Maximum Renyi Entropy Principle for Systems with Power-Law Hamiltonians

Abstract

The Renyi distribution ensuring the maximum of Renyi entropy is investigated for a particular case of a power-law Hamiltonian. Both Lagrange parameters $\alpha$ and $\beta$ can be eliminated. 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 resulting 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 the behavior of the stochastic process. The Renyi distribution for such $q$ becomes a power-law distribution with the exponent $-(\kappa +1)$. When $q=1/(1+\kappa )+ϵ$ ($0<ϵ\ll 1$) there appears a horizontal head part of the Renyi distribution that precedes the power-law part. Such a picture corresponds to some observed phenomena.