Dual thermodynamic ensembles, relative entropies, and excess free energy

Abstract

It has long been known that the relative entropy of a non-equilibrium ensemble to the corresponding equilibrium ensemble is the excess free energy. We show that the reverse relative entropy also has a thermodynamic interpretation: it is the excess free energy of a dual ensemble in which the roles of energy and entropy are interchanged.

Figures (1)

• Figure 1: Thermodynamic duality of non-canonical ensembles. Ensemble ${\text{B}}$ is out-of-thermodynamic equilibrium and non-canonical, whereas ${\text{A}}$ is the equilibrium ensemble the system would relax to if left undisturbed. Ensembles ${\text{B}}$,${\text{C}}$ and ${\text{A}}$,${\text{D}}$ share the same probability distributions respectively, while ${\text{A}}$, ${\text{B}}$ and ${\text{C}}$, ${\text{D}}$ share the same energy spectrum. Consequently, whereas the relative entropy $D({\text{B}}\|{\text{A}})$ is the excess free energy of ensemble ${\text{B}}$, the reverse relative entropy $D({\text{A}}\|{\text{B}})= D({\text{D}}\|{\text{C}})$ is the excess free energy of the thermodynamically dual ensemble ${\text{D}}$.