A superstatistical measure of distance from canonical equilibrium

Non-equilibrium systems in steady states are commonly described by generalized statistical mechanical theories such as non-extensive statistics and superstatistics. Superstatistics assumes that the inverse temperature β = 1/(kB T) follows some pre-established statistical distribution, however, it has been previously proved (Physica A 505, 864-870 [2018]) that β 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. This supports the use of the mutual information as a descriptor of complexity and correlation in complex systems, also providing in some cases a sound basis for the use of Tsallis' entropic index q as a measure of distance from equilibrium, being in those cases a proxy for D.