Large deviations of a long-time average in the Ehrenfest Urn Model

TL;DR

This paper analyzes large deviations in the Ehrenfest urn model, deriving exact rate functions for the probability of observing atypical long-time averages, using Donsker-Varadhan theory and WKB approximation, with and without interactions.

Contribution

It provides exact calculations of large deviation rate functions for the Ehrenfest urn model, including interactions, and compares two analytical methods for these calculations.

Findings

Exact rate functions for large deviations derived

WKB approximation is asymptotically exact for large N

Time history analysis reveals dominant system trajectories

Abstract

Since its inception in 1907, the Ehrenfest urn model (EUM) has served as a test bed of key concepts of statistical mechanics. Here we employ this model to study large deviations of a time-additive quantity. We consider two continuous-time versions of the EUM with $K$ urns and $N$ balls: without and with interactions between the balls in the same urn. We evaluate the probability distribution $\mathcal{P}_T(\bar{n}= a N)$ that the average number of balls in one urn over time $T$, $\bar{n}$, takes any specified value $aN$, where $0\leq a\leq 1$. For long observation time, $T\to \infty$, a Donsker-Varadhan large deviation principle holds: $-\ln \mathcal{P}_T(\bar{n}= a N) \simeq T I(a,N,K,\dots)$, where $\dots$ denote additional parameters of the model. We calculate the rate function $I(a,N,K, \dots)$ exactly by two different methods due to Donsker and Varadhan and compare the exact results with those obtained with a variant of WKB approximation (after Wentzel, Kramers and Brillouin). In the absence of interactions the WKB prediction for $I(a,N,K, \dots)$ is exact for any $N$. In the presence of interactions the WKB method gives asymptotically exact results for $N\gg 1$. The WKB method also uncovers the (very simple) time history of the system which dominates the contribution of different time histories to $\mathcal{P}_T(\bar{n}= a N)$.