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Hydrodynamic Limit for the SSEP with a Slow Membrane

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Tertuliano Franco at Federal University of Bahia
Tertuliano Franco
Mariana Tavares
Mariana Tavares
Mariana Tavares
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Abstract and Figures

In this paper we consider a symmetric simple exclusion process on the d-dimensional discrete torus TNd{\mathbb {T}}^d_N with a spatial non-homogeneity given by a slow membrane. The slow membrane is defined here as the boundary of a smooth simple connected region Λ\Lambda on the continuous d-dimensional torus Td{\mathbb {T}}^d. In this setting, bonds crossing the membrane have jump rate α/Nβ\alpha /N^\beta and all other bonds have jump rate one, where α>0\alpha >0, β∈[0,∞]\beta \in [0,\infty ], and N∈NN\in {\mathbb {N}} is the scaling parameter. In the diffusive scaling we prove that the hydrodynamic limit presents a dynamical phase transition, that is, it depends on the regime of β\beta . For β∈[0,1)\beta \in [0,1), the hydrodynamic equation is given by the usual heat equation on the continuous torus, meaning that the slow membrane has no effect in the limit. For β∈(1,∞]\beta \in (1,\infty ], the hydrodynamic equation is the heat equation with Neumann boundary conditions, meaning that the slow membrane ∂Λ\partial \Lambda divides Td{\mathbb {T}}^d into two isolated regions Λ\Lambda and Λ∁\Lambda ^\complement . And for the critical value β=1\beta =1, the hydrodynamic equation is the heat equation with certain Robin boundary conditions related to the Fick’s Law.
The region in gray represents Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}, and the white region represents its complement Λ∁\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ^\complement $$\end{document}. The grid represents N-1TNd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{-1}{\mathbb {T}}_N^d$$\end{document}, the discrete torus embedded on the continuous torus Td\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^d$$\end{document}. By ζ→(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {\zeta }(u)$$\end{document} we denote the normal exterior unitary vector to Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} at the point u∈∂Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \partial \Lambda $$\end{document}
The region in gray represents Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}, and the white region represents its complement Λ∁\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ^\complement $$\end{document}. The grid represents N-1TNd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{-1}{\mathbb {T}}_N^d$$\end{document}, the discrete torus embedded on the continuous torus Td\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^d$$\end{document}. By ζ→(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {\zeta }(u)$$\end{document} we denote the normal exterior unitary vector to Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} at the point u∈∂Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in \partial \Lambda $$\end{document}
… 
Illustration (in dimension 2) of a polygonal path joining the sites x and y=x+j1e1+j2e2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=x+j_1e_1+j_2 e_2$$\end{document}, with j1=j2=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_1=j_2=3$$\end{document}. Note the embedding in the continuous torus Td\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^d$$\end{document}
Illustration (in dimension 2) of a polygonal path joining the sites x and y=x+j1e1+j2e2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=x+j_1e_1+j_2 e_2$$\end{document}, with j1=j2=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_1=j_2=3$$\end{document}. Note the embedding in the continuous torus Td\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^d$$\end{document}
… 
Illustration in dimension two of CN[x,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_N[x,2]$$\end{document}. The sites in CN[x,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_N[x,2]$$\end{document} are those laying in the gray region
Illustration in dimension two of CN[x,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_N[x,2]$$\end{document}. The sites in CN[x,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_N[x,2]$$\end{document} are those laying in the gray region
… 
