On maximum entropy principle, superstatistics, power-law distribution and Renyi parameter

The equilibrium distributions of probabilities are derived on the basis of the maximum entropy principle (MEP) for the Renyi and Tsallis entropies. New S-forms for the Renyi and Tsallis distribution functions are found which are normalised with corresponding entropies in contrast to the usual Z-forms normalised with partition functions Z. The superstatistics based on the Gibbs distribution of energy fluctuations gives rise to a distribution function of the same structure that the Renyi and Tsallis distributions have. The long-range “tail” of the Renyi distribution is the power-law distribution with the exponent −s expressed in terms of the free Renyi parameter q as s=1/(1−q). The condition s>0 gives rise to the requirement q<1. The parameter q can be uniquely determined with the use of a further extension of MEP as the condition for maximum of the difference between the Renyi and Boltzmann entropies for the same power-law distribution dependent on q. It is found that the maximum is realized for q within the range from 0.25 to 0.5 and the exponent s varies from 1.3 to 2 in dependence on parameters of stochastic systems.