Metastability of reversible condensed zero range processes on a finite set

Abstract

Abstract. Let r : S × S → R + be the jump rates of an irreducible random walk on a finite set S, reversible with respect to some probability measure m.Consider a zero range process on S in which a particle jumps from a site x, occupied by k particles, to a site y at rate g(k)r(x, y). Let N stand for the total number of particles. In the stationary state, as N ↑ ∞, 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 S whose transition rates are proportional to the capacities of the underlying random walk.

Emergent motion of condensates in mass-transport models

“…In the usual asymmetric ZRP [ Fig. 1(a)], it is known that the condensate is static up to time scales of order L b and then it relocates to a random site due to fluctuations [14,[18][19][20]. There is a striking qualitative difference in the dynamics of model (1) and (2) …”

“…In the ZRP, where condensation takes place when a macroscopic fraction of particles occupies a single site, the resulting condensate does not drift in the thermodynamic limit [14,[18][19][20]. It is shown below that this is related to the fact that the steady state of the ZRP is a product measure.…”

Condensation for random variables conditioned by the value of their sum

“…The contribution of the dip region is of crucial importance for the analysis of the stationary dynamics of the condensate [12]. The conclusions given in [12] have been confirmed by rigorous mathematical studies [27,28,29,30].…”

Metastable Markov chains

“…Dynamics which display this behavior are said to "visit points". This class includes condensing zero-range processes [16,84,4,116], random walks in a potential field [91,92,93] or models in which the valleys are singletons as the inclusion process [25] or random walks evolving among random traps [63,62,77,78], but it does not contain the example of Section 1. For such dynamics, in which the entropy plays a role in the metastable behavior, a different approach is needed.…”