Claudio Landim, Aniura Milanés, Stefano Olla
We prove nonequilibrium fluctuations for the boundary driven symmetric simple exclusion process. We deduce from this result the stationary fluctuations.
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We prove the nonequilibrium fluctuations of one-dimensional, boundary driven, weakly asymmetric exclusion processes through a microscopic Cole-Hopf transformation.
We examine the entropy of stationary nonequilibrium measures of boundary driven symmetric simple exclusion processes. In contrast with the Gibbs--Shannon entropy \cite{B, DLS2}, the entropy of nonequilibrium stationary states differs from the entropy of local equilibrium states.
For a sequence of i.i.d. random variables $\{ξ_x : x\in \bb Z\}$ bounded above and below by strictly positive finite constants, consider the nearest-neighbor one-dimensional simple exclusion process in which a particle at $x$ (resp. $x+1$) jumps to $x+1$ (resp. $x$) at rate $ξ_x$. We examine a quenched nonequilibrium central limit theorem for the position of a tagged particle in the exclusion process with bond disorder $\{ξ_x : x\in \bb Z\}$. We prove that the position of the tagged particle converges under diffusive scaling to a Gaussian process if the other particles are initially distribute…
We prove a nonequilibirum central limit theorem for the position of a tagged particle in the one-dimensional nearest-neighbor symmetric simple exclusion process under diffusive scaling starting from a Bernoulli product measure associated to a smooth profile $ρ_0:\bb R\to [0,1]$.
Nonequilibrium Joint Fluctuations for Current and Occupation Time in The Symmetric Exclusion Process
本文研究一维非平衡 SSEP 中电流与占据时间的联合涨落，给出极限高斯过程协方差公式，并通过相关性估计、离散梯度估计和 Fourier 型 Kipnis–Varadhan 引理证明紧性。
Nonequilibrium moderate deviations from hydrodynamics of simple exclusion processes
本文证明了从非平衡态出发的一维环上简单排他过程流体动力学极限的中等偏差原理，扩展了平衡态结果，利用指数紧性和替换引理及Gronwall不等式等工具。\n
We consider the symmetric simple exclusion process on Z^d, for d>= 5, and study the regularity of the quasi-stationary measures of the dynamics conditionned on not occupying the origin. For each ρ\in ]0,1[, we establish uniqueness of the density of quasi-stationary measures in L^2(d\nur), where \nur is the stationary measure of density ρ. This, in turn, permits us to obtain sharp estimates for P_{\nur}(τ>t), where τis the first time the origin is occupied.
Nonequilibrium fluctuations for the occupation time of the SSEP in $d \geq 2$
本文研究多维SSEP中占据时间的非平衡波动，通过鞅方法和相关性估计证明了其不变性原理。
The asymmetric exclusion process is an idealised stochastic model of transport, whose exact solution has given important insight into a general theory of nonequilibrium statistical physics. In this work, we consider a totally asymmetric exclusion process with multiple species of particles on a one-dimensional lattice in contact with reservoirs. We derive the exact nonequilibrium phase diagram for the system in the long time limit. We find two new phenomena in certain regions of the phase diagram: when the density of a species becomes zero throughout the system, and dynamical localisation when …
From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval
本文证明了区间上KPZ不动点的平稳测度是稳态下开放非对称简单排它过程高度增量过程的标度极限，且随系统尺寸趋于无穷大参数会适当改变。
The processes described in the title always have reversible stationary distributions. In this paper, we give sufficient conditions for the existence of, and for the nonexistence of, nonreversible stationary distributions. In the case of an i.i.d. environment, these combine to give a necessary and sufficient condition for the existence of nonreversible stationary distributions.
Stationary Distribution of open Asymmetric Simple Exclusion Processes on an Interval as a marginal of a two-layer ensemble
本文研究区间上开放边界ASEP的平稳分布，在Liggett条件下将其表示为两层系综顶层的边缘分布，该表示在扇形区域和激波区域均成立，并给出了两层权函数的递推关系。
We prove a law of large numbers for the empirical density of one-dimensional, boundary driven, symmetric exclusion processes with different types of non-reversible dynamics at the boundary. The proofs rely on duality techniques.
