A superstatistical measure of distance from canonical equilibrium

Non-equilibrium systems in steady states are commonly described by generalized statistical mechanical frameworks such as superstatistics, which assumes that the inverse temperature β = 1 / ( k B T ) is an unknown quantity having some pre-established statistical distribution. The uncertainty in β is usually understood as the fluctuation of a physical observable, however, it has been previously proved (Davis and Gutiérrez 2018 Physica A 505 864–70) that β in a superstatistical model cannot be associated to an observable function B ( Γ ) of the microstates Γ . In this work, we provide an information-theoretical interpretation of this theorem by introducing a new quantity D , the mutual information between β and Γ . Our results show that D is also a measure of departure from canonical equilibrium, and reveal a minimum, non-zero uncertainty about β given Γ for every non-canonical superstatistical ensemble. The behavior of D is illustrated in the case of a collisionless plasma described by kappa distributions, revealing the precise sense in which the spectral index κ can be understood as a measure of distance from equilibrium.