Return distributions in dog-flea model revisited

A recent study of coherent noise model for the system size independent case provides an exact relation between the exponent $\tau$ of avalanche size distribution and the $q$ value of appropriate $q$-Gaussian that fits the return distribution of the model. This relation is applied to Ehrenfest’s historical dog-flea model by treating the fluctuations around the thermal equilibrium as avalanches. We provide a clear numerical evidence that the relation between the exponent $\tau$ of fluctuation length distribution and the $q$ value of appropriate $q$-Gaussian obeys this exact relation when the system size is large enough. This allows us to determine the value of $q$-parameter \emph{a priori} from one of the well known exponents of such dynamical systems. Furthermore, it is shown that the return distribution in dog-flea model gradually approaches to $q$-Gaussian as the system size increases and this tendency can be analyzed by a well defined analytical expression.