Temperature fluctuations and mixtures of equilibrium states in the canonical ensemble

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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 χ² 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.