Generalized Boltzmann factors and the maximum entropy principle

Details Generalized Boltzmann factors and the maximum entropy principle Generalized Boltzmann factors and the maximum entropy principle

2006-02-16

arxiv.org/abs/cond-mat/0602389v3

Physics

Condensed Matter

Statistical Mechanics

4 pages no figures

Scientific paper

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.

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