Metastable Behavior of Reversible, Critical Zero-Range Processes

Abstract

We 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.

A Resolvent Approach to Metastability

“…Furthermore, in the last part of the article, we show that these conditions are in force for reversible, critical zero-range dynamics. In particular, we are able to extend the results presented in [33] to the case in which the process starts from a configuration instead of a measure spread over a well E x N . Comments.…”

“…The metastable behavior of condensing zero-range processes has a long history [8,31,2,44,41,33]. The critical case, examined here and in [33], presents a major difference with respect to the super-critical case considered before.…”

“…The metastable behavior of condensing zero-range processes has a long history [8,31,2,44,41,33]. The critical case, examined here and in [33], presents a major difference with respect to the super-critical case considered before. While in the super-critical case, when entering a well, the process visits all its configurations before visiting a new well, this is no longer true in the critical case.…”

“…We apply it, in the second part of the article, to derive the metastable behavior of critical zero-range processes starting from points. This is a model which does not satisfy the conditions of [6], and whose metastable behavior could only be derived when the process starts from measures spread over a well [33].…”

“…Replacing the Poisson equation with resolvent equations has a significant advantage, as the solutions of the later equation are bounded. It permits, in particular, to prove L ∞ estimates instead of the L 2 estimates derived in [33]. This, in turn, allows to start the process from a fixed configuration instead of a measure spread over the sets E x N .…”