  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.
