Maximum Renyi Entropy Principle for Systems with Power-Law Hamiltonians

The Renyi distribution ensuring the maximum of Renyi entropy is investigated for a particular case of a power-law Hamiltonian. Both Lagrange parameters α and β can be eliminated. It is found that β does not depend on a Renyi parameter q and can be expressed in terms of an exponent κ 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+κ) 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 -(κ+1). When q=1/(1+κ)+ϵ (0<ϵ≪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.