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In this paper, a dynamical process in a statistical thermodynamic system ofspins exhibiting a phase transition is described on a contact manifold, wheresuch a dynamical process is a process that a metastable equilibrium stateevolves into the most stable symmetry broken equilibrium state. Metastable andthe most stable equilibrium states in the symmetry broken phase or orderedphase are assumed to be described as pruned projections of Legendresubmanifolds of contact manifolds, where these pruned projections of thesubmanifolds express hysteresis and pseudo-free energy curves. Singularitiesassociated with phase transitions are naturally arose in this framework as hasbeen suggested by Legendre singularity theory. Then a particular contactHamiltonian vector field is proposed so that a pruned segment of the projectedLegendre submanifold is a stable fixed point set in a region of a contactmanifold, and that another pruned segment is a unstable fixed point set. Thiscontact Hamiltonian vector field is identified with a dynamical processdeparting from a metastable equilibrium state to the most stable equilibriumone. To show the statements above explicitly an Ising type spin model withlong-range interactions, called the Husimi-Temperley model, is focused, wherethis model exhibits a phase transition.

journal of mathematical physics

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