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2.0

Jun 28, 2018
06/18

by
C. Landim; P. Lemire

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We consider the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field evolving on a large but finite torus. We obtain sharp estimates for the transition time, we characterize the set of critical configurations, and we prove the metastable behavior of the dynamics as the temperature vanishes.

Topics: Probability, Statistical Mechanics, Mathematics, Condensed Matter

Source: http://arxiv.org/abs/1512.09286

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6.0

Jun 30, 2018
06/18

by
C. Landim; M. Loulakis; M. Mourragui

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We consider continuous-time Markov chains which display a family of wells at the same depth. We show that in an appropriate time-scale the state of the process can be represented as a time-dependent convex combination of mestastable states, each of which is supported on one well. The time dependence of the convex combination is given in terms of the distribution of a reduced Markov chain.

Topics: Probability, Statistical Mechanics, Condensed Matter, Mathematics

Source: http://arxiv.org/abs/1703.09481

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10.0

Jun 28, 2018
06/18

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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In the context of driven diffusive systems, for thermodynamic transformations over a large but finite time window, we derive an expansion of the energy balance. In particular, we characterize the transformations which minimize the energy dissipation and describe the optimal correction to the quasi-static limit. Surprisingly, in the case of transformations between homogeneous equilibrium states of an ideal gas, the optimal transformation is a sequence of inhomogeneous equilibrium states.

Topics: Statistical Mechanics, Condensed Matter

Source: http://arxiv.org/abs/1506.05691

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4.0

Jun 30, 2018
06/18

by
C. Landim; R. Misturini; K. Tsunoda

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Let $\Xi$ be an open and bounded subset of $\bb R^d$, and let $F:\Xi\to\bb R$ be a twice continuously differentiable function. Denote by $\Xi_N$ th discretization of $\Xi$, $\Xi_N = \Xi \cap (N^{-1} \bb Z^d)$, and denote by $X_N(t)$ the continuous-time, nearest-neighbor, random walk on $\Xi_N$ which jumps from $\bs x$ to $\bs y$ at rate $ e^{-(1/2) N [F(\bs y) - F(\bs x)]}$. We examine in this article the metastable behavior of $X_N(t)$ among the wells of the potential $F$.

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1408.6704

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Jul 24, 2013
07/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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In this paper we present a self-contained macroscopic description of diffusive systems interacting with boundary reservoirs and under the action of external fields. The approach is based on simple postulates which are suggested by a wide class of microscopic stochastic models where they are satisfied. The description however does not refer in any way to an underlying microscopic dynamics: the only input required are transport coefficients as functions of thermodynamic variables, which are...

Source: http://arxiv.org/abs/0807.4457v2

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Sep 18, 2013
09/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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Experiments show that macroscopic systems in a stationary nonequilibrium state exhibit long range correlations of the local thermodynamic variables. In previous papers we proposed a Hamilton-Jacobi equation for the nonequilibrium free energy as a basic principle of nonequilibrium thermodynamics. We show here how an equation for the two point correlations can be derived from the Hamilton-Jacobi equation for arbitrary transport coefficients for dynamics with both external fields and boundary...

Source: http://arxiv.org/abs/0705.2996v1

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45

Sep 17, 2013
09/13

by
C. Landim; R. M. Sued; G. Valle

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We consider the asymmetric exclusion process. We start from a profile which is constant along the drift direction and prove that the density profile, under a diffusive rescaling of time, converges to the solution of a parabolic equation.

Source: http://arxiv.org/abs/math/0305398v1

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Jun 27, 2018
06/18

by
J. Beltrán; M. Jara; C. Landim

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We prove uniqueness of a martingale problem with boundary conditions on a simplex associated to a differential operator with an unbounded drift. We show that the solution of the martingale problem remains absorbed at the boundary once it attains it, and that, after hitting the boundary, it performs a diffusion on a lower dimensional simplex, similar to the original one. We also prove that in the diffusive time scale condensing zero-range processes evolve as this absorbed diffusion.

