Geometric structure of the non-equilibrium thermodynamics of homogeneous systems

Abstract

A geometric model for non-equilibrium thermodynamics is constructed as a codimension one submanifold of a contact space in which the action of the contact form represents the second law of thermodynamics. This model generalizes that of the thermostatic system as a Legendre submanifold of a contact space. A thermodynamic system in the contact manifold is locally the graph of a generalized energy function defined on the symplectic manifold of all 2 n conjugate variables. Non-equilibrium processes are paths on this graph. The Gibbs contact form acting on admissible non-equilibrium paths reproduces the Clausius-Duhem inequality and measures dissipation. A gradient relaxation process is defined, in which the dissipation is maximal for fixed control variables. The Kelvin-Voigt model and Newton's law of cooling are shown to be such gradient relaxation processes. Affinities are defined in terms of the generalized energy function, and the classical non-equilibrium linear Onsager relations are generalized for gradient relaxation processes.