Temperature fluctuations and mixtures of equilibrium states in the canonical ensemble

TL;DR

This paper investigates the microscopic origins of temperature fluctuations in a finite perfect gas system and their connection to non-Gibbsian distributions, challenging the interpretation of $q$-exponentials as simple mixtures of Gibbs states.

Contribution

It provides a detailed microscopic analysis showing that inverse temperature fluctuations follow a chi-squared-like distribution, leading to non-Gibbsian mixed states, and compares these results with previous studies.

Findings

Inverse temperature probability density resembles a chi-squared distribution.

Mixed Gibbs distribution derived is non-Gibbsian.

Results relate to turbulence and nuclear scattering experiments.

Abstract

It has been suggested recently that `$q$-exponential' distributions which form the basis of Tsallis' non-extensive thermostatistical formalism may be viewed as mixtures of exponential (Gibbs) distributions characterized by a fluctuating inverse temperature. In this paper, we revisit this idea in connection with a detailed microscopic calculation of the energy and temperature fluctuations present in a finite vessel of perfect gas thermally coupled to a heat bath. We find that the probability density related to the inverse temperature of the gas has a form similar to a $\chi^2$ density, and that the `mixed' Gibbs distribution inferred from this density is non-Gibbsian. These findings are compared with those obtained by a number of researchers who worked on mixtures of Gibbsian distributions in the context of velocity difference measurements in turbulent fluids as well as secondaries distributions in nuclear scattering experiments.