# On the possible distributions of temperature in nonequilibrium steady states

## TL;DR

This paper investigates how broad temperature distributions emerge in nonequilibrium steady states described by superstatistics, showing that temperature correlations with the environment are key and defining a unique inverse temperature function.

## Contribution

It establishes a unique microscopic inverse temperature compatible with superstatistics and clarifies its dependence solely on the environment, extending previous impossibility results.

## Key findings

- Broad temperature distributions arise from subsystem-environment correlations.
- A unique inverse temperature function $eta$ is identified, depending only on the environment.
- Constraints on joint system-environment ensembles compatible with superstatistics are derived.

## Abstract

Superstatistics is a framework in nonequilibrium statistical mechanics that successfully describes a wide variety of complex systems, including hydrodynamic turbulence, weakly-collisional plasmas, cosmic rays, power grid fluctuations, among several others. In this work we analyze the class of nonequilibrium steady-state systems consisting of a subsystem and its environment, and where the subsystem is described by the superstatistical framework. In this case we provide an answer to the mechanism by which a broad distribution of temperature arises, namely due to correlation between subsystem and environment. We prove that there is a unique microscopic definition $\mathcal{B}$ of inverse temperature compatible with superstatistics, in the sense that all moments of $\mathcal{B}$ and $\beta=1/(k_B T)$ coincide. The function $\mathcal{B}$ however, cannot depend on the degrees of freedom of the system itself, only on the environment, in full agreement with our previous impossibility theorem [Physica A \textbf{505}, 864-870 (2018)]. The present results also constrain the possible joint ensembles of system and environment compatible with superstatistics.