From Stochastic Lattice Dynamics Models to Partial Differential Equations

Ira Livshits
Carnegie Mellon University, USA

Shlomo Ta'asan
Carnegie Mellon University, USA


In this talk we consider stochastic lattice dynamics models, such as the Ising model, and we derive partial differential equations that describe the behavior of these models on large (macroscopic) scales in terms of spins densities. This is our first step in constructing space-time multiscale techniques for the analysis and the efficient simulation of lattice models. Our numerical approach is based on using coloring schemes that track the spins propagation, and it leads to derivation of finite difference equations describing the dynamics of the spins density on large space-time scales. The methods have been applied to Kawasaki model where a non-linear diffusion processes governs the densities dynamics. Our numerical results agree with the expected theoretical ones.

A related free boundary problem is considered as well. It describes the large scale dynamics of the macroscopic interface defined between large clusters of spins of different signs.

The method seems to be general and can be used in the study of macroscopic behavior for a variety of stochastic dynamics models.