Illustration in dimension two of C[u,ε]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C[u,\varepsilon ]$$\end{document}, which is represented by the region in gray, while B[u,ε]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B[u,\varepsilon ]$$\end{document} is represented by the square delimited by the dashed line. Note that C[u,ε]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C[u,\varepsilon ]$$\end{document} is the continuous counterpart of CN[x,ℓ]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_N[x,\ell ]$$\end{document} defined in (5.9)
Illustration in dimension two of C[u,ε]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C[u,\varepsilon ]$$\end{document}, which is represented by the region in gray, while B[u,ε]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B[u,\varepsilon ]$$\end{document} is represented by the square delimited by the dashed line. Note that C[u,ε]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C[u,\varepsilon ]$$\end{document} is the continuous counterpart of CN[x,ℓ]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_N[x,\ell ]$$\end{document} defined in (5.9)
… 
Illustration of sites x,y,z∈ΓN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y,z\in \Gamma _N$$\end{document}. We note that two adjacent edges to x are slow bonds, and two adjacent edges are not. Besides, any opposite vertex to x will be of the form x±ej\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\pm e_j$$\end{document}
+1
Illustration of sites x,y,z∈ΓN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y,z\in \Gamma _N$$\end{document}. We note that two adjacent edges to x are slow bonds, and two adjacent edges are not. Besides, any opposite vertex to x will be of the form x±ej\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\pm e_j$$\end{document}
… 
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Journal of Statistical Physics (2019) 175:233–268
https://doi.org/10.1007/s10955-019-02254-y
Hydrodynamic Limit for the SSEP with a Slow Membrane
Tertuliano Franco1·Mariana Tavares1
Received: 21 September 2018 / Accepted: 16 February 2019 / Published online: 21 February 2019
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract
In this paper we consider a symmetric simple exclusion process on the d-dimensional discrete
torus Td
Nwith a spatial non-homogeneity given by a slow membrane. The slow membrane
is defined here as the boundary of a smooth simple connected region on the continuous
d-dimensional torus Td. In this setting, bonds crossing the membrane have jump rate α/Nβ
and all other bonds have jump rate one, where α>0, β∈[0,∞],andN∈Nis the scaling
parameter. In the diffusive scaling we prove that the hydrodynamic limit presents a dynamical
phase transition, that is, it depends on the regime of β.Forβ∈[0,1), the hydrodynamic
equation is given by the usual heat equation on the continuous torus, meaning that the slow
membrane has no effect in the limit. For β∈(1,∞], the hydrodynamic equation is the heat
equation with Neumann boundary conditions, meaning that the slow membrane ∂ divides
Tdinto two isolated regions and . And for the critical value β=1, the hydrodynamic
equation is the heat equation with certain Robin boundary conditions related to the Fick’s
Law.
Keywords Hydrodynamic limit ·Exclusion process ·Non-homogeneous environment ·
Slow bonds
Mathematics Subject Classification 60K35 ·35K55
1 Introduction
A central question of Statistical Mechanics is about how microscopic interactions determine
the macroscopic behavior of a given system. Under this guideline, an entire area on scaling
limits of interacting random particle systems has been developed, see [10] and references
therein.
Communicated by Abhishek Dhar.
BTertuliano Franco
tertu@ufba.br
Mariana Tavares
tavaresaguiar57@gmail.com
1UFBA, Instituto de Matemática, Campus de Ondina, Av. Adhemar de Barros, S/N.,
Salvador CEP 40170-110, Brazil
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... The microscopic models they considered are either defined on a ring with a slow site/slow bond, or defined on a segment with slow boundaries. The results have also been extended to higher dimensions in [8,14,16]. ...
... By [8,Section 5.3], to prove Lemma 4.3, we only need to prove the following result. To prove Lemma 4.4, we need the following lemma. ...
... where the supremum is carried over all functionsH ∈ C 0,1 ([0, T ]×R d ) with compact support contained in [0, T ] × ([−M, M] d \{|u 1 | ≤ 1 M }). Based on the above lemma, the proof of Lemma 4.3 follows from the same analysis as that given in the proof of[8, Lemma 5.7], where a crucial step is the utilization of Riesz's representation theorem, the details of which we omit here. ...
