The Renyi entropy as a "free entropy" for complex systems

TL;DR

This paper introduces a 'free entropy' based on Renyi entropy for complex systems, deriving a distribution that explains power-law behaviors observed in such systems, and explores its thermodynamic implications.

Contribution

It derives a new entropy measure called 'free entropy' using Renyi entropy, linking it to power-law distributions in complex systems and thermodynamic equilibrium.

Findings

Renyi entropy maximization yields a power-law distribution.

The optimal Renyi parameter q is 1/(1+κ).

The resulting distribution aligns with observed phenomena in complex systems.

Abstract

The Boltzmann entropy $S^{(B)}$ is true in the case of equal probability of all microstates of a system. In the opposite case it should be averaged over all microstates that gives rise to the Boltzmann--Shannon entropy (BSE). Maximum entropy principle (MEP) for the BSE leads to the Gibbs canonical distribution that is incompatible with power--low distributions typical for complex system. This brings up the question: Does the maximum of BSE correspond to an equilibrium (or steady) state of the complex system? Indeed, the equilibrium state of a thermodynamic system which exchange heat with a thermostat corresponds to maximum of Helmholtz free energy rather than to maximum of average energy, that is internal energy $U$. Following derivation of Helmholtz free energy the Renyi entropy is derived as a cumulant average of the Boltzmann entropy for systems which exchange an entropy with the thermostat. The application of MEP to the Renyi entropy gives rise to the Renyi distribution for an isolated system. It 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 $U$. 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 behavior of the stochastic process. The Renyi distribution for such $q$ becomes a power--law distribution with the exponent $-(\kappa +1)$. Such a picture corresponds to some observed phenomena in complex systems.