On the definition of fluctuating temperature

Abstract

The Maxwell distribution is derived from the $F$-distribution in the limit where one of the degrees of freedom of the $χ^2$ variates tends to infinity. The estimator of the temperature is consistent, and, hence coincides with the temperature of the heat reservoir in the asymptotic limit; it is also unbiased and efficient. Consequently, there is a contradiction between indentifying the Lagrange multiplier in the variational formalism that Tsallis and co-workers use to maximize his nonadditive entropy with respect to escort expectation values in order to derive the Student $t$- and $r$-distributions and the physical meaning of these variables. Only in the asymptotic limit when these distributions become the $χ^2$-distributions of MBG statistics can the Lagrange multiplier be interpreted as the inverse temperature. Hence, there is no generalization of the $χ^2$-distributions that can be made which involves interpreting the Lagrange multiplier as the inverse temperature. The frequency interpretation of the fluctuating temperature is contrasted with the Bayesian approach that treats a parameter to be estimated as a random variable which is equipped with a probability distribution.