Hydrodynamics of the generalized $N$-urn Ehrenfest model

TL;DR

This paper analyzes a generalized N-urn Ehrenfest model, deriving its hydrodynamic limit, non-equilibrium fluctuations, and large deviation principles, revealing the macroscopic behavior and fluctuations of the system under scaling.

Contribution

It introduces a hydrodynamic limit, fluctuation analysis, and large deviation principles for a generalized N-urn Ehrenfest model with independent random walks.

Findings

Empirical measure converges to a deterministic density governed by an integral equation.

Non-equilibrium fluctuations follow a measure-valued generalized Ornstein-Uhlenbeck process.

Large deviation principle established under product form transition rates.

Abstract

In this paper we are concerned with a generalized $N$-urn Ehrenfest model, where balls keeps independent random walks between $N$ boxes uniformly laid on $[0, 1]$. After a proper scaling of the transition rates function of the aforesaid random walk, we derive the hydrodynamic limit of the model, i.e., the law of large numbers which the empirical measure of the model follows, under an assumption where the initial number of balls in each box independently follows a Poisson distribution. We show that the empirical measure of the model converges weakly to a deterministic measure with density driven by an integral equation. Furthermore, we derive non-equilibrium fluctuation of the model, i.e, the central limit theorem from the above hydrodynamic limit. We show that the non-equilibrium fluctuation of the model is driven by a measure-valued time-inhomogeneous generalized O-U process. At last, we prove a large deviation principle from the hydrodynamic limit under an assumption where the transition rates function from $[0, 1]\times [0, 1]$ to $[0, +\infty)$ of the aforesaid random walk is a product of two marginal functions from $[0, 1]$ to $[0, +\infty)$.