Two-parameter deformations of logarithm, exponential, and entropy: A consistent framework for generalized statistical mechanics

TL;DR

This paper introduces a two-parameter class of generalized logarithms and entropies, derived from a consistent maximum entropy framework, applicable to anomalous systems with preserved mathematical and physical properties.

Contribution

It develops a novel two-parameter family of generalized logarithms and entropies, expanding the theoretical foundation for generalized statistical mechanics.

Findings

The entropies are positive, continuous, symmetric, and Lesche stable.

The framework includes Boltzmann-Shannon and known one-parameter entropies.

Distribution functions derived are applicable to anomalous systems.

Abstract

A consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace-form entropy and by requiring that physically motivated mathematical properties are preserved. The emerging differential-functional equation yields a two-parameter class of generalized logarithms, from which entropies and power-law distributions follow: these distributions could be relevant in many anomalous systems. Within the specified range of parameters, these entropies possess positivity, continuity, symmetry, expansibility, decisivity, maximality, concavity, and are Lesche stable. The Boltzmann-Shannon entropy and some one parameter generalized entropies already known belong to this class. These entropies and their distribution functions are compared, and the corresponding deformed algebras are discussed.