Temperature is not an observable in superstatistics

Abstract

Superstatistics (Beck and Cohen, 2003) is a formalism that attempts to explain the presence of distributions other than the Boltzmann–Gibbs distributions in Nature, typically powerlaw behavior, for systems out of equilibrium such as fluids under turbulence, plasmas and gravitational systems. Superstatistics postulates that those systems are found in a superposition of canonical ensembles at different temperatures, and sometimes the physical interpretation is one of local thermal equilibrium in the sense of an inhomogeneous temperature distribution in different regions of space or instants of time. Here we show that, in order for superstatistics to be internally consistent, it is impossible to define a phase-space function or microscopic observable B(p, q) corresponding oneto-one to the local value of β = 1/kBT . Thus, unlike energy which is defined by a phasespace function H(p, q) (the Hamiltonian), temperature is not a microscopic observable. An important consequence of our proof is that, in Superstatistics, the identification of temperature with the kinetic energy is limited to the expectation of β and cannot be used to measure the different temperatures in local thermal equilibrium or its fluctuations.