Limiting Behaviors of High Dimensional Stochastic Spin Ensembles

Abstract

Lattice spin models in statistical physics are used to understand magnetism. Their Hamiltonians are a discrete form of a version of a Dirichlet energy, signifying a relationship to the Harmonic map heat flow equation. The Gibbs distribution, defined with this Hamiltonian, is used in the Metropolis-Hastings (M-H) algorithm to generate dynamics tending towards an equilibrium state. In the limiting situation when the inverse temperature is large, we establish the relationship between the discrete M-H dynamics and the continuous Harmonic map heat flow associated with the Hamiltonian. We show the convergence of the M-H dynamics to the Harmonic map heat flow equation in two steps: First, with fixed lattice size and proper choice of proposal size in one M-H step, the M-H dynamics acts as gradient descent and will be shown to converge to a system of Langevin stochastic differential equations (SDE). Second, with proper scaling of the inverse temperature in the Gibbs distribution and taking the lattice size to infinity, it will be shown that this SDE system converges to the deterministic Harmonic map heat flow equation. Our results are not unexpected, but show remarkable connections between the M-H steps and the SDE Stratonovich formulation, as well as reveal trajectory-wise out of equilibrium dynamics to be related to a canonical PDE system with geometric constraints.

Nonlocal stochastic-partial-differential-equation limits of spatially correlated noise-driven spin systems derived to sample a canonical distribution

“…Building on our previous work [9] that studied the limiting dynamics of a geometric MH process with white noise in the proposal, we fill a missing gap in the above results showing strong convergence of trajectories started far from equilibrium to a non-local SPDE in a geometric setting, with the underlying dynamics of the process designed to sample an (non product form) invariant measure using colored noise with a given covariance structure. Similar to [20], we derive a drift term that implicitly is driven by a non-local diffusion operator.…”

“…In section II we layout the vector notation we adapt for the paper. In section III we review our results from [9] pointing out a few interesting facts that will be in contrast to the colored noise case. We extend these results to the case of colored noise in section IV, outlining the derivation of the limiting SDE system from the MH dynamics in section IV A (details of the proof in Appendix A), discussing the correct projection of the noise onto the tangent plane of the underlying geometry to ensure the SDE system samples the desired distribution (1) in section IV B, proving the invariant measure of the MH dynamics converges to this same invariant measure in section IV C, and discuss the Fourier representation of the non-local SPDE (5) in section IV D with an outline the well-posedness in Appendix C when the noise is trace class.…”

“…Here we present an overview of our previous work, [9], pointing out a few interesting facts that will be in contrast to the colored noise case. We remind the reader that though we limit our discussion here to the cases d = 1, m = 2 for ease of exposition, all results here extend to d ≥ 1, m ≥ 1 with small modifications.…”

“…The idea behind the convergence is to equate one Metropolis Hastings step to one Euler-Maruyama numerical integration step of the Itô SDE (21). Following [9,20] we consider the leading order in proposal size ε terms for the drift and diffusion of one MH step. At various points we drop higher order terms that are random variables, which are capable of taking on arbitrarily large values, but with small probability.…”

“…For ease of exposition and physical importance, we will restrict ourselves to m = 2 and work only with spins defined as vectors on S 2 . We build on our recent work [9] which showed that on a general torus in any dimension, T d , the MH dynamics for a system of spatiallyuncorrelated "white" noise driven spins with confining geometry converged to the dynamics of an SDE system as ε, the proposal size, went to zero. We also considered the N → ∞ limit of the dynamics while quenching the noise (β = N γ for γ sufficiently large) to arrive at the harmonic map heat flow equation…”