Fundamental temperature exclusively determines the validity of superstatistics

The theory of superstatistics is a generalization of Boltzmann-Gibbs statistical mechanics which admits temperature fluctuations, and generates non-canonical ensembles from the distribution function of these fluctuations. Recently, some results have been presented showing that superstatistics is not universally applicable, but several conditions on the so-called fundamental inverse temperature function $\beta_F$ must be met by any superstatistical model. In this work we provide a set of neccessary and sufficient conditions for a non-equilibrium steady state model to be expressible by superstatistics, showing that $\beta_F$ by itself determines the existence of a superstatistical distribution of temperature.