Search ResultsShowing 1-20 of 33
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arXiv:2412.13515 (Published 2024-12-18)
$Γ$-expansion of the measure-current large deviations rate functional of non-reversible finite-state Markov chains
Comments: 14 pagesCategories: math.PRConsider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Let $I_n$ be the measure-current large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under a hypothesis on the jump rates, we prove that $I_n$ can be written as $I_n = \mathbf I^{(0)} \,+\, \sum_{1\le p\le \mathfrak q} (1/\theta^{(p)}_n) \, \mathbf I^{(p)}$ for some rate functionals $\mathbf I^{(p)}$. The weights $\theta^{(p)}_n$ correspond to the time-scales at which the sequence of Markov chains $X^{(n)}_t$ evolve among the metastable wells, and the rate functionals $\mathbf I^{(p)}$ characterise the asymptotic Markovian dynamics among these wells. This expansion provides therefore an alternative description of the metastable behavior of a sequence of Markovian dynamics.
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arXiv:2412.04015 (Published 2024-12-05)
Linear fluctuation of interfaces in Glauber-Kawasaki dynamics
Categories: math.PR, cond-mat.stat-mechIn this article, we find a scaling limit of the space-time mass fluctuation field of Glauber + Kawasaki particle dynamics around its hydrodynamic mean curvature interface limit. Here, the Glauber rates are scaled by $K=K_N$, the Kawasaki rates by $N^2$ and space by $1/N$. We start the process so that the interface $\Gamma_t$ formed is stationary that is, $\Gamma_t$ is `flat'. When the Glauber rates are balanced on $T^d$, $\Gamma_t=\Gamma=\{x: x_1=0\}$ is immobile and the hydrodynamic limit is given by $\rho(t,v) = \rho_+$ for $v_1\in (0,1/2)$ and $\rho(t,v)= \rho_-$ for $v_1\in (-1/2,0)$ for all $t\ge 0$, where $v=(v_1,\ldots,v_d)\in T^d$ identified with $[-1/2,1/2)^d$. Since in the formation the boundary region about the interface has width $O(1/\sqrt{K_N})$, we will scale the $v_1$ coordinate in the fluctuation field by $\sqrt{K_N}$ so that the scaling limit will capture information `near' the interface. We identify the fluctuation limit as a Gaussian field when $K_N\uparrow \infty$ and $K_N= O(\sqrt{\log(N)})$ in $d\leq 2$. In the one dimensional case, the field limit is given by ${\bf e}(v_1) B_t$ where $B_t$ is a Brownian motion and ${\bf e}$ is the normalized derivative of a decreasing `standing wave' solution $\phi$ of $\partial^2_{v_1} \phi - V'(\phi)=0$ on $R$, where $V'$ is the homogenization of the Glauber rates. In two dimensions, the limit is ${\bf e}(v_1)Z_t(v_2)$ where $Z_t$ is the solution of a one dimensional stochastic heat equation. The appearance of the function ${\bf e}(\cdot)$ in the limit field indicates that the interface fluctuation retains the shape of the transition layer $\phi$.
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arXiv:2411.17653 (Published 2024-11-26)
Exclusion processes with non-reversible boundary: hydrodynamics and large deviations
Comments: 44 pagesWe consider a one-dimensional exclusion dynamics in mild contact with boundary reservoirs. In the diffusive scale, the particles' density evolves as the solution of the heat equation with non-linear Robin boundary conditions. For appropriate choices of the boundary rates, these partial differential equations have more than one stationary solution. We prove the dynamical large deviations principle.
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arXiv:2309.05546 (Published 2023-09-11)
Metastability and time scales for parabolic equations with drift 1: the first time scale
Comments: 60 pages, 3 figuresConsider the elliptic operator given by $$ \mathscr{L}_{\epsilon}f= {b} \cdot \nabla f + \epsilon \Delta f $$ for some smooth vector field $ b\colon \mathbb R^d \to\mathbb R^d$ and a small parameter $\epsilon>0$. Consider the initial-valued problem $$ \left\{ \begin{aligned} &\partial_ t u_\epsilon = \mathscr L_\epsilon u_\epsilon,\\ &u_\epsilon (0, \cdot) = u_0(\cdot), \end{aligned} \right. $$ for some bounded continuous function $u_0$. Denote by $\mathcal M_0$ the set of critical points of $b$, $\mathcal M_0 =\{x\in \mathbb R^d : b(x)=0\}$, assumed to be finite. Under the hypothesis that $ b = -(\nabla U + \ell)$, where $ \ell$ is a divergence-free field orthogonal to $\nabla U$, the main result of this article states that there exist a time-scale $\theta^{(1)}_\epsilon$, $\theta^{(1)}_\epsilon \to \infty$ as $\epsilon \rightarrow 0$, and a Markov semigroup $\{p_t : t\ge 0\}$ defined on $\mathcal M_0$ such that $$ \lim_{\epsilon\to 0} u_\epsilon (t\theta^{(1)}_\epsilon, x) =\sum_{m'\in \mathcal M_0} p_t(m, m')\, u_0( m'), $$ for all $t>0$ and $ x$ in the domain of attraction of $m$ for the ODE $\dot{x}(t)= b( x(t))$. The time scale $\theta^{(1)}$ is critical in the sense that, for all time scale $\varrho_\epsilon$ such that $\varrho_\epsilon \to \infty$, $\varrho_\epsilon/\theta^{(1)}_\epsilon \to 0$, $$ \lim_{\epsilon\to 0} u_\epsilon (\varrho_\epsilon, x)=u_0(m) $$ for all $x \in \mathcal D(m)$. Namely, $\theta_\epsilon^{(1)}$ is the first scale at which the solution to the initial-valued problem starts to change. In a companion paper [Landim, Lee, Seo, forthcoming] we extend this result finding all critical time-scales at which the solution $u_\epsilon$ evolves smoothly in time and we show that the solution $u_\epsilon$ is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of $b$.
