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.

Appendix: Local Limit Theorems

In this appendix we state relevant limit theorems for triangular arrays of independent non identical random variables that are key to results on the equivalence of ensembles for spatially inhomogeneous systems.

Details and a proof of the Lindeberg-Feller central limit theorem can be found in, for example, [77]. The local central limit theorem can be found in [47] and [55]. For each L, let ξ _x,L , 1≤x≤L, be independent non identical random variables whose law depends on the x and the number of random variables L (a triangular array of random variables).

Theorem A.1

(The Lindeberg-Feller central limit theorem)

Suppose $\mathbb{E}[\xi_{x,L}] = 0$, and

• (i) $\sum_{x=1}^{L} \mathbb{E}[\xi_{x,L}^{2}] \to1$ as L→∞.

• (ii) For all ϵ>0, $\sum_{x=1}^{L}\mathbb {E} [\vert \xi_{x,L}\vert ^{2}{\bf1}_{\vert\xi _{x,L}\vert> \epsilon} ] \to0$ as n→∞.

Then $\sum_{x=1}^{L}\xi_{x,L}$ converges in distribution to the standard normal as n→∞.

For example we typically apply the central limit theorem to the centered and standardized sum of the single site occupations under the grand canonical measures, i.e.

$$\xi_{x,L} = \frac{\eta_x-R_x(\phi)}{\sqrt{L\,\sum_y\textrm {Var}_\phi(\eta_y)}} $$

where η _x has law $\nu^{x}_{\phi}$ not depending on the system size L. Then (i) follows by definition and (ii) follows by a dominated convergence argument if the second moments are uniformly bounded.

Now we assume that η _x,L , 1≤x≤L, is a triangular array of independent integer valued random variables where η _x,L has law P _x,L . The Bernoulli part decomposition of the random variables η _x,L is expressed in terms of,

$$q(P_{x,L}) = \sum_{n} \bigl( P_{x,L}[n] \wedge P_{x,L}[n+1] \bigr). $$

Define $Q_{L} = \sum_{x=1}^{L} q(P_{x,L})$.

Theorem A.2

(Local central limit theorem [47])

Let $\varSigma_{L} = \sum_{x=1}^{L}\eta_{x,L}$. Suppose there exist sequences B _L >0 and A _L , L≥1, such that B _L →∞, $\limsup B^{2}_{L}/Q_{L} < \infty$, and (Σ _L −A _L )/B _L converges in distribution to the standard normal. Then,

$$\begin{aligned} \sup_{n}\biggl \vert B_L P[ \varSigma_L = n] - \varPhi \biggl(\frac {n-A_L}{B_L} \biggr)\biggr \vert \to0 \quad\textrm{as} \ L\to \infty, \end{aligned}$$

(97)

where Φ is the standard normal density.

An alternative form of the local limit theorem can be found in [55]. In our cases A _L ∼L and $B_{L} \sim\sqrt{L}$, and the main condition is to show that Q _L ∼L. In fact, using the structure of the marginals (ϕλ _x )ⁿ w _x (n) with uniform regularity of the w _x (n) (22) it is easy to see that in fact Q _L ∼L.