Dynamical and stationary non-equilibrium fluctuations for a boundary-driven exclusion process with energy
本文研究边界驱动下具有能量排斥过程的动力学与平稳涨落，提出无需两点相关函数估计的新证明，发现动力学涨落由耦合广义奥恩斯坦-乌伦贝克过程描述。
The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli ρmeasure as initial conditions, 0<ρ<1, is stationary in space and time. Let N_t(j) be the number of particles which have crossed the bond from j to j+1 during the time span [0,t]. For j=(1-2ρ)t+2w(ρ(1-ρ))^{1/3} t^{2/3} we prove that the fluctuations of N_t(j) for large t are of order t^{1/3} and we determine the limiting distribution function F_w(s), which is a generalization of the GUE Tracy-Widom distribution. The family F_w(s) of distribution functions have been obtained before by Baik …
We describe the extremal translation invariant stationary (ETIS) states of the facilitated exclusion process on $\mathbb{Z}$. In this model all particles on sites with one occupied and one empty neighbor jump at each integer time to the empty neighbor site, and if two particles attempt to jump into the same empty site we choose one randomly to succeed. The ETIS states are qualitatively different for densities $ρ<1/2$, $ρ=1/2$, and $1/2<ρ<1$, but in each density region we find states which may be grouped into families, each of which is in natural correspondence with the set of all ergodic measu…
Stationary States of the One-dimensional Facilitated Asymmetric Exclusion Process
该研究刻画了一维助推非对称排斥过程的平移不变定态，证明低密度下定态与跳变速率参数无关，高密度下存在唯一与硬核粒子Gibbs测度同构的定态。
We consider totally asymmetric simple exclusion processes with n types of particle and holes ($n$-TASEPs) on $\mathbb {Z}$ and on the cycle $\mathbb {Z}_N$. Angel recently gave an elegant construction of the stationary measures for the 2-TASEP, based on a pair of independent product measures. We show that Angel's construction can be interpreted in terms of the operation of a discrete-time $M/M/1$ queueing server; the two product measures correspond to the arrival and service processes of the queue. We extend this construction to represent the stationary measures of an n-TASEP in terms of a sys…
We consider a one-dimensional symmetric simple exclusion process in contact with slowed reservoirs: at the left (resp. right) boundary, particles are either created or removed at rates given by $α/n$ or $(1-α)/n$ (resp. $β/n$ or $(1-β)/n$) where $α, β>0$ and $n$ is a scaling parameter. We obtain the non-equilibrium fluctuations and consequently the non-equilibrium stationary fluctuations.
A simple symmetric exclusion process driven by an asymmetric tracer particle
研究周期性一维晶格上由非对称示踪粒子驱动的排他过程，给出了稳态精确解、非平衡配分函数及流和关联公式，并发现非平衡自由能在热力学极限下无法良好定义。
Approximating the stationary distribution of the ASEP with open boundaries
本文研究带开放边界ASEP的平稳分布，通过将其投影到随系统长度增长的子区间上，并依据边界参数给出条件，证明投影分布在总变差距离下接近乘积测度。
We give a combinatorial description of the stationary measure for a totally asymmetric exclusion process (TASEP) with second class particles, on either Z or on the cycle Z_N. The measure is the image by a simple operation of the uniform measure on some larger finite state space. This reveals a combinatorial structure at work behind several results on the TASEP with second class particles.
We consider the symmetric exclusion process on the $d$-dimensional lattice with translational invariant and ergodic initial data. It is then known that as $t$ diverges the distribution of the process at time $t$ converges to a Bernoulli product measure. Assuming a summable decay of correlations of the initial data, we prove a quantitative version of this convergence by obtaining an explicit bound on the Ornstein $\bar d$-distance. The proof is based on the analysis of a two species exclusion process with annihilation.
