The maximization of Tsallis entropy with complete deformed functions and the problem of constraints

TL;DR

This paper addresses the incomplete nature of q-deformed functions in Tsallis entropy maximization, proposing a complete formulation that favors ordinary probability distributions over escort distributions, with implications for statistical mechanics.

Contribution

It introduces a complete definition of q-deformed functions via dual mapping, clarifies the use of ordinary distributions in Tsallis entropy maximization, and links duality to escort distributions.

Findings

Complete q-functions are derived using dual mapping.

Maximization of Tsallis entropy favors ordinary probability distributions.

Escort distributions can be obtained through ordinary averaging in specific cases.

Abstract

We first observe that the (co)domains of the q-deformed functions are some subsets of the (co)domains of their ordinary counterparts, thereby deeming the deformed functions to be incomplete. In order to obtain a complete definition of $q$-generalized functions, we calculate the dual mapping function, which is found equal to the otherwise \textit{ad hoc} duality relation between the ordinary and escort stationary distributions. Motivated by this fact, we show that the maximization of the Tsallis entropy with the complete $q$-logarithm and $q$-exponential implies the use of the ordinary probability distributions instead of escort distributions. Moreover, we demonstrate that even the escort stationary distributions can be obtained through the use of the ordinary averaging procedure if the argument of the $q$-exponential lies in (-$\infty$, 0].