Maximum Renyi entropy principle for systems with power--law Hamiltonian

Abstract

The Renyi distribution ensuring the maximum of a Renyi entropy is investigated for a particular case of a power--law Hamiltonian. Both Lagrange parameters, $\alpha$ and $\beta$ can be excluded. It is found that $\beta$ does not depend on a Renyi parameter $q$ and can be expressed in terms of an exponent $\kappa$ of the power--law Hamiltonian and an average energy $U$. The Renyi entropy for the resulted Renyi distribution reaches its maximal value at $q=1/(1+\kappa)$ that can be considered as the most probable value of $q$ when we have no additional information on behaviour of the stochastic process. The Renyi distribution for such $q$ becomes a power--law distribution with the exponent $-(\kappa +1)$. When $q=1/(1+\kappa)+\epsilon$ ($0<\epsilon\ll 1$) there appears a horizontal "head" part of the Renyi distribution that precedes the power--law part. Such a picture corresponds to observables.