Stationary fluctuations for the facilitated exclusion process
本文利用FEP与零程过程的映射，推导了一维FEP在对称、弱不对称和不对称情形下的平稳涨落，并在对称情形下通过系综等价性给出了直接证明。
We introduce a two-dimensional, distribution-valued field which we call the quadratic field associated to the one-dimensional Ornstein-Uhlenbeck process. We show that the stationary quadratic fluctuations of the simple exclusion process, when rescaled in the diffusive scaling, converge to this quadratic field. We show that this quadratic field evaluated at the diagonal corresponds to the Wick-renormalized square of the Ornstein-Uhlenbeck process, and we use this new representation in order to prove some small and large-time properties of it.
There has been significant progress recently in our understanding of the stationary measures of the exclusion process on $Z$. The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on $Z^d$ and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other sta…
Laws of large numbers, starting from certain nonequilibrium measures, have been shown for the integrated current across a bond, and a tagged particle in one-dimensional symmetric nearest-neighbor simple exclusion [Ann. Inst. Henri Poincare Probab. Stat. 42 (2006) 567-577]. In this article, we prove corresponding large deviation principles and evaluate the rate functions, showing different growth behaviors near and far from their zeroes which connect with results in [J. Stat. Phys. 136 (2009) 1-15].
We consider a chain composed of $N$ coupled harmonic oscillators in contact with heat baths at temperature $T_\ell$ and $T_r$ at sites 1 and $N$ respectively. The oscillators are also subjected to non-momentum conserving bulk stochastic noises. These make the heat conductivity satisfy Fourier's law. Here we describe some new results about the hydrodynamical equations for typical macroscopic energy and displacement profiles, as well as their fluctuations and large deviations, in two simple models of this type.
We study two versions of the asymmetric exclusion process (ASEP) -- an ASEP on a semi-infinite lattice with an open left boundary, and an ASEP on a finite lattice with open left and right boundaries -- and we demonstrate a surprising relationship between their stationary measures. The semi-infinite ASEP was first studied by Liggett and then Grosskinsky, while the finite ASEP had been introduced earlier by Spitzer and Macdonald-Gibbs-Pipkin. We show that the finite correlation functions involving the first L sites for the stationary measures on the semi-infinite ASEP can be obtained as a nonphy…
Large deviation principle for the stationary measures of open asymmetric simple exclusion processes
本文严格证明了开放边界非对称简单排斥过程在扇区相图区域平稳测度高度轮廓的大偏差原理，利用两层表达式和TASEP的LDP，首次给出该结果的数学证明。
We give a partly new proof of the fluctuation bounds for the second class particle and current in the stationary asymmetric simple exclusion process. One novelty is a coupling that preserves the ordering of second class particles in two systems that are themselves ordered coordinatewise.
We show that the joint probability generating function of the stationary measure of a finite state asymmetric exclusion process with open boundaries can be expressed in terms of joint moments of Markov processes called quadratic harnesses. We use our representation to prove the large deviations principle for the total number of particles in the system. We use the generator of the Markov process to show how explicit formulas for the average occupancy of a site arise for special choices of parameters. We also give similar representations for limits of stationary measures as the number of sites t…
Stationary fluctuations for an exclusion process with mass and energy conservation
本文引入一种同时守恒质量和能量的新型排斥过程，通过非线性涨落流体力学揭示其普适类，并严格证明在合适参数下涨落场收敛到非耦合随机Burgers方程，建立了宏观电流雅可比矩阵可对角化的数学基础。
Exclusion processes with non-reversible boundary: hydrodynamics and large deviations
该研究针对一维排斥动力学与边界库的温和接触问题，揭示了扩散尺度下粒子密度遵循带非线性Robin边界条件的热方程演化规律，证明了其动力学大偏差原理，且边界速率适当选择时偏微分方程存在多个平稳解。
We prove here that the height function associated to non-simple exclusion processes with arbitrary jump-length converges to the solution of the Kardar-Parisi-Zhang SPDE under suitable scaling and renormalization. This extends the work of Dembo-Tsai'16 for arbitrary jump-length and Goncalves-Jara '17 for the non-stationary regime. Thus we answer a ``Big Picture Question" from the AIM workshop on KPZ and also expand on the almost empty set of non-integrable and non-stationary particle systems for which weak KPZ universality is proven. We use an approximate microscopic Cole-Hopf transform as in D…
We consider a symmetric exclusion process on a discrete interval of $S$ points with various boundary conditions at the endpoints. We study the asymptotic decay of correlations as $S\to\infty$. The main result is asymptotic independence of a stationary distribution whem the points are far away from each other. We develop a new recurrent probabilistic approach which is an alternative to Derrida's algebraic technique.
