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C. Landim

Publications and source records attributed to C. Landim.

At least 37 records · Page 2Linked to original sources
arXiv2015

Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus

We consider the Kawasaki dynamics at inverse temperature $β$ for the Ising lattice gas on a two-dimensional square of length $2L+1$ with periodic boundary conditions. We assume that initially the particles form a square of length $n$, which may increase, as well as $L$, with $β$. We show that in a proper time scale the particles form almost always a square and that the center of mass of the square evolves as a Brownian motion when the temperature vanishes.

math.PR↗
arXiv2015

A martingale problem for an absorbed diffusion: the nucleation phase of condensing zero range processes

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.

math.PR↗
arXiv2014

A topology for limits of Markov chains

In the investigation of limits of Markov chains, the presence of states which become instantaneous states in the limit may prevent the convergence of the chain in the Skorohod topology. We present in this article a weaker topology adapted to handle this situation. We use this topology to derive the limit of random walks among random traps and sticky zero-range processes.

math.PR↗
arXiv2014

Metastability of reversible random walks in potential fields

Let $Ξ$ be an open and bounded subset of $\bb R^d$, and let $F:Ξ\to\bb R$ be a twice continuously differentiable function. Denote by $Ξ_N$ th discretization of $Ξ$, $Ξ_N = Ξ\cap (N^{-1} \bb Z^d)$, and denote by $X_N(t)$ the continuous-time, nearest-neighbor, random walk on $Ξ_N$ 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 $F$.

math.PR↗
arXiv2013

A Martingale approach to metastability

We presented in \cite{bl2,bl7} an approach to derive the metastable behavior of continuous-time Markov chains. We assumed in these articles that the Markov chains visit points in the time scale in which it jumps among the metastable sets. We replace this condition here by assumtpions on the mixing times and on the relaxation times of the chains reflected at the boundary of the metastable sets.

math.PR↗
arXiv2013

Tunneling of the Kawasaki dynamics at low temperatures in two dimensions

Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature $β$ on a two dimensional torus $Λ_L=\{0,..., L-1\}^2$ . 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 $e^{2β}$ the process is close to a Markov process on $Λ_L$ which jumps from any site $\mb x$ to any other site $\mb y\not =\mb x$ at a strictly positive rate which can be expressed in terms of the jump rates of simple random walks.

math.PR↗
arXiv2012

Clausius inequality and optimality of quasi static transformations for nonequilibrium stationary states

Nonequilibrium stationary states of thermodynamic systems dissipate a positive amount of energy per unit of time. If we consider transformations of such states that are realized by letting the driving depend on time, the amount of energy dissipated in an unbounded time window becomes then infinite. Following the general proposal by Oono and Paniconi and using results of the macroscopic fluctuation theory, we give a natural definition of a renormalized work performed along any given transformation. We then show that the renormalized work satisfies a Clausius inequality and prove that equality is achieved for very slow transformations, that is in the quasi static limit. We finally connect the renormalized work to the quasi potential of the macroscopic fluctuation theory, that gives the probability of fluctuations in the stationary nonequilibrium ensemble.

cond-mat.stat-mech↗
arXiv2012

Thermodynamic transformations of nonequilibrium states

We consider a macroscopic system in contact with boundary reservoirs and/or under the action of an external field. We discuss the case in which the external forcing depends explicitly on time and drives the system from a nonequilibrium state to another one. In this case the amount of energy dissipated along the transformation becomes infinite when an unbounded time window is considered. Following the general proposal by Oono and Paniconi and using results of the macroscopic fluctuation theory, we give a natural definition of a renormalized work. We then discuss its thermodynamic relevance by showing 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. The latter result provides a characterization of the quasi potential that does not involve rare fluctuations.

cond-mat.stat-mech↗
arXiv2012

Universality of trap models in the ergodic time scale

Consider a sequence of possibly random graphs $G_N=(V_N, E_N)$, $N\ge 1$, whose vertices's have i.i.d. weights $\{W^N_x : x\in V_N\}$ with a distribution belonging to the basin of attraction of an $α$-stable law, $0<α<1$. Let $X^N_t$, $t \ge 0$, be a continuous time simple random walk on $G_N$ which waits a \emph{mean} $W^N_x$ exponential time at each vertex $x$. Under considerably general hypotheses, we prove that in the ergodic time scale this trap model converges in an appropriate topology to a $K$-process. We apply this result to a class of graphs which includes the hypercube, the $d$-dimensional torus, $d\ge 2$, random $d$-regular graphs and the largest component of super-critical Erdös-Rényi random graphs.

math.PR↗
arXiv2012

A Markovian growth dynamics on rooted binary trees evolving according to the Gompertz curve

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 $β>0$. 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 $β(n-k)/n$, 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 $t$ and let $T_n = β^{-1} n \ln(n /\ln 4) + (\ln 2)/(2 β)$. We prove that $2^{-n} Z_n(T_n + n τ)$, $τ\in\bb R$, converges to the Gompertz curve $\exp (- (\ln 2) e^{-βτ})$. We also prove a central limit theorem for the martingale associated to $Z_n(t)$.

q-bio.CB↗
arXiv2012

Metastability for a non-reversible dynamics: the evolution of the condensate in totally asymmetric zero range processes

Let $\bb T_L = \bb Z/L \bb Z$ be the one-dimensional torus with $L$ points. For $α>0$, let $g: \bb N\to \bb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k) = [k/(k-1)]^α$, $k\ge 2$. 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 $α>1$, as $N\uparrow\infty$, all particles but a finite number accumulate on one single site. We show in this article that in the time scale $N^{1+α}$ the site which concentrates almost all particles evolves as a random walk on $\bb T_L$ whose transition rates are proportional to the capacities of the underlying random walk, extending to the asymmetric case the results obtained in \cite{bl3} for reversible zero-range processes on finite sets.

math.PR↗
arXiv2010

Lagrangian phase transitions in nonequilibrium thermodynamic systems

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 exclusion process when the drift due to the external field is opposite to the one due to the external reservoirs, and strong enough.

cond-mat.stat-mech↗
arXiv2009

Quenched scaling limits of trap models

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, $W=\sum_{i\ge 1} w_i δ_{x_i}$, 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}.

math.PR↗
arXiv2008

Towards a nonequilibrium thermodynamics: a self-contained macroscopic description of driven diffusive systems

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 experimentally accessible. The basic postulates are local equilibrium which allows a hydrodynamic description of the evolution, the Einstein relation among the transport coefficients, and a variational principle defining the out of equilibrium free energy. Associated to the variational principle there is a Hamilton-Jacobi equation satisfied by the free energy, very useful for concrete calculations. Correlations over a macroscopic scale are, in our scheme, a generic property of nonequilibrium states. Correlation functions of any order can be calculated from the free energy functional which is generically a non local functional of thermodynamic variables. Special attention is given to the notion of equilibrium state from the standpoint of nonequilibrium.

cond-mat.stat-mech↗
arXiv2008

Meta-stability and condensed zero-range processes on finite sets

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.

math.PR↗