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arXiv:2308.10895 (Published 2023-08-21)
Dynamic and static large deviations of a one dimensional SSEP in weak contact with reservoirs
Categories: math.PR, cond-mat.stat-mechWe derive a formula for the quasi-potential of one-dimensional symmetric exclusion process in weak contact with reservoirs. The interaction with the boundary is so weak that, in the diffusive scale, the density profile evolves as the one of the exclusion process with reflecting boundary conditions. In order to observe an evolution of the total mass, the process has to be observed in a longer time-scale, in which the density profile becomes immediately constant.
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arXiv:2211.02593 (Published 2022-11-04)
Large deviations for diffusions: Donsker-Varadhan meet Freidlin-Wentzell
Categories: math.PR, cond-mat.stat-mechWe consider a diffusion process on $\mathbb R^n$ and prove a large deviation principle for the empirical process in the joint limit in which the time window diverges and the noise vanishes. The corresponding rate function is given by the expectation of the Freidlin-Wentzell functional per unit of time. As an application of this result, we obtain a variational representation of the rate function for the Gallavotti-Cohen observable in the small noise and large time limits.
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Thermodynamics of nonequilibrium driven diffusive systems in mild contact with boundary reservoirs
Journal: Journal of Statistical Physics (2022) 188:19Categories: math.PR, cond-mat.stat-mechKeywords: nonequilibrium driven diffusive systems, boundary reservoirs, mild contact, renormalized work, thermodynamicTags: journal articleWe consider macroscopic systems in weak contact with boundary reservoirs and under the action of external fields. We present an explicit formula for the Hamiltonian of such systems, from which we deduce the equation of motions, the action functional, the hydrodynamic equation for the adjoint dynamics, and a formula for the quasi-potential. We examine the case in which the external forcing depends on time and drives the system from one nonequilibrium state to another. We extend the results presented in [6] on thermodynamic transformations for systems in strong contact with boundary reservoirs to the present situation. In particular, we propose a natural definition of renormalized work, and show that it satisfies a Clausius inequality, and that quasi-static transformations minimize the renormalized work. In addition, we connect the renormalized work to the quasi-potential describing the fluctuations in the stationary nonequilibrium ensemble.
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arXiv:2111.05892 (Published 2021-11-10)
Concurrent Donsker-Varadhan and hydrodynamical large deviations
Categories: math.PR, cond-mat.stat-mechWe consider the weakly asymmetric exclusion process on the $d$-dimensional torus. We prove a large deviations principle for the time averaged empirical density and current in the joint limit in which both the time interval and the number of particles diverge. This result is obtained both by analyzing the variational convergence, as the number of particles diverges, of the Donsker-Varadhan functional for the empirical process and by considering the large time behavior of the hydrodynamical rate function. The large deviations asymptotic of the time averaged current is then deduced by contraction principle. The structure of the minimizers of this variational problem corresponds to the possible occurrence of dynamical phase transitions.
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arXiv:2008.03076 (Published 2020-08-07)
The stochastic heat equation as the limit of a stirring dynamics perturbed by a voter model
Categories: math.PRWe prove that in dimension $d\le 3$ a modified density field of a stirring dynamics perturbed by a voter model converges to the stochastic heat equation.
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arXiv:2006.04214 (Published 2020-06-07)
Metastable Behavior of reversible, Critical Zero-Range Processes
Categories: math.PRWe prove that the position of the condensate of reversible, critical zero-range processes on a finite set $S$ evolves, in a suitable time scale, as a continuous-time Markov chain on $S$ whose jump rates are proportional to the capacities of the underlying random walk which describes the jumps of particles in the zero-range dynamics.