In the boundary driven symmetric simple exclusion process consider an open set $\ms O$ of density profiles which does not contain the stationary density profile. We prove that the first time the empirical measure visits the set $\ms O$ converges to an exponential distribution.
The main subject of the thesis is the study of stationary nonequilibrium states trough the use of microscopic stochastic models that encode the physical interaction in the rules of Markovian dynamics for particles configurations. These models are known as interacting particles systems and are simple enough to be treated analytically but also complex enough to capture essential physical behaviours. The thesis is organized in two parts. The part 1 is devoted to the microscopic theory of the stationary states. We characterize these states developing some general structures that have an interest i…
Limit fluctuations of stationary measure of totally asymmetric simple exclusion process with open boundaries on the coexistence line
该研究针对共存线上开放完全非对称简单排他过程的平稳测度，分析其高度函数的极限涨落，证明经随机中心化与√n归一化后，高度函数的二阶极限为两个独立布朗运动的随机混合。
We prove that the stationary measure associated to a boundary driven exclusion process in any dimension satisfies a large deviation principle with rate function given by the quasi potential of the Freidlin and Wentzell theory.
From asymmetric simple exclusion processes with open boundaries to stationary measures of open KPZ fixed point: the shock region
本文在开边界 ASEP 的激波区域中证明了高度函数标度极限依分布收敛到开 KPZ 不动点的平稳测度，证明基于 Askey-Wilson 符号测度的矩阵乘积拟设积分表示，并利用 Bryc-Zatitskii 的最近结果建立对偶公式。
Nonequilibrium fluctuations and moderate deviations for the occupation time of the SSEP with Glauber dynamics
研究SSEP与Glauber动力学的非平衡测度起始下的occupation时间波动，利用鞅方法和相对熵方法证明二维中心极限定理和一维样本路径中等偏差原理。
We establish an invariance principle for a one-dimensional random walk in a dynamical random environment given by a speed-change exclusion process. The jump probabilities of the walk depend on the configuration of the exclusion in a finite box around the walker. The environment starts from equilibrium. After a suitable space-time rescaling, the random walk converges to a sum of two independent processes, a Brownian motion and a Gaussian process with stationary increments.
Moderate deviations for the facilitated exclusion process in equilibrium
本文推导了一维促进排斥过程在平稳测度下对称与非对称情形的涨落场中度偏差原理，核心是通过基于对数Sobolev不等式证明Boltzmann-Gibbs原理的超指数版本。
We prove that the variance of the current across a characteristic is of order t^{2/3} in a stationary asymmetric simple exclusion process, and that the diffusivity has order t^{1/3}. The proof proceeds via couplings to show the corresponding results for the expected deviations and variance of a second class particle.
We introduce the mathematical theory of the particle systems that interact via permutations, where the transition rates are assigned not to the jumps from a site to a site, but to the permutations themselves. This permutation processes can be viewed as a generalization of the symmetric exclusion processes, where particles interact via transpositions. The duality and coupling techniques for the processes are described, the needed conditions for them to apply are established. The stationary distributions of the permutation processes are explored for translation invariant cases.
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