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We introduce the voter model on the infinite integer lattice with a slow membrane and investigate its hydrodynamic behavior and nonequilibrium fluctuations. The voter model is one of the classical interacting particle systems with state space {0,1}Zd{0,1}Zd\{0,1\}^{\mathbb Z^d}. In our model, a voter adopts one of its neighbors’ opinion at rate one except for neighbors crossing the hyperplane {x:x1=1/2}{x:x1=1/2}\{x:x_1 = 1/2\}, where the rate is αN-βαN−β\alpha N^{-\beta } and thus is called a slow membrane. Above, α>0andβ≥0α>0 and β≥0\alpha >0 \ \textrm{and} \ \beta \ge 0 are given parameters and the positive integer N is a scaling parameter. We consider the limit N→∞N→∞N \rightarrow \infty and prove that the hydrodynamic limits are given by the heat equation without or with Robin/Neumann conditions depending on the values of ββ\beta . We also consider the nonequilibrium fluctuations, where the limit is described by generalized Ornstein–Uhlenbeck processes with certain boundary conditions corresponding to the hydrodynamic equation.
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... In this article, we combined the main features of the models presented in [7] and [12]. More specifically, we consider a symmetric exclusion process where particles move on a d−dimensional lattice and there exists a slow barrier hindering some of the (possibly long) jumps. ...
... Next, we add a barrier hindering the jumps between Z d−1 ×Z * − and Z In the same way as it is done in [7] and [12], the slowing factor is αn −β , with n ≥ 1, α > 0 and β ≥ 0. We stress that only (some of the) movements affecting the last coordinate are affected. For instance, for d = 2, the jump rate from (−2, −2) to (−2, 2) is multiplied by αn −β ; on the other hand jumps from (−2, −3) to (−2, −4), from (−1, −3) to (2, −3) and from (2, 1) to (2, −1) are not realized through slow bonds. ...
... We observe that this work is a non-trivial generalization of the results in [7,12]. Indeed, in [7,12], the initial distribution needs to satisfy an entropy bound with respect to a reference measure, the Bernoulli product measure of constant parameter (this is analogous to the case where h in Definition 2.4 is taken as being constant). ...
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We obtain the hydrodynamic limit of symmetric long-jumps exclusion in Zd\mathbb{Z}^d (for d≥1d \geq 1), where the jump rate is inversely proportional to a power of the jump's length with exponent γ+1\gamma+1, where γ≥2\gamma \geq 2. Moreover, movements between Zd−1×Z−∗\mathbb{Z}^{d-1} \times \mathbb{Z}_{-}^{*} and Zd−1×N\mathbb{Z}^{d-1} \times \mathbb N are slowed down by a factor αn−β\alpha n^{-\beta} (with α>0\alpha>0 and β≥0\beta\geq 0). In the hydrodynamic limit we obtain the heat equation in Rd\mathbb{R}^d without boundary conditions or with Neumann boundary conditions, depending on the values of β\beta and γ\gamma. The (rather restrictive) condition in \cite{casodif} (for d=1) about the initial distribution satisfying an entropy bound with respect to a Bernoulli product measure with constant parameter is weakened or completely dropped.
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... The microscopic models they considered are either defined on a ring with a slow site/slow bond, or defined on a segment with slow boundaries. The results have also been extended to higher dimensions in [7,14]. ...
... [5,Section 7.2], it is easy to prove the weak solutions to the above PDEs (2.5) and (2.6) are unique. We also refer the readers to [7,Section 7] for the uniqueness of weak solutions to the above PDEs if the underlying space is the torus T d . ...
... By [7,Section 5.3], to prove Lemma 5.2, we only need to prove the following result. ...
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... In [9] the analogue of our model in case of nearest-neighbour jumps was analysed in the d-dimensional torus and the resulting hydrodynamic equation was the heat equation with Neumann, Robin or no boundary conditions. We believe that our results could be easily extended to the d-dimensional setting by combining our results with the results of [9], but we leave this to a future work. ...