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arXiv:2006.00583 (Published 2020-05-31)
On hydrodynamic limits in Sinai-type random environments
Comments: 30 pagesWe investigate the hydrodynamical behavior of a system of random walks with zero-range interactions moving in a common `Sinai-type' random environment on a one dimensional torus. The hydrodynamic equation found is a quasilinear SPDE with a `rough' random drift term coming from a scaling of the random environment and a homogenization of the particle interaction. Part of the motivation for this work is to understand how the space-time limit of the particle mass relates to that of the known single particle Brox diffusion limit. In this respect, given the hydrodynamic limit shown, we describe formal connections through a two scale limit.
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arXiv:1806.07631 (Published 2018-06-20)
Metastability of the two-dimentional Blume-Capel model with zero chemical potential and small magnetic field on a large torus
Categories: math.PRWe consider the Blume-Capel model with zero chemical potential and small magnetic field in a two-dimensional torus whose length increaseswith the inverse of the temeprature. We prove the mestastable behavior and that starting from a configuration with only negative spins, the process visits the configuration with only 0-spins on its way to the ground state which is the configuration with all spins equal to +1.
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arXiv:1607.01925 (Published 2016-07-07)
Metastability of non-reversible mean-field Potts model with three spins
Comments: 34 pages, 12 figuresWe examine a non-reversible, mean-field Potts model with three spins on a set with $N\uparrow\infty$ points. Without an external field, there are three critical temperatures and five different metastable regimes. The analysis can be extended by a perturbative argument to the case of small external fields. We illustrate the case of large external fields with some phenomena which are not present in the absence of external field.
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arXiv:1606.07227 (Published 2016-06-23)
Static large deviations for a reaction-diffusion model
Comments: 40 pagesCategories: math.PRWe consider the superposition of a symmetric simple exclusion dynamics, speeded-up in time, with a spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We prove the large deviations principle for the empirical measure under the stationary state. We deduce from this result that the stationary state is concentrated on the stationary solutions of the hydrodynamic equation which are stable.
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arXiv:1605.01009 (Published 2016-05-03)
Metastability of non-reversible random walks in a potential field, the Eyring-Kramers transition rate formula
Comments: 48 pages, 5 figuresWe consider non-reversible random walks evolving on a potential field in a bounded domain of $\mathbb{R}^d$. We describe the complete metastable behavior of the random walk among the landscape of valleys, and we derive the Eyring-Kramers formula for the mean transition time from a metastable set to a stable set.
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arXiv:1508.07818 (Published 2015-08-31)
Hydrostatics and dynamical large deviations for a reaction-diffusion model
Categories: math.PRWe consider the superposition of a symmetric simple exclusion dynamics, speeded-up in time, with a spin-flip dynamics in a one-dimensional interval with periodic boundary conditions. We prove the hydrostatics and the dynamical large deviation principle.
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arXiv:1504.05771 (Published 2015-04-22)
Nonequilibrium fluctuations of one-dimensional boundary driven weakly asymmetric exclusion processes
Categories: math.PR, cond-mat.stat-mechWe prove the nonequilibrium fluctuations of one-dimensional, boundary driven, weakly asymmetric exclusion processes through a microscopic Cole-Hopf transformation.
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arXiv:1305.2867 (Published 2013-05-13)
Entropy of non-equilibrium stationary measures of boundary driven TASEP
Comments: 42 pagesKeywords: boundary driven tasep, non-equilibrium stationary measures, equilibrium stationary state converges, asymmetric simple exclusion processes, driven totally asymmetric simpleTags: journal articleWe examine the entropy of non-equilibrium stationary states of boundary driven totally asymmetric simple exclusion processes. As a consequence, we obtain that the Gibbs-Shannon entropy of the non equilibrium stationary state converges to the Gibbs-Shannon entropy of the local equilibrium state. Moreover, we prove that its fluctuations are Gaussian, except when the mean displacement of particles produced by the bulk dynamics agrees with the particle flux induced by the density reservoirs in the maximal phase regime.
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arXiv:1111.2445 (Published 2011-11-10)
A Dirichlet principle for non reversible Markov chains and some recurrence theorems
Categories: math.PRWe extend the Dirichlet principle to non-reversible Markov processes on countable state spaces. We present two variational formulas for the solution of the Poisson equation or, equivalently, for the capacity between two disjoint sets. As an application we prove a some recurrence theorems. In particular, we show the recurrence of two-dimensional cycle random walks under a second moment condition on the winding numbers.
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arXiv:1011.1199 (Published 2010-11-04)
Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear rate zero-range processes
Comments: 27 pagesSubjects: 60K35Nonequilibrium fluctuations of a tagged, or distinguished particle in a class of one dimensional mean-zero zero-range systems with sublinear, increasing rates are derived. In Jara-Landim-Sethuraman (2009), processes with at least linear rates are considered. A different approach to establish a main "local replacement" limit is required for sublinear rate systems, given that their mixing properties are much different. The method discussed also allows to capture the fluctuations of a "second-class" particle in unit rate, symmetric zero-range models.