A superstatistical measure of distance from canonical equilibrium

Abstract

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 $\beta =1/(k_{B}T)$ follows some pre-established statistical distribution, however, it has been previously proved (Physica A 505, 864-870 [2018]) that $\beta$ cannot be associated to an observable function $B(\mathbf{\Gamma })$ of the microstates $\mathbf{\Gamma }$. In this work, we provide an information-theoretical interpretation of this theorem by introducing a new quantity $\mathcal{D}$, the mutual information between $\beta$ and $\mathbf{\Gamma }$. Our results show that $\mathcal{D}$ is also a measure of departure from canonical equilibrium, and reveal a minimum, non-zero uncertainty about $\beta$ given $\mathbf{\Gamma }$ 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 $\mathcal{D}$.

References in corpus (24)

• Superstatistics
• A Statistical Measure of Complexity
• Kappa distributions: theory and applications in space plasmas
• Nonextensive statistical mechanics and economics
• Superstatistics: Theory and Applications
• The nonextensive parameter and Tsallis distribution for self-gravitating systems
• Superstatistics, thermodynamics, and fluctuations
• Jeans' gravitational instability and nonextensive kinetic theory
• Nonadditive Entropies Yield Probability Distributions with Biases not Warranted by the Data
• Generalized statistical mechanics for superstatistical systems
• Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks
• Bayesian approach to superstatistics
• Single-particle velocity distributions of collisionless, steady-state plasmas must follow Superstatistics
• Universality of market superstatistics
• Entropy, Information, and the Updating of Probabilities
• On the Relation between Kappa Distribution Functions and the Plasma Beta Parameter in the Earth Magnetosphere: THEMIS observations
• Universal Non-Gaussian Velocity Distribution in Violent Gravitational Processes
• Temperature is not an observable in superstatistics
• Complexity measures, emergence, and multiparticle correlations
• Superstatistics and temperature fluctuations
• Kappa distribution from particle correlations in non-equilibrium, steady-state plasmas
• Comment on "Black Hole Entropy: A Closer Look"
• On the possible distributions of temperature in nonequilibrium steady states
• The foundations of statistical physics: entropy, irreversibility, and inference

Cited by in corpus (1)

• Kappa distributions in the framework of superstatistics