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

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 ${\mathcal{P}}_T(\bar{n}=aN)$ that the average number of balls in one urn over time $T$, $\bar{n}$, takes any specified value $aN$, where $0\le a\le 1$. For long observation time, $T\to \mathrm{\infty }$, 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 $I(a,N,K, \dots)$isexactforany$N$. 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 ${\mathcal{P}}_T(\bar{n}=aN)$.