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: with and without interactions between the balls in the same urn. We evaluate the probability distribution P_T (n = aN) that the average number of balls in one urn over time T, n, takes any specified value aN, where 0 ≤ a ≤ 1. For long observation time, T → ∞, a Donsker-Varadhan large deviation principle holds: -ln P_T (n = aN) ≃ TI(a,N,K, ⋯), where ⋯ denote additional parameters of the model. We calculate the rate function I(a,N,K, ⋯) 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, ⋯) is exact for any N. In the presence of interactions the WKB method gives asymptotically exact results for N ≥ 1. The WKB method also uncovers the (very simple) time history of the system which dominates the contribution of different time histories to P_T(N = aN).