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

TL;DR

This paper investigates the behavior of Glauber dynamics near a cusp in a mean field Ising model, introducing a contact Hamiltonian flow to approximate it, but finds fundamental limitations in this approach.

Contribution

It develops a contact geometric framework to approximate Glauber dynamics near a cusp and identifies inherent discrepancies in scaling laws.

Findings

Contact Hamiltonian flow captures some features of Glauber dynamics

Discrepancy in relaxation time scaling laws between models

Limitations of contact geometric approach near singularities

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.