Large Deviations for the Spatial and Energy Distributions of Systems of Classical Ensemble

TL;DR

This paper explores large deviation principles for the spatial and energy distributions in classical ensembles, linking statistical mechanics with thermodynamic processes and deriving conditions for equilibrium and irreversibility.

Contribution

It connects large deviation theory with thermodynamic processes, deriving equilibrium conditions and entropy inequalities for classical ensembles.

Findings

Large deviation probabilities for spatial and energy distributions are established.

Differentials of entropies relate to thermodynamic variables in reversible processes.

Inequalities of entropy production are proposed for irreversible processes.

Abstract

Boltzmann-Sanov and Cramer-Chernoff's theorems provide large deviation probabilities, entropy, and rate functions for the spatial distribution of systems and the total internal energy of an ensemble respectively. By the method of Lagrange's undetermined multipliers, the results of both theorems, the differentials of entropies or rate functions with respect to numbers of systems, establish the statistical equilibrium condition. We connect the large deviation statistics to the reversible and irreversible processes of the spatial arrangements of the systems and the energy exchange of the ensemble with the heat reservoir. We obtain the equalities between the differentials of entropies and other thermodynamic variables in the reversible processes and suggest inequalities of entropy productions for the irreversible processes.