Renyi Thermostatistics and Self-Organization

Abstract

Taking into account extremum of a Helmholtz free energy in the equilibrium state of a thermodynamic system the Renyi entropy is derived from the Boltzmann entropy by the same way as the Helmholtz free energy from the Hamiltonian. The application of maximum entropy principle to the Renyi entropy gives rise to the Renyi distribution. The $q$-dependent Renyi thermodynamic entropy is defined as the Renyi entropy for Renyi distribution. A temperature and free energy are got for a Renyi thermostatistics. Transfer from the Gibbs to Renyi thermostatistics is found to be a phase transition at zero value of an order parameter $\eta=1-q$. It is shown that at least for a particular case of the power-law Hamiltonian $H=C\sum_i x_i^\kappa$ this entropy increases with $\eta$. Therefore in the new entropic phase at $\eta>0$ the system tends to develop into the most ordered state at $\eta=\eta_{max}=\kappa/(1+\kappa)$. The Renyi distribution at $\eta_{max}$ becomes a pure power-law distribution.