Fundamental temperature in the superstatistical description of non-equilibrium steady states

Among the statistical mechanical frameworks able to describe systems in non-equilibrium steady states such as collisionless plasmas, self-gravitating systems and other complex systems, superstatistics have gained recent attention. Superstatistics postulates a superposition of canonical systems with inverse temperatures β described by a probability distribution depending on the external conditions. Unfortunately, the uncertainty about β cannot be attributed to fluctuations of a phase space function, and this suggests that the distribution of β is purely of statistical nature and must be inferred rather than measured. This lack of direct observability of the superstatistical temperature then becomes a conceptual issue in need of resolution. In this work we address this issue, showing that all the information relevant to determine the superstatistical β is contained in the recently proposed fundamental temperature βF, a model-dependent function of the energy. In this way, a mapping can be constructed from functions of β to new functions of βF such that their expectation values coincide. Our results provide new tools to access the superstatistical temperature from energy values, and we illustrate this by computing the superstatistical inverse temperature distribution of the q-canonical ensemble without the use of Laplace inversion.