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.

Contact geometric approach to Glauber dynamics near a cusp and its limitation

“…where ϕ(u) = β −1 ln (2 cosh(βu)) and the first equation represents the so-called self-consistent equation (see e.g. [7]). The real parameter β > 0 is the inverse temperature, and b > 0 is determined by the strength of the interaction and the geometry of the model.…”

“…First, we design a contact Hamiltonian which roughly speaking 'imitates' Glauber dynamics, see section 3. We refer the reader to [4,6,7] for earlier steps in this direction. Since the magnetic field is constant, such a Hamiltonian necessarily has a form H(z, q) (i.e.…”

“…Indeed, look at the regions bounded by the stable and metastable branches, and recall that the contact evolution of the energy z is given by ż = H. Since the stable branch S − lies above M − , ∂H/∂z is necessarily negative at S − , and hence M − become unstable. We refer the reader to [7] where metastable states were treated as dynamically unstable ones.…”

“…Here, roughly speaking, relaxation is a dynamical process starting from a nonequilibrium state to a point of the equilibrium state set in thermodynamic phase space. At equilibrium, for the Ising model with mean field type interactions, the equation of state is explicitly derived by calculating the partition function in the thermodynamic limit, and the system at equilibrium exhibits a first-order phase transition together with metastable states (see [7] for this derivation). For Glauber dynamics, various scaling relations have been proposed, and one of them is the scaling of the relaxation time near the critical point [1,11].…”

“…Contact Hamiltonian flows model processes of nonequilibrium thermodynamics (see e.g. [3,4,[6][7][8][9]15]). In this paper, we focus on designing a contact Hamiltonian system whose dynamics captures the equilibria and their stability patterns of the Glauber-Suzuki-Kubo ordinary differential equation (ODE) (5) near critical points, show time-scales near critical points for the phase transition, and compare this with other proposals in the literature.…”