Generalized Boltzmann factors and the maximum entropy principle

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.