Abstract

The theory of superstatistics is a generalization of Boltzmann-Gibbs statistical mechanics that allows for temperature fluctuations and builds steady-state ensembles from the distribution of these fluctuations. Although it has been widely used for non-equilibrium steady states in complex systems, recent work has shown that superstatistics is not always applicable because certain conditions must be satisfied by the so-called fundamental inverse temperature function /3F. In this work, we present a complete set of sufficient conditions under which a steadystate model can be represented using superstatistics. We show that /3F alone, along with its derivatives, fully determines the existence and form of the underlying temperature distribution. Moreover, we provide explicit expressions for the moments and cumulants of the conditional distribution of /3 given the energy E, in terms of /3F, and demonstrate that superstatistical models different from the canonical require /3F to be infinitely differentiable, which excludes all polynomial cases. These results strengthen the theoretical foundations of superstatistics and offer a practical way to assess its relevance in real-world applications, such as turbulence, finance, and plasma physics.