\alpha_r=0.02$, $\\beta_r=0.01$, $\\alpha_b=0.01$, $\\beta_b=0.02$]{\n \\includegraphics[width=0.275\\textwidth]{{figures\/Regime2Mass}.pdf}\n }\n \\hspace{0.2cm}\n \\subfigure[$\\alpha_r=0.6$, $\\beta_r=0.8$, $\\alpha_b=0.7$, $\\beta_b=0.9$]{\n \\includegraphics[width=0.275\\textwidth]{{figures\/Regime3Mass}.pdf}\n }\n \\hspace{0.2cm}\n \\subfigure[$\\alpha_r=0.1$, $\\beta_r=0.2$, $\\alpha_b=0.2$,\\quad $\\beta_b=0.4$]{\n \\includegraphics[width=0.275\\textwidth]{{figures\/Regime4Mass}.pdf}\n }\n \\caption{Evolution of the total mass for different flux boundary conditions and with $D_r=D_b = 0.1$ and $\\lambda = 0.01$ in all cases.}\n \\label{fig:mass}\n\\end{figure}\n\n\n\n\n\\begin{figure}\n \\centering\n \\subfigure[$t=0$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/PeriodicDensitiesAtTime_t=0.00}.pdf}\n }\n \\subfigure[$t=0.4$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/PeriodicDensitiesAtTime_t=0.50}.pdf}\n }\n \\subfigure[$t=1$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/PeriodicDensitiesAtTime_t=1.00}.pdf}\n }\n \\subfigure[$t=3.9$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/PeriodicDensitiesAtTime_t=3.90}.pdf}\n }\n \\caption{Periodic boundary conditions with $D_r=D_b = 0.01$ and $\\lambda = 0.01$. All models converge to constant stationary solution.}\n \\label{fig:periodic}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\subfigure[$t=0$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime6DensitiesAtTime_t=0.00}.pdf}\n }\n \\subfigure[$t=0.5$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime6DensitiesAtTime_t=0.50}.pdf}\n }\n \\subfigure[$t=2$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime6DensitiesAtTime_t=2.00}.pdf}\n }\n \\subfigure[$t=10$]{\n \\includegraphics[width=0.23\\textwidth]{{figures\/Regime6DensitiesAtTime_t=10.00}.pdf}\n }\n \\label{fig:confined}\n \\caption{No flux boundary conditions with $D_r = D_b = 0.1$, $\\lambda = 0.01$ and an confining potential $V_r = V_b = 5(x-0.5)^2$. }\n\\end{figure}\n\n\n\n\n\\section*{Acknowledgements}\nThe work of MTW was partly supported by the Austrian Academy of Sciences New Frontier's grant NFG-0001. JFP thanks the DAAD for support via the PPP project 57447206. MBu acknowledges partial financial support by European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 777826 (NoMADS) and the German Science Foundation (DFG) through CRC TR 154 \"Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks\", Subproject C06. Maria Bruna was partially supported by a Royal Society University Research Fellowship (grant number URF\/R1\/180040) and a Humboldt Research Fellowship from the Alexander von Humboldt Foundation.\n\n\\printbibliography\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nModern High Energy Physics (HEP) experiments are designed to detect large amount of data with very high rate. In addition to that weak signatures of new physics must be searched in complex background condition. In order to reach these achievements, new computing paradigms must be adopted. A novel approach is based on the use of high parallel computing devices, like Graphics Processing Units (GPU), which delivers such high performance solutions to be used in HEP. In particular, a massive parallel computation based on General Purpose Graphics Processing Units (GPGPU)~\\cite{nvidia} could dramatically speed up the algorithms for charged particle tracking and fitting, allowing their use for fast decision taking and triggering. \nIn this paper we describe a tracking recognition algorithm based on the Hough Transform~\\cite{hough:paper,hough:hep1,hough:hep2} and its implementation on Graphics Processing Units (GPU). \n\n\n\n\n\\section{Tracking with the Hough Transform}\n\nThe Hough Transform (HT) is a pattern recognition technique for features extraction in image processing, and in our case we will use a HT based algorithm to extract the tracks parameters from the hits left by charged particles in the detector. A preliminary result on this study has been already presented in~\\cite{tipp}. Our model is based on a cylindrical multi-layer silicon detector installed around the interaction point of a particle collider, with the detector axis on the beam-line. \nThe algorithm works in two serial steps. In the first part, for each hit having coordinates $(x_H,y_H,z_H)$ the algorithm computes all the circles in the $x-y$ transverse plane passing through that hit and the interaction point, where the circle equation is $x^2+y^2-2Ax-2By=0$, and $A$ and $B$ are the two parameters corresponding to the coordinates of the circle centre. The circle detection is performed taking into account also the longitudinal ($\\theta$) and polar ($\\phi$) angles. For all the $\\theta$, $\\phi$, $A$, $B$, satisfying the circle equation associated to a given hit, the corresponding $M_H(A,B,\\theta,\\phi)$ Hough Matrix (or Vote Matrix) elements are incremented by one. After computing all the hits, all the $M_H$ elements above a given threshold would correspond to real tracks. Thus, the second step is a local maxima search among the $M_H$ elements.