Nonequilibrium in Thermodynamic Formalism: the Second Law, gases and Information Geometry
share
In Nonequilibrium Thermodynamics and Information Theory, the relative entropy (or, KL divergence) plays a very important role. Consider a Hölder Jacobian J and the Ruelle (transfer) operator ℒ_log J. Two equilibrium probabilities μ_1 and μ_2, can interact via a discrete-time Thermodynamic Operation described by the action of the dual of the Ruelle operator ℒ_log J^*. We argue that the law μ→ℒ_log J^*(μ), producing nonequilibrium, can be seen as a Thermodynamic Operation after showing that it's a manifestation of the Second Law of Thermodynamics. We also show that the change of relative entropy satisfies D_K L (μ_1,μ_2) - D_K L (ℒ_log J^*(μ_1),ℒ_log J^*(μ_2))= 0. Furthermore, we describe sufficient conditions on J,μ_1 for getting h(ℒ_log J^*(μ_1))≥ h(μ_1), where h is entropy. Recalling a natural Riemannian metric in the Banach manifold of Hölder equilibrium probabilities we exhibit the second-order Taylor formula for an infinitesimal tangent change of KL divergence; a crucial estimate in Information Geometry. We introduce concepts like heat, work, volume, pressure, and internal energy, which play here the role of the analogous ones in Thermodynamics of gases. We briefly describe the MaxEnt method.
READ FULL TEXT