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... This type of models is interesting because it presents different behaviors in the macroscopic level depending on how slow the dynamics are put. Fruitful results has been obtained on the hydrodynamic limit(see, for instance [6,1,4,5,8,13,14,15,16]), fluctuations of density field(see [7,10,11]) and large deviations from the hydrodynamic limit(see [9,12]). ...
... Nevertheless, when the invariant measure is product, the Varadhan's entropy method still works for deriving the hydrodynamic limit of high dimensional models. This is the case of the model in [4], where the authors study the SSEP with a slow membrane in dimension d ≥ 2. ...
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... The symmetric simple exclusion process (SSEP) with a slow bond was introduced in [5] by Franco, Gonçalves and Neumann to derive from microscopic systems PDEs with boundary conditions, which has become a popular topic recently [1,8,11]. The process evolves on the discrete ring with N sites, where N is the scaling parameter. ...
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Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus Td. In the discrete torus N-1TdN, we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on Λ in the following way: if both sites are in Λ or Λc, the exchange rate is 1; if one site is in Λ and the other site is in Λc, and the direction of the bond connecting the sites is ej, then the exchange rate is defined as N-1 times the absolute value of the inner product between ej and the normal exterior vector to ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.
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We consider the exclusion process in the one-dimensional discrete torus with N points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance N−βN^{-\beta}, with β∈[0,∞)\beta\in[0,\infty). We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter β\beta. If β∈[0,1)\beta\in [0,1), the hydrodynamic limit is given by the usual heat equation. If β=1\beta=1, it is given by a parabolic equation involving an operator ddxddW\frac{d}{dx}\frac{d}{dW}, where W is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic point related to the slow bond. If β∈(1,∞)\beta\in(1,\infty), it is given by the heat equation with Neumann's boundary conditions, meaning no passage through the slow bonds in the continuum.
View
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Hydrodynamic Limit for a Type of Exclusion Process with Slow Bonds in Dimension d ≥ 2
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Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus T d . In the discrete torus N-1T d N , we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on Λ in the following way: if both sites are in Λ or Λc, the exchange rate is 1; if one site is in Λ and the other site is in Λc, and the direction of the bond connecting the sites is e j , then the exchange rate is defined as N-1 times the absolute value of the inner product between e j and the normal exterior vector to ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.
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Fix a polynomial Phi of the form Phi(alpha) = alpha + Sigma(2 <= j <= m)a(j)alpha(j) with Phi '(1) > 0. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes on T-d, with conductances given by special class of functions W, is described by the unique weak solution of the non-linear parabolic partial differential equation partial derivative(t)rho = Sigma(d)(k=1) partial derivative(xk) partial derivative(Wk) Phi(rho). We also derive some properties of the operator Sigma(d)(k=1) partial derivative(xk) partial derivative(Wk).
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Phase transition in equilibrium fluctuations of symmetric slowed exclusion
Article
  • Jan 2013
  • STOCH PROC APPL
We analyze the equilibrium fluctuations of the density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is αn−β\alpha n^-\beta, where α,β≥0\alpha,\beta\geq{0} and n is the scaling parameter. Depending on the regime of β\beta, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value β=1\beta=1, starting a tagged particle near the slow bond, we obtain a family of gaussian processes indexed in α\alpha, interpolating a fractional brownian motion of Hurst exponent 1/4 and the degenerate process equal to zero.
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Equilibrium fluctuations for exclusion processes with conductances in random environments
Article
  • Nov 2009
  • STOCH PROC APPL
Fix a function W:ℝ d →ℝ such that W(x 1 ,⋯,x d )=∑ k=1 d W k (x k ), where d≥1, and each function W k :ℝ→ℝ is strictly increasing, right continuous with left limits. We prove the equilibrium fluctuations for exclusion processes with conductances, induced by W, in random environments, when the system starts from an equilibrium measure. The asymptotic behavior of the empirical distribution is governed by the unique solution of a stochastic differential equation taking values in a certain nuclear Fréchet space.
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