Stochastic Dynamics in Spatially Extended Magnetic Systems
Models for spatially extended magnetic systems are typically N nearest-neighbor coupled unit vectors that evolve stochastically in time due to unresolved thermal effects, and sample the Gibbs distribution defined by a Hamiltonian quantifying this nearest-neighbor interaction and effects of an external field. Interesting questions arise as the number of vectors goes to infinity while scaling other parameters that effect the temperature and damping in the system. I will present a scaling limit allowing the dynamics of a Metropolis Hastings Algorithm to converge to a deterministic PDE. By adding correlated (colored) noise to the M-H proposal step, the simple accept/reject step no longer samples the Gibbs distribution due to the interaction of the noise and geometry in the problem. However, a limiting Stochastic PDE, that is non-local in its deterministic part, does sample the Gibbs distribution for a specific choice of projecting the noise into the tangent planes of the spin vectors.