Tsallis statistics with normalized q-expectation values is thermodynamically stable: illustrations

TL;DR

This paper analyzes the stability of Tsallis statistics with normalized q-expectation values, demonstrating its thermodynamic stability across all temperatures and emphasizing the importance of probability cut-offs for accurate solutions.

Contribution

The study clarifies the stability of Tsallis statistics with normalized q-expectation values and corrects previous methods for computing thermodynamic quantities.

Findings

The iterative procedure may not always yield correct temperature dependence.

The $eta o eta'$ transformation provides correct results.

Normalized Tsallis statistics is stable for all temperature ranges.

Abstract

We present a study of both the ``Iterative Procedure'' and the ``$\beta \to \beta'$ transformation'', proposed by Tsallis et al (Physica A261, 534) to find the probabilities $p_i$ of a system to be in a state with energy $\epsilon_i$, within the framework of a generalized statistical mechanics. Using stability and convexity arguments, we argue that the iterative procedure does not always provide the right temperature dependence of thermodynamic observables. In addition, we show how to get the correct answers from the ``$\beta \to \beta'$ transformation''. Our results provide an evidence that the Tsallis statistics with normalized q-expectation values is stable for all ranges of temperatures. We also show that the cut-off in the computation of probabilities is required to achieve the stable solutions.