Topics: Mathematics, Probability

Source: http://arxiv.org/abs/1505.00980

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3.0

Jun 28, 2018
06/18

by
C. Landim; T. Xu

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Consider a sequence $(\eta^N(t) :t\ge 0)$ of continuous-time, irreducible Markov chains evolving on a fixed finite set $E$, indexed by a parameter $N$. Denote by $R_N(\eta,\xi)$ the jump rates of the Markov chain $\eta^N_t$, and assume that for any pair of bonds $(\eta,\xi)$, $(\eta',\xi')$ $\arctan \{R_N(\eta,\xi)/R_N(\eta',\xi')\}$ converges as $N\uparrow\infty$. Under a hypothesis slightly more restrictive (cf. \eqref{mhyp} below), we present a recursive procedure which provides a sequence...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1512.06597

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56

Sep 19, 2013
09/13

by
M. D. Jara; C. Landim; S. Sethuraman

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We prove a non-equilibrium functional central limit theorem for the position of a tagged particle in mean-zero one-dimensional zero-range process. The asymptotic behavior of the tagged particle is described by a stochastic differential equation governed by the solution of the hydrodynamic equation.

Source: http://arxiv.org/abs/math/0703226v1

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Sep 21, 2013
09/13

by
A. Faggionato; M. Jara; C. Landim

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Consider a system of particles performing nearest neighbor random walks on the lattice $\ZZ$ under hard--core interaction. The rate for a jump over a given bond is direction--independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an $\a$--stable law, $0

Source: http://arxiv.org/abs/0709.0306v1

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Sep 23, 2013
09/13

by
Cedric Bernardin; C. Landim

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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.

Source: http://arxiv.org/abs/1105.0494v1

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104

Jul 20, 2013
07/13

by
O. Benois; C. Landim; M. Mourragui

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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.

Source: http://arxiv.org/abs/1304.0805v1

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Sep 22, 2013
09/13

by
C. Landim; J. Quastel; M. Salmhofer; H. T. Yau

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We prove that the diffusion coefficient for the asymmetric exclusion process diverges at least as fast as $t^{1/4}$ in dimension $d=1$ and $(\log t)^{1/2}$ in $d=2$. The method applies to nearest and non-nearest neighbor asymmetric exclusion processes.

Source: http://arxiv.org/abs/math/0201317v1

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Sep 18, 2013
09/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona--Lasinio; C. Landim

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This paper provides an introduction to some stochastic models of lattice gases out of equilibrium and a discussion of results of various kinds obtained in recent years. Although these models are different in their microscopic features, a unified picture is emerging at the macroscopic level, applicable, in our view, to real phenomena where diffusion is the dominating physical mechanism. We rely mainly on an approach developed by the authors based on the study of dynamical large fluctuations in...

Source: http://arxiv.org/abs/0705.1247v1

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7.0

Jun 28, 2018
06/18

by
E. Chavez; C. Landim

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We consider a one-dimensional, weakly asymmetric, boundary driven exclusion process on the interval $[0,N]\cap Z$ in the super-diffusive time scale $N^2 \epsilon^{-1}_N$, where $1\ll \epsilon^{-1}_N \ll N^{1/4}$. We assume that the external field and the chemical potentials, which fix the density at the boundaries, evolve smoothly in the macroscopic time scale. We derive an equation which describes the evolution of the density up to the order $\epsilon_N$.

Topics: Mathematics, Statistical Mechanics, Condensed Matter, Probability

Source: http://arxiv.org/abs/1509.05801

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36

Sep 23, 2013
09/13

by
B. Derrida; C. Enaud; C. Landim; S. Olla

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We investigate the fluctuations around the average density profile in the weakly asymmetric exclusion process with open boundaries in the steady state. We show that these fluctuations are given, in the macroscopic limit, by a centered Gaussian field and we compute explicitly its covariance function. We use two approaches. The first method is dynamical and based on fluctuations around the hydrodynamic limit. We prove that the density fluctuations evolve macroscopically according to an autonomous...

Source: http://arxiv.org/abs/cond-mat/0511275v1

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Sep 18, 2013
09/13

by
C. Landim; J. A. Ramirez; H. -T. Yau

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It was proved \cite{EMYa, QY} that stochastic lattice gas dynamics converge to the Navier-Stokes equations in dimension $d=3$ in the incompressible limits. In particular, the viscosity is finite. We proved that, on the other hand, the viscosity for a two dimensional lattice gas model diverges faster than $\log \log t$. Our argument indicates that the correct divergence rate is $(\log t)^{1/2}$. This problem is closely related to the logarithmic correction of the time decay rate for the velocity...

Source: http://arxiv.org/abs/math/0505090v1

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3.0

Jun 30, 2018
06/18

by
C. Landim; M. Mariani; I. Seo

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We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles to provide a sharp estimate for the transition times between two different wells for non-reversible diffusion processes. This estimate permits to describe the metastable behavior of the system.

