Generalized Boltzmann factors and the maximum entropy principle

TL;DR

This paper extends the Boltzmann-Gibbs entropy framework to a broader class of distribution functions, providing a generalized entropy that maintains thermodynamic consistency and aligns with maximum entropy principles, including Tsallis entropy.

Contribution

It introduces a generalized entropy based on arbitrary distribution functions and their inverses, unifying classical and non-extensive entropies within a common framework.

Findings

Generalized entropy reproduces classical thermodynamics

Observed distributions are solutions to maximum entropy principle

Includes Tsallis entropy as a special case

Abstract

We generalize the usual exponential Boltzmann factor to any reasonable and potentially observable distribution function, $B(E)$. By defining generalized logarithms $\Lambda$ as inverses of these distribution functions, we are led to a generalization of the classical Boltzmann-Gibbs entropy, $S_{BG}= -\int d \epsilon \omega(\epsilon) B(\epsilon) \log B(\epsilon)$ to the expression $S\equiv -\int d \epsilon \omega(\epsilon) \int_0^{B(\epsilon)} dx \Lambda (x)$, which contains the classical entropy as a special case. We demonstrate that this entropy has two important features: First, it describes the correct thermodynamic relations of the system, and second, the observed distributions are straight forward solutions to the Jaynes maximum entropy principle with the ordinary (not escort!) constraints. Tsallis entropy is recovered as a further special case.