Gibbs' theorem for open systems with incomplete statistics

TL;DR

This paper extends Gibbs' theorem to open systems with incomplete statistics, demonstrating that their stationary distributions maximize entropy and proposing a new entropy measure for self-organization.

Contribution

It generalizes Gibbs' theorem to open systems with incomplete statistics and introduces a renormalized entropy as a measure of self-organization.

Findings

Stationary distributions in open systems maximize entropy.

Inverse power law distributions are associated with incomplete statistics.

Renormalized entropy can measure self-organization.

Abstract

Gibbs' theorem, which is originally intended for canonical ensembles with complete statistics has been generalized to open systems with incomplete statistics. As a result of this generalization, it is shown that the stationary equilibrium distribution of inverse power law form associated with the incomplete statistics has maximum entropy even for open systems with energy or matter influx. The renormalized entropy definition given in this paper can also serve as a measure of self-organization in open systems described by incomplete statistics.