Return distributions in dog-flea model revisited

TL;DR

This paper revisits the dog-flea model, demonstrating that the relation between avalanche size distribution exponent and return distribution fits a $q$-Gaussian, with the return distribution approaching this form as system size increases.

Contribution

It establishes a quantitative relation between avalanche exponents and $q$-Gaussian parameters in the dog-flea model, supported by numerical evidence and analytical expressions.

Findings

The $q$-Gaussian fit improves with increasing system size.

The relation between $ au$ and $q$ holds for large systems.

Return distributions tend toward $q$-Gaussian as size grows.

Abstract

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.