Constrained Forms of the Tsallis Entropy Function and Local Equilibria

TL;DR

This paper derives constrained forms of the Tsallis entropy function using Lagrangian methods, enabling local analysis of states and system-dependent forms, with applications to systems with equispaced energy levels.

Contribution

It introduces a systematic derivation of constrained Tsallis entropy forms for various constraints, highlighting their local properties and dependence on system structure.

Findings

Constrained Tsallis functions reduce to Shannon entropy at q=1.

Each form allows local examination of states relative to maximum entropy.

Application demonstrated on a system with equispaced energy levels.

Abstract

The Lagrangian technique of Niven (2004, Physica A, 334(3-4): 444) is used to determine the constrained forms of the Tsallis entropy function - i.e. Lagrangian functions in which the probabilities of each state are independent - for each constraint type reported in the literature (here termed the Mark I, II and III forms). In each case, a constrained form of the Tsallis entropy function exists, which at q=1 reduces to its Shannon equivalent. Since they are fully constrained, each constrained Tsallis function can be "dismembered" to give its partial or local form, providing the means to independently examine each state i relative to its local stationary (maximum entropy) position. The Mark II and III functions depend on q, the probability, the stationary probability, and the respective q-partition function; in contrast the Mark I form depends only on the first three parameters. The Mark II and III forms therefore depend on the structure of the system. The utility of the dismemberment method is illustrated for a system with equispaced energy levels.