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

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.