Topics: Probability, Statistical Mechanics, Condensed Matter, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1701.00985

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104

Sep 18, 2013
09/13

by
J. Beltrán; C. Landim

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We recover the Navier-Stokes equation as the incompressible limit of a stochastic lattice gas in which particles are allowed to jump over a mesoscopic scale. The result holds in any dimension assuming the existence of a smooth solution of the Navier-Stokes equation in a fixed time interval. The proof does not use non-gradient methods or the multi-scale analysis due to the long range jumps.

Source: http://arxiv.org/abs/math/0611721v1

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91

Jul 22, 2013
07/13

by
C. Landim; R. D. Portugal; B. F. Svaiter

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Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields model. Fix $n\ge 1$ and $\beta>0$. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate $\beta(n-k)/n$, where $k$ is the distance from the node to the root. Denote by $Z_n(t)$ the number of nodes with no descendants at time $t$...

Source: http://arxiv.org/abs/0807.1750v2

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49

Sep 19, 2013
09/13

by
P. A. Ferrari; C. Landim; H. Thorisson

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We give a deterministic algorithm to construct a graph with no loops (a tree or a forest) whose vertices are the points of a d-dimensional stationary Poisson process S, subset of R^d. The algorithm is independent of the origin of coordinates. We show that (1) the graph has one topological end --that is, from any point there is exactly one infinite self-avoiding path; (2) the graph has a unique connected component if d=2 and d=3 (a tree) and it has infinitely many components if d\ge 4 (a...

Source: http://arxiv.org/abs/math/0209395v2

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Sep 22, 2013
09/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in non--equilibrium, namely for non reversible systems. In this paper we consider a simple example of a non--equilibrium situation, the symmetric simple exclusion process in which we let the system exchange particles with the boundaries at two different rates. We prove a dynamical large deviation principle for the empirical density which describes the...

Source: http://arxiv.org/abs/cond-mat/0307280v1

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145

Jul 19, 2013
07/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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In previous papers we have introduced a natural nonequilibrium free energy by considering the functional describing the large fluctuations of stationary nonequilibrium states. While in equilibrium this functional is always convex, in nonequilibrium this is not necessarily the case. We show that in nonequilibrium a new type of singularities can appear that are interpreted as phase transitions. In particular, this phenomenon occurs for the one-dimensional boundary driven weakly asymmetric...

Source: http://arxiv.org/abs/1005.1489v3

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32

Sep 21, 2013
09/13

by
M. Jara; C. Landim; A. Teixeira

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Fix a strictly positive measure $W$ on the $d$-dimensional torus $\bb T^d$. For an integer $N\ge 1$, denote by $W^N_x$, $x=(x_1, ..., x_d)$, $0\le x_i 1$, if $W$ is a finite discrete measure, $W=\sum_{i\ge 1} w_i \delta_{x_i}$, we prove that the random walk which jumps from $x/N$ uniformly to one of its neighbors at rate $(W^N_x)^{-1}$ has a metastable behavior, as defined in \cite{bl1}, described by the $K$-process introduced in \cite{fm1}.

Source: http://arxiv.org/abs/0902.3334v1

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59

Sep 20, 2013
09/13

by
J. Beltran; C. Landim

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We propose a definition o meta-stability and obtain sufficient conditions for a sequence of Markov processes on finite state spaces to be meta-stable. In the reversible case, these conditions reduce to estimates of the capacity and the measure of certain meta-stable sets. We prove that a class of condensed zero-range processes with asymptotically decreasing jump rates is meta-stable.

Source: http://arxiv.org/abs/0802.2171v1

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47

Sep 21, 2013
09/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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In this paper we formulate a dynamical fluctuation theory for stationary non equilibrium states (SNS) which covers situations in a nonlinear hydrodynamic regime and is verified explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Our results include the modification of the Onsager-Machlup theory in the SNS, a general Hamilton-Jacobi equation for the macroscopic entropy and a non equilibrium, non linear fluctuation...

Source: http://arxiv.org/abs/cond-mat/0104153v1

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44

Sep 23, 2013
09/13

by
J. Beltrán; C. Landim

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Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature $\beta$ on a two dimensional torus $\Lambda_L=\{0,..., L-1\}^2$ . We prove the tunneling behavior of the process among the states of minimal energy. More precisely, assume that there are $n^2\ll L$ particles and that the initial state is the configuration in which all sites of the square $\mb x + \{0,..., n-1\}^2$ are occupied. We show that in the time scale $e^{2\beta}$ the process is close...

