Monte Carlo simulations for spinodal decomposition

This paper addresses the phenomenon of spinodal decomposition for the Cahn- Hilliard equation. Namely, we are interested in why most solutions to the C ahn-Hilliard equation which start near a homogeneous equilibrium u(o),= mu in the spinodal interval exhibit phase separation with a characteristic wav elength when exiting a ball of radius R in a Hilbert space centered at u(0) . There are two mathematical explanations for spinodal decomposition, due t o Grant and to Maier-Paape and Wanner. In this paper, we numerically compar e these two mathematical approaches. In fact, we are able to synthesize the understanding we gain from our numerics with the approach of Maier-Paape a nd Wanner, leading to a better understanding of the underlying mechanism fo r this behavior. With this new approach, we can explain spinodal decomposit ion for a longer time and larger radius than either of the previous two app roaches. A rigorous mathematical explanation is contained in a separate pap er. Our approach is to use Monte Carlo simulations to examine the dependenc e of R, the radius to which spinodal decomposition occurs, as a function of the parameter epsilon of the governing equation. We give a description of the dominating regions on the surface of the ball by estimating certain den sities of the distributions of the exit points. We observe, and can show ri gorously, that the behavior of most solutions originating near the equilibr ium is determined completely by the linearization for an unexpectedly long time. We explain the mechanism for this unexpectedly linear behavior, and s how that for some exceptional solutions this cannot be observed. We also de scribe the dynamics of these exceptional solutions.