Maximum entropy principle and power-law tailed distributions

TL;DR

This paper explores how to extend the maximum entropy principle to generalized statistical mechanics, resulting in new theories with power-law distributions that maintain stability and core features of classical thermodynamics.

Contribution

It introduces a framework for selecting generalized entropies and distributions that preserve the twofold link of ordinary statistical mechanics, leading to stable power-law tail distributions.

Findings

New generalized logarithms and exponentials define coherent statistical theories.

These theories produce power-law tailed distributions.

The generalized entropies are thermodynamically and Lesche stable.

Abstract

In ordinary statistical mechanics the Boltzmann-Shannon entropy is related to the Maxwell-Bolzmann distribution $p_i$ by means of a twofold link. The first link is differential and is offered by the Jaynes Maximum Entropy Principle. The second link is algebraic and imposes that both the entropy and the distribution must be expressed in terms of the same function in direct and inverse form. Indeed, the Maxwell-Boltzmann distribution $p_i$ is expressed in terms of the exponential function, while the Boltzmann-Shannon entropy is defined as the mean value of $-\ln(p_i)$. In generalized statistical mechanics the second link is customarily relaxed. Here we consider the question if and how is it possible to select generalized statistical theories in which the above mentioned twofold link between entropy and the distribution function continues to hold, such as in the case of ordinary statistical mechanics. Within this scenario, there emerge new couples of direct-inverse functions, i.e. generalized logarithms $\Lambda(x)$ and generalized exponentials $\Lambda^{-1}(x)$, defining coherent and self-consistent generalized statistical theories. Interestingly, all these theories preserve the main features of ordinary statistical mechanics, and predict distribution functions presenting power-law tails. Furthermore, the obtained generalized entropies are both thermodynamically and Lesche stable.