An amended MaxEnt formulation for deriving Tsallis factors, and associated issues

TL;DR

This paper proposes a modified MaxEnt approach involving dual constraints and reference distributions to derive generalized escort distributions and Levy-like solutions, connecting Tsallis and Rényi entropies with nonextensive statistical mechanics.

Contribution

It introduces an amended MaxEnt formulation with dual constraints, deriving new Levy-like distributions and establishing a duality with Legendre structures in nonextensive statistics.

Findings

Derived generalized escort distributions associated with new equilibrium.

Formulated dual functions for parameter identification.

Established duality between solutions and Legendre structure.

Abstract

An amended MaxEnt formulation for systems displaced from the conventional MaxEnt equilibrium is proposed. This formulation involves the minimization of the Kullback-Leibler divergence to a reference $Q$ (or maximization of Shannon $Q$-entropy), subject to a constraint that implicates a second reference distribution $P\_{1}$ and tunes the new equilibrium. In this setting, the equilibrium distribution is the generalized escort distribution associated to $P\_{1}$ and $Q$. The account of an additional constraint, an observable given by a statistical mean, leads to the maximization of R\'{e}nyi/Tsallis $Q$-entropy subject to that constraint. Two natural scenarii for this observation constraint are considered, and the classical and generalized constraint of nonextensive statistics are recovered. The solutions to the maximization of R\'{e}nyi $Q$-entropy subject to the two types of constraints are derived. These optimum distributions, that are Levy-like distributions, are self-referential. We then propose two `alternate' (but effectively computable) dual functions, whose maximizations enable to identify the optimum parameters. Finally, a duality between solutions and the underlying Legendre structure are presented.