Source: http://arxiv.org/abs/1109.2776v2

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38

Sep 20, 2013
09/13

by
J. Beltrán; C. Landim

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We proposed in \cite{bl2} a new approach to prove the metastable behavior of reversible dynamics based on potential theory and local ergodicity. In this article we extend this theory to nonreversible dynamics based on the Dirichlet principle proved in \cite{gl2}.

Source: http://arxiv.org/abs/1205.0445v1

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Sep 18, 2013
09/13

by
M. D. Jara; C. Landim

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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 $\rho_0:\bb R\to [0,1]$.

Source: http://arxiv.org/abs/math/0505091v1

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Sep 19, 2013
09/13

by
P. A. Ferrari; A. Galves; C. Landim

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We give bounds on the rate of convergence to equilibrium of the symmetric simple exclusion process in $\Z^d$. Our results include the existent results in the literature. We get better bounds and larger class of initial states via a unified approach. The method includes a comparison of the evolution of n interacting particles with n independent ones along the whole time trajectory.

Source: http://arxiv.org/abs/math/9912008v1

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Sep 22, 2013
09/13

by
M. D. Jara; C. Landim

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For a sequence of i.i.d. random variables $\{\xi_x : x\in \bb Z\}$ 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 $\xi_x$. We examine a quenched nonequilibrium central limit theorem for the position of a tagged particle in the exclusion process with bond disorder $\{\xi_x : x\in \bb Z\}$. We prove that the position of the tagged...

Source: http://arxiv.org/abs/math/0603653v1

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Sep 22, 2013
09/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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We generalize to non equilibrium states Onsager's minimum dissipation principle. We also interpret this principle and some previous results in terms of optimal control theory. Entropy production plays the role of the cost necessary to drive the system to a prescribed macroscopic configuration.

Source: http://arxiv.org/abs/cond-mat/0310072v1

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Sep 21, 2013
09/13

by
C. Landim

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Let $\bb T_L = \bb Z/L \bb Z$ be the one-dimensional torus with $L$ points. For $\alpha >0$, let $g: \bb N\to \bb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k) = [k/(k-1)]^\alpha$, $k\ge 2$. Consider the totally asymmetric zero range process on $\bb T_L$ in which a particle jumps from a site $x$, occupied by $k$ particles, to the site $x+1$ at rate $g(k)$. Let $N$ stand for the total number of particles. In the stationary state, if $\alpha >1$, as $N\uparrow\infty$, all particles but a...

Source: http://arxiv.org/abs/1204.5987v1

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Sep 20, 2013
09/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles. More precisely, we establish a large deviation principle for a space-time fluctuation $j$ of the empirical current with a rate functional $\mc I (j)$. We then estimate the probability of a fluctuation of the average current over a large time interval; this probability can be obtained by solving a variational problem for the functional...

Source: http://arxiv.org/abs/cond-mat/0506664v1

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Sep 18, 2013
09/13

by
C. Landim

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We prove Gaussian tail estimates for the transition probability of $n$ particles evolving as symmetric exclusion processes on $\bb Z^d$, improving results obtained in \cite{l}. We derive from this result a non-equilibrium Boltzmann-Gibbs principle for the symmetric simple exclusion process in dimension 1 starting from a product measure with slowly varying parameter.

Source: http://arxiv.org/abs/math/0505089v1

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Sep 21, 2013
09/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.

Source: http://arxiv.org/abs/math/0602557v1

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Sep 17, 2013
09/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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We formulate a dynamical fluctuation theory for stationary non equilibrium states (SNS) which is tested explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Within this theory we derive the following results: the modification of the Onsager-Machlup theory in the SNS; a general Hamilton-Jacobi equation for the macroscopic entropy; a non equilibrium, non linear fluctuation dissipation relation valid for a wide class of...

Source: http://arxiv.org/abs/cond-mat/0108040v2

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Sep 21, 2013
09/13

by
L. Bertini; A. De Sole; D. Gabrielli; G. Jona-Lasinio; C. Landim

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We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach to density fluctuations developed in previous articles. More precisely, we derive large deviation estimates for the space--time fluctuations of the empirical current which include the previous results. Large time asymptotic estimates for the fluctuations of the time average of the current, recently established by Bodineau and Derrida, can be derived in a more general setting. There are models...

Source: http://arxiv.org/abs/cond-mat/0407161v2