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

2012/04/26 by Cláudio Landim, Landim, C. · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #60K35 82C20 82B26 #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1204.5987

openalex publication_date 2012/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \bb T_L = \bb Z/L \bb Z be the one-dimensional torus with L points. For α>0, let g: \bb N→ \bb R₊ be given by g(0)=0, g(1)=1, g(k) = [k/(k-1)]^α, k≥ 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∞, 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 \citebl3 for reversible zero-range processes on finite sets.

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• A Martingale approach to metastability

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