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 beta=1/(kBT) is an unknown quantity having some pre-established statistical distribution. The uncertainty in beta is usually understood as the fluctuation of a physical observable, however, it has been previously proved (Davis and Guti & eacute;rrez 2018 Physica A 505 864-70) that beta in a superstatistical model cannot be associated to an observable function B(Gamma) of the microstates Gamma. In this work, we provide an information-theoretical interpretation of this theorem by introducing a new quantity D , the mutual information between beta and Gamma. Our results show that D is also a measure of departure from canonical equilibrium, and reveal a minimum, non-zero uncertainty about beta given Gamma 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 kappa can be understood as a measure of distance from equilibrium.