Analysis of self-organized criticality in Ehrenfest's dog-flea model

TL;DR

This paper investigates self-organized criticality in Ehrenfest's dog-flea model through simulations, revealing q-Gaussian fluctuation distributions and confirming a theoretical relation between avalanche exponents and q-values.

Contribution

It provides numerical evidence linking avalanche size exponents to q-Gaussian parameters, validating recent theoretical predictions in nonextensive statistical mechanics.

Findings

Fluctuation distributions follow q-Gaussian shape at large system sizes.

The relation between avalanche exponent and q-value matches theoretical predictions.

Finite size effects are minimized to observe critical behavior.

Abstract

The self-organized criticality in Ehrenfest's historical dog-flea model is analyzed by simulating the underlying stochastic process. The fluctuations around the thermal equilibrium in the model are treated as avalanches. We show that the distributions for the fluctuation length differences at subsequent time steps are in the shape of a $q$-Gaussian (the distribution which is obtained naturally in the context of nonextensive statistical mechanics) if one avoids the finite size effects by increasing the system size. We provide a clear numerical evidence that the relation between the exponent $\tau$ of avalanche size distribution obtained by maximum likelihood estimation and the $q$ value of appropriate q-Gaussian obeys the analytical result recently introduced by Caruso et al. [Phys. Rev. E \textbf{75}, 055101(R) (2007)]. This rescues the q parameter to remain as a fitting parameter and allows us to determine its value a priori from one of the well known exponents of such dynamical systems.