Applying incomplete statistics to nonextensive systems with different $q$ indices

TL;DR

This paper extends nonextensive statistics to systems with varying $q$ indices by proposing a generalized nonadditivity rule based on incomplete information theory, enabling broader applicability of $q$-entropy.

Contribution

It introduces a new nonadditivity rule for entropy that accommodates systems with different $q$ values, based on the zeroth law of thermodynamics.

Findings

Proposes a generalized nonadditivity rule for entropy.

Shows the rule leads uniquely to $q$-entropy.

Extends applicability of nonextensive statistics to non-uniform $q$ systems.

Abstract

The nonextensive statistics based on the $q$-entropy $S_q=-\frac{\sum_{i=1}^v(p_i-p_i^q)}{1-q}$ has been so far applied to systems in which the $q$ value is uniformly distributed. For the systems containing different $q$'s, the applicability of the theory is still a matter of investigation. The difficulty is that the class of systems to which the theory can be applied is actually limited by the usual nonadditivity rule of entropy which is no more valid when the systems contain non uniform distribution of $q$ values. In this paper, within the framework of the so called incomplete information theory, we propose a more general nonadditivity rule of entropy prescribed by the zeroth law of thermodynamics. This new nonadditivity generalizes in a simple way the usual one and can be proved to lead uniquely to the $q$-entropy.