Exponential or Power Law? How to Select a Stable Distribution of Probability in a Physical System ^†

A mapping of non-extensive statistical mechanics with non-additivity parameter $q\ne 1$ into Gibbs’ statistical mechanics exists (E. Vives, A. Planes, PRL 88 2, 020601 (2002)) which allows generalization to $q\ne 1$ both of Einstein’s formula for fluctuations and of the ’general evolution criterion’ (P. Glansdorff, I. Prigogine, Physica 30 351 (1964)), an inequality involving the time derivatives of thermodynamical quantities. Unified thermodynamic description of relaxation to stable states with either Boltzmann ( $q=1$ ) or power-law ( $q\ne 1$ ) distribution of probabilities of microstates follows. If a 1D (possibly nonlinear) Fokker-Planck equation describes relaxation, then generalized Einstein’s formula predicts whether the relaxed state exhibits a Boltzmann or a power law distribution function. If this Fokker-Planck equation is associated to the stochastic differential equation obtained in the continuous limit from a 1D, autonomous, discrete, noise-affected map, then we may ascertain if a a relaxed state follows a power-law statistics—and with which exponent—by looking at both map dynamics and noise level, without assumptions concerning the (additive or multiplicative) nature of the noise and without numerical computation of the orbits. Results agree with the simulations (J. R. Sánchez, R. Lopez-Ruiz, EPJ 143.1 (2007): 241–243) of relaxation leading to a Pareto-like distribution function.