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

Abstract

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 finite number accumulate on one single site. We show in this article that in the time scale $N^{1+\alpha }$ 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.

References in corpus (1)

• Dynamics of the condensate in zero-range processes

Cited by in corpus (1)

• Condensation in the inhomogeneous zero-range process: an interplay between interaction and diffusion disorder