Klimontovich`s S theorem in nonextensive formalism and the problem of constraints

TL;DR

This paper extends Klimontovich's S theorem within nonextensive thermostatistics, addressing the issue of energy renormalization and escort distributions to correctly compare entropy values in nonextensive systems.

Contribution

It introduces a nonextensive generalization of Klimontovich's S theorem, incorporating ordinary probability and relative entropy for improved entropy comparison.

Findings

The generalized S theorem is applicable to nonextensive systems.

Energy renormalization is essential in the nonextensive formalism.

Illustration with the Van der Pol oscillator demonstrates the theorem's utility.

Abstract

Ordinary Boltzmann-Gibbs entropy is inadequate to be used in systems depending on a control parameter that yield different mean energy values. Such systems fail to give the correct comparison between the off-equilibrium and equilibrium entropy values. Klimontovich's S theorem solves this problem by renormalizing energy and making use of escort distributions. Since nonextensive thermostatistics is a generalization of Boltzmann-Gibbs entropy, it too exhibits this same deficiency. In order to remedy this, we present the nonextensive generalization of Klimontovich's S theorem. We show that this generalization requires the use of ordinary probability and the associated relative entropy in addition to the renormalization of energy. Lastly, we illustrate the generalized S theorem for the Van der Pol oscillator.