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

Abstract

We study a nonequilibrium mean field Ising model in the low temperature phase regime, where metastable equilibrium states develop a cuspidal (spinodal) singularity. We focus on celebrated Glauber dynamics, and design a contact Hamiltonian flow which captures some of its rough features in this regime. We prove, however, that there is an inevitable discrepancy between the scaling laws for the relaxation time in the Glauber and the contact Hamiltonian dynamical systems.

From the Fokker–Planck equation to a contact Hamiltonian system

“…• Extend the present theory to describe systems with phase transitions [19,20], where singularities are developed on free-energy curves. • Explore how symplectic topology and contact topology [33] can be applied to the present theory.…”

“…There are several approaches to express nonequilibrium thermodynamic processes by means of contact geometry, [7,15,20]. In section 3.2 the class of contact Hamiltonians of the form…”

“…From the expression of p j (t; q), one can express an approximated expectation value of B 1 , as well as the exact expectation value that has been defined in (20). To express the approximated one for a given B 1 ∈ ΓΛ 1 M at t, introduce…”

“…In this subsection the contact geometric description of the Fokker-Planck equation has been discussed, where the variable z is chosen to be the moment generating function, and the equilibrium distribution function does not depend on q and B. Meanwhile in the literature dynamics of the minus of a free-energy F in ( 6) is often considered [8,15,20]. To compare the present choice of z with the existing one, the case with z = −F is considered in section A.4.…”

Contact topology and non-equilibrium thermodynamics

“…the mean field Glauber) scenario incorporating the phase transitions. This line of research was further advanced in [16].…”