Nonequilibrium thermodynamic process with hysteresis and metastable states -- A contact Hamiltonian with unstable and stable segments of a Legendre submanifold

Abstract: In this paper, a dynamical process in a statistical thermodynamic system of spins exhibiting a phase transition is described on a contact manifold, where such a dynamical process is a process that a metastable equilibrium state evolves into the most stable symmetry broken equilibrium state. Metastable and the most stable equilibrium states in the symmetry broken phase or ordered phase are assumed to be described as pruned projections of Legendre submanifolds of contact manifolds, where these pruned projections of the submanifolds express hysteresis and pseudo-free energy curves. Singularities associated with phase transitions are naturally arose in this framework as has been suggested by Legendre singularity theory. Then a particular contact Hamiltonian vector field is proposed so that a pruned segment of the projected Legendre submanifold is a stable fixed point set in a region of a contact manifold, and that another pruned segment is a unstable fixed point set. This contact Hamiltonian vector field is identified with a dynamical process departing from a metastable equilibrium state to the most stable equilibrium one. To show the statements above explicitly an Ising type spin model with long-range interactions, called the Husimi-Temperley model, is focused, where this model exhibits a phase transition.