On the possible distributions of temperature in nonequilibrium steady states

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 B of inverse temperature compatible with superstatistics, in the sense that all moments of B and β = 1/(kBT ) coincide. The function 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 505, 864-870 (2018)]. The present results also constrain the possible joint ensembles of system and environment compatible with superstatistics.

Conditional maximum entropy and Superstatistics

“…We have complemented the framework initiated in Ref. [14], by connecting it with the maximum entropy principle under a conditional energy expectation constraint, H S,G = Ē(G). In this formalism, knowledge of Ē(G), the canonical caloric curve ε(β) and the density of states Ω G of the environment completely determines the most unbiased form of superstatistics, with f (β) or ρ(E) given by Eqs.…”

“…Recently [14], a mechanism leading to superstatistics has been explored where the original (target) system with microstates x is in contact with an environment with microstates y and Hamiltonian G(y), in such a way that Eq. 3 is recovered by integration over the microstates of the environment, that is,…”

“…3 and 4 are simultaneously true, then Ref. [14] proves that this can be accomplished if and only if the conditional distribution of the system given a fixed environment is canonical, with a temperature which is exclusively a property of the environment. That is,…”

The $q$-canonical ensemble as a consequence of Bayesian superstatistics

“…This is a particular class of models defined by a distribution of inverse temperature P (β|S). The central equation of superstatistics, as presented in recent works [14,15], is the joint distribution of microstates Γ and inverse temperature β, which is given by the product rule of probability, P (Γ, β|S) = P (Γ|β)P (β|S),…”

Conditional maximum entropy and superstatistics

“…Recently [30], a mechanism leading to superstatistics has been explored where the original (target) system with microstates x is in contact with an environment with microstates y and Hamiltonian G( y), in such a way that equation ( 3) is recovered by integration over the microstates of the environment, that is,…”