Condensation in Stochastic Particle Systems with Stationary Product Measures

Abstract

We study stochastic particle systems with stationary product measures that exhibit a condensation transition due to particle interactions or spatial inhomogeneities. We review previous work on the stationary behaviour and put it in the context of the equivalence of ensembles, providing a general characterization of the condensation transition for homogeneous and inhomogeneous systems in the thermodynamic limit. This leads to strengthened results on weak convergence for subcritical systems, and establishes the equivalence of ensembles for spatially inhomogeneous systems under very general conditions, extending previous results which focused on attractive and finite systems. We use relative entropy techniques which provide simple proofs, making use of general versions of local limit theorems for independent random variables. This paper is dedicated to Herbert Spohn in honour of his 65th birthday.

Zero-range process with finite compartments: Gentile's statistics and glassiness

“…All these findings suggest several courses of action. In particular, it would be interesting to study physical effects induced by the finite site capacities in realm of more general transport models with factorized steady states [3,[35][36][37].…”

Equivalence of Ensembles, Condensation and Glassy Dynamics in the Bose–Hubbard Hamiltonian

“…Next, let us consider the event B = γ ∈ Γ ρ,ε,V : max x∈V γ(x) ≥ (V ∆ε) 1/2 + V α with some 0 < α < 1/2 to be specified later. As before, we use Lemma 3 to get the bound |B| ≤ e cV 1/3 ln V +max p∈B s(p) for some c < +∞, withB = {p ∈ P ρ,ε,V : p = p(γ, ·) for some γ ∈ B} where p(γ, ·) is defined by (7). Given p ∈B, there exists n 0 ≥ (V ∆ε) 1/2 + V α such that p(n 0 ) ≥ 1/V .…”

Emergence of particle clusters in a one-dimensional model: connection to condensation processes

“…, y N ); also β = J/T is the (dimensionless) inverse temperature, the function δ(x) is a Dirac delta, and Z y (β) is the partition function for this representation of the system. Distributions of this form are familiar from zero-range processes and from mass transport models [1,2,5,4]. One sees from (4) that this probability density diverges as y i → 0, and that the distribution will not be normalisable for β ≥ 1.…”

“…For our purposes, observe that if lim →0 Z y (β) = Z y (β) then for any y we have P (y) → P (y) in (4). Physical observables in the system are calculated as averages with respect to P (y): if the observable of interest is bounded in magnitude then this analysis is sufficient to ensure that it converges to a finite limit as → 0.…”

“…Even in much simpler model systems such as exclusion processes or zero-range processes in one dimension, unexpected phenomena continue to surprise physicists and mathematicians, including condensation [1,2,3,4,5,6,7] and unusual fluctuation phenomena [8,9,10,11,12].…”