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Condensation in stochastic lattice gases with size-dependent stationary weights

Joshua Blank, Paul Chleboun, Stefan Grosskinsky, Watthanan Jatuviriyapornchai

Abstract

We consider stochastic lattice gases with stationary product weights and a polynomial perturbation vanishing with the system size that leads to condensation. If the density of particles exceeds a critical value the system phase separates into a bulk with homogeneous distribution of particles and a condensed phase. Depending on parameter values, the latter consists of a single macroscopic cluster or a diverging number of independent clusters on a smaller scale. We establish the condensation transition via the equivalence of ensembles and the main novelty is a derivation of the cluster size distribution using size-biased sampling, generalizing previous work on zero-range and inclusion processes. Simulations of zero-range processes illustrate our theoretical results on the condensate scale and size distribution.

Condensation in stochastic lattice gases with size-dependent stationary weights

Abstract

We consider stochastic lattice gases with stationary product weights and a polynomial perturbation vanishing with the system size that leads to condensation. If the density of particles exceeds a critical value the system phase separates into a bulk with homogeneous distribution of particles and a condensed phase. Depending on parameter values, the latter consists of a single macroscopic cluster or a diverging number of independent clusters on a smaller scale. We establish the condensation transition via the equivalence of ensembles and the main novelty is a derivation of the cluster size distribution using size-biased sampling, generalizing previous work on zero-range and inclusion processes. Simulations of zero-range processes illustrate our theoretical results on the condensate scale and size distribution.
Paper Structure (10 sections, 7 theorems, 74 equations, 4 figures)

This paper contains 10 sections, 7 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The model and main results
  3. Definitions
  4. Main results
  5. Examples of lattice gases
  6. Proofs of the main results
  7. Proof of Theorem \ref{['cluster']}
  8. Preparation for the Proof of Lemma \ref{['corollary_quotients']}
  9. Scale of $Z_{L,N}$ and Proof of Lemma \ref{['corollary_quotients']}
  10. Proof of Theorem \ref{['thm3']}

Key Result

Theorem 1

Consider a lattice gas with stationary product weights $w_L$ as defined in weights and wmoments. Then for any finite set of sites $\Delta\subset\Lambda$ (for $L$ large enough), denoting $\pi_{L,N}^\Delta$ the marginal of $\pi_{L,N}$ on $\Delta$, we have in the thermodynamic limit thermo with $N/L\to where $\rho_c$ and $R(\phi)$ are given in wmoments and zrlim, respectively.

Figures (4)

  • Figure 1: Phase diagram of lattice gases with stationary product weights of the form \ref{['weights']}. Rigorous results of Theorem \ref{['cluster']} and Theorem \ref{['thm3']} apply in the red regions (excluding the transition line) and additional results bib_CGG22 for inclusion-type models hold for $\kappa =-1$, including a hierarchical Poisson-Dirichlet distribution PD$(\alpha )$ at the transition point (cf. Section \ref{['sec_ips']}).
  • Figure 2: Accumulated density profiles $H_{\eta}(k) = \sum_{x=1}^k \eta_{x}$ on $\Lambda = \{1, \dots\ L\}$ for two independent realizations $\eta ,\eta'$ of $\pi_{L,N}$ with weights $w_L$ given in \ref{['weights']} and parameters $L = 512$, $N=1024$, $\kappa = 0.5$, $\gamma =1$ and $\theta = 0.1$. Bulk weights are $w(n)=2^{-(n+1)}$ and typical profiles show various clusters with a background at density $\rho_c =1$\ref{['wmoments']}.
  • Figure 3: Size-biased empirical tail distribution $\bar{\pi}_{L,N}(s)=\frac{1}{N}\, \sum_{x \in \Lambda} \eta_x \mathbbm{1}\{\eta_x > C_L s\}$ on scale $C_L$ averaged over $48$ realizations of $\pi_{L,N}$ with weights $w_L$ given in \ref{['weights']}, parameters $L = 512$, $\gamma =1$ and $\theta = 0.1$ and bulk weights $w(n)=2^{-(n+1)}$. The tails illustrate the result in Theorem \ref{['cluster']} and are compared to predicted asymptotic Gamma distributions, corrected for finite-size effects by using $R_L (1)$\ref{['rl']} truncated at $n\leq\rho L$ instead of $\rho_c =1$\ref{['weights']}. Note that finite-size effects are significantly larger for $\kappa =1$ with cluster scale $C_L \propto L^{1/3}$ than for $\kappa =0$ with $C_L \propto L^{1/2}$.
  • Figure 4: Size-biased empirical tail distribution $\bar{\pi}_{L,N}(s)=\frac{1}{N}\, \sum_{x \in \Lambda} \eta_x \mathbbm{1}\{\eta_x > C_L s\}$ on scale $C_L =(\rho -\rho_c )L$ averaged over $16$ realizations of $\pi_{L,N}$ each with weights $w_L$ given in \ref{['weights']}, parameters $L = 512$, $\kappa =-1$, $\gamma =2$ and $\theta = 0.1$ and bulk weights $w(n)=2^{-(n+1)}$. The tails illustrate the result in Theorem \ref{['thm3']} and are compared to the predicted asymptotic step function for the tail. The fraction of mass in the condensate is $(\rho -\rho_c )/\rho =0.75$ and $0.5$, respectively, denoted by dashed red lines.

Theorems & Definitions (12)

  • Theorem 1: Equivalence of ensembles
  • proof
  • Theorem 2: Distribution of the condensate I
  • Theorem 3: Distribution of the condensate II
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Remark 1
  • ...and 2 more