\n\n\nIn our test, we used a dataset of 100 simulated events ($pp$ collisions at LHC energy, Minimum Bias sample with tracks having transverse momentum $p_T>500$ MeV), each event containing up to 5000 particle hits on a cylindrical 12-layer silicon detector centred on the nominal collision point. The four hyper-dimensions of the Hough space have been binned in $4 \\times 16 \\times 1024 \\times 1024$ along the corresponding $A,B,\\theta,\\phi$ parameters. \n\n\nThe algorithm performance compared to a $\\chi^2$ fit method is shown in Fig.~\\ref{hough:perf}: the $\\rho=\\sqrt{A^2+B^2}$ and $\\varphi=\\tan^{-1}(B\/A)$ are shown together with the corresponding resolutions.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=.8\\textwidth]{rinaldi_lorenzo_fig1.pdf}}\n\\caption{Hough Transform algorithm compared to $\\chi^2$ fit. (a) $\\rho$ distribution; (b) $\\varphi$ distribution; (c) $\\rho$ resolution; (d) $\\varphi$ resolution. }\\label{hough:perf}\n\\end{figure}\n\n\n\\section{SINGLE-GPU implementation}\n\n\nThe HT tracking algorithm has been implemented in GPGPU splitting the code in two kernels, for Hough Matrix filling and searching local maxima on it. Implementation has been performed both in CUDA~\\cite{nvidia} and OpenCL~\\cite{opencl}. GPGPU implementation schema is shown in Fig.~\\ref{GPGPU:schema}. \n\n\n\n\nConcerning the CUDA implementation, for the $M_H$ filling kernel, we set a 1-D grid over all the hits, the grid size being equal to the number of hits of the event. Fixed the ($\\theta,\\phi$) values, a thread-block has been assigned to the $A$ values, and for each $A$, the corresponding $B$ is evaluated. The $M_H(A,B,\\theta,\\phi)$ matrix element is then incremented by a unity with an {\\tt atomicAdd} operation. The $M_H$ initialisation is done once at first iteration with {\\tt cudaMallocHost} (pinned memory) and initialised on device with {\\tt cudaMemset}. \nIn the second kernel, the local maxima search is carried out using a 2-D grid over the $\\theta,\\phi$ parameters, the grid dimension being the product of all the parameters number over the maximum number of threads per block ($N_{\\phi} \\times N_{\\theta} \\times N_A \\times N_B$)\/{\\tt maxThreadsPerBlock}, and 2-D threadblocks, with {\\tt dimXBlock}=$N_A$ and {\\tt dimYBlock=MaxThreadPerBlock}\/$N_A$. Each thread compares one $M_H(A,B,\\theta,\\phi)$ element to its neighbours and, if the biggest, it is stored in the GPU shared memory and eventually transferred back. With such big arrays the actual challenge lies in optimizing array allocation and access and indeed for\nthis kernel a significant speed up has been achieved by tuning matrix access in a coalesced fashion, thus allowing to gain a crucial computational speed-up.\n\\begin{figure}[hb]\n\\centerline{\\includegraphics[width=.6\\textwidth]{rinaldi_lorenzo_fig2.pdf}}\n\\caption{GPGPU implementation schema of the two Hough Transform algorithm kernels.}\\label{GPGPU:schema}\n\\end{figure}\nThe OpenCL implementation has been done using a similar structure used for CUDA. Since in OpenCL there is no direct pinning memory, a device buffer is mapped to an already existing $memallocated$ host buffer ({\\tt clEnqueueMapBuffer}) and dedicated kernels are used for matrices initialisation in the device memory. The memory host-to-device buffer allocation is performed concurrently and asynchronously, saving overall transferring time. \n\n\n\n\n\\subsection{SINGLE-GPU results}\n\n\n\\begin{footnotesize}\n\\begin{table}[h]\n\\centerline{\\begin{tabular}{ | l | c | c | c |}\n\\hline\nDevice & NVIDIA & NVIDIA & NVIDIA \\\\\nspecification & GeForce GTX770 & Tesla K20m & Tesla K40m \\\\\n\\hline\nPerformance (Gflops) & 3213 & 3542 & 4291 \\\\\nMem. Bandwidth (GB\/s) & 224.2 & 208 & 288 \\\\\nBus Connection & PCIe3 & PCIe3 & PCIe3 \\\\\nMem. Size (MB) & 2048 & 5120 & 12228 \\\\\nNumber of Cores & 1536 & 2496 & 2880 \\\\\nClock Speed (MHz) & 1046 & 706 & 1502 \\\\\n\\hline\n\\end{tabular}}\n\\caption{Computing resources setup.