Non-local SPDE limits of spatially-correlated-noise driven spin systems derived to sample a canonical distribution

TL;DR

This paper derives a non-local stochastic PDE as a limit of a spin system driven by spatially correlated noise, showing it samples the canonical distribution and analyzing the effects of noise correlation and symmetry failure.

Contribution

It introduces a novel non-local stochastic PDE limit for correlated-noise driven spin systems, demonstrating well-posedness and correct sampling of the Gibbs distribution.

Findings

The derived PDE is mathematically well-posed.

The PDE correctly samples the canonical distribution.

Numerical simulations confirm convergence and dynamics.

Abstract

We study the macroscopic behavior of a stochastic spin ensemble driven by a discrete Markov jump process motivated by the Metropolis-Hastings algorithm where the proposal is made with spatially correlated (colored) noise, and hence fails to be symmetric. However, we demonstrate a scenario where the failure of proposal symmetry is a higher order effect. Hence, from these microscopic dynamics we derive as a limit as the proposal size goes to zero and the number of spins to infinity, a non-local stochastic version of the harmonic map heat flow (or overdamped Landau-Lipshitz equation). The equation is both mathematically well-posed and samples the canonical/Gibbs distribution related to the kinetic energy. The failure of proposal symmetry due to interaction between the confining geometry of the spin system and the colored noise is in contrast to the uncorrelated, white-noise, driven system. Specifically, the choice of projection of the noise to conserve the magnitude of the spins is crucial to maintaining the proper equilibrium distribution. Numerical simulations are included to verify convergence properties and demonstrate the dynamics.