Fundamental temperature exclusively determines the validity of superstatistics

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 βF. In this work, we present a complete set of sufficient conditions under which a steady-state model can be represented using superstatistics. We show that βF 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 β given the energy E, in terms of βF, and demonstrate that superstatistical models different from the canonical require βF 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.