}\n\\label{tab:gpus}\n\\end{table}\n\\end{footnotesize}\n\nThe test has been performed using the NVIDIA~\\cite{nvidia} GPU boards listed in table~\\ref{tab:gpus}. The GTX770 board is mounted locally on a desktop PC, the Tesla K20 and K40 are installed in the INFN-CNAF HPC cluster. \n\n\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=1.05\\textwidth]{rinaldi_lorenzo_fig3.pdf}}\n\\caption{Execution timing as a function of the number of analysed hits. (a) Total execution time for all devices; (b) Total execution time for GPU devices only; (c) $M_H$ filling time for all devices; (d) $M_H$ filling timing for GPU devices only; (e) local maxima search timing for all devices; (f) local maxima search timing for GPU devices only; (g) device-to-host transfer time (GPUS) and I\/O time (CPU). }\\label{allgpu}\n\\end{figure}\n\n\n\n\nThe measurement of the execution time of all the algorithm components has been carried out as a function of the number of hits to be processed, and averaging the results over 100 independent runs. The result of the test is summarised in Fig.~\\ref{allgpu}.\nThe total execution time comparison between GPUs and CPU is shown in Fig.~\\ref{allgpu}a, while in Fig.~\\ref{allgpu}b the details about the execution on different GPUs are shown. The GPU execution is up to 15 times faster with respect to the CPU implementation, and the best result is obtained for the CUDA algorithm version on the GTX770 device. The GPUs timing are less dependent on the number of the hits with respect to CPU timing.\n\n\nThe kernels execution on GPUs is even faster with respect to CPU timing, with two orders of magnitude GPU-CPU speed up, as shown in Figs.~\\ref{allgpu}c and \\ref{allgpu}e. When comparing the kernel execution on different GPUs (Figs.~\\ref{allgpu}d) and~\\ref{allgpu}f), CUDA is observed to perform slightly better than OpenCL. Figure~\\ref{allgpu}g shows the GPU-to-CPU data transfer timings for all devices together with the CPU I\/O timing, giving a clear idea of the dominant part of the execution time.\n\n\n\n\\section{MULTI-GPU implementation}\n\n\n\nAssuming that the detector model we considered could have multiple readout boards working independently, it is interesting to split the workload on multiple GPUs. We have done this by splitting the transverse plane in four sectors to be processed separately, since the data across sectors are assumed to be read-out independently. \nHence, a single HT is executed for each sector, assigned to a single GPU, and eventually the results are merged when each GPU finishes its own process. The main advantage is to reduce the load on a single GPU by using lightweight Hough Matrices and output structures. Only CUDA implementation has been tested, using the same workload schema discussed in Sec. 3, but using four $M_H(A,B,\\theta)$, each matrix processing the data of a single $\\phi$ sector.\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=.8\\textwidth]{rinaldi_lorenzo_fig4.pdf}}\n\\caption{Execution timing as a function of the number of the hits for multi-GPU configuration. (a) Total execution time; (b) $M_H$ filling timing; (c) local maxima search timing; (d) device-to-host transfer time.}\\label{multigpu}\n\\end{figure}\n\n\n\\subsection{MULTI-GPU results}\n\nThe multi-GPU results are shown in Fig.~\\ref{multigpu}. The test has been carried out in double configuration, separately, with two NVIDIA Tesla K20 and two NVIDIA Tesla K40. The overall execution time is faster with double GPUs in both cases, even if timing does not scale with the number of GPUs. An approximate half timing is instead observed when comparing kernels execution times. On the other hand, the transferring time is almost independent on the number of GPUs, this leading the overall time execution.\n\n\n\n\n\\section{Conclusions}\n\nA pattern recognition algorithm based on the Hough Transform has been successfully implemented on CUDA and OpenCL, also using multiple devices. The results presented in this paper show that the employment of GPUs in situations where time is critical for HEP,\nlike triggering at hadron colliders, can lead to significant and encouraging speed-up. Indeed the problem by itself offers wide room for a parallel approach to computation: this is reflected in the results shown where the speed-up is around 15 times better than what achieved with a normal CPU. There are still many handles for optimising the performance, also taking into account the GPU architecture and board specifications. \nNext steps of this work go towards an interface to actual experimental frameworks, including the management of the experimental data structures and testing with more graphics accelerators and coprocessor.\n\n\n\n\n\n\n\n\n\n\\begin{footnotesize}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}