Generalized Boltzmann factors and the maximum entropy principle: Entropies for complex systems

Abstract

We generalize the usual exponential Boltzmann factor to any reasonable and potentially observable distribution function, B ( E ) . By defining generalized logarithms Λ as inverses of these distribution functions, we are led to a generalization of the classical Boltzmann–Gibbs entropy ( S BG = - ∫ d ε ω ( ε ) B ( ε ) ln B ( ε ) ) to the expression S ≡ - ∫ d ε ω ( ε ) ∫ 0 B ( ε ) d x Λ ( x ) , which contains the classical entropy as a special case. We show that this is the unique modification of entropy which is compatible with the maximum entropy principle for arbitrary, non-exponential distribution functions. 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 straightforward solutions to the Jaynes maximum entropy principle with the ordinary (not escort!) constraints. Tsallis entropy is recovered as a further special case.