Monte Carlo Simulations for Spinodal Decomposition
 Evelyn Sander,
 Thomas Wanner
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This paper addresses the phenomenon of spinodal decomposition for the Cahn–Hilliard equation. Namely, we are interested in why most solutions to the Cahn–Hilliard equation which start near a homogeneous equilibrium u _{0}≡μ in the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R in a Hilbert space centered at u _{0}. There are two mathematical explanations for spinodal decomposition, due to Grant and to MaierPaape and Wanner. In this paper, we numerically compare these two mathematical approaches. In fact, we are able to synthesize the understanding we gain from our numerics with the approach of MaierPaape and Wanner, leading to a better understanding of the underlying mechanism for this behavior. With this new approach, we can explain spinodal decomposition for a longer time and larger radius than either of the previous two approaches. A rigorous mathematical explanation is contained in a separate paper. Our approach is to use Monte Carlo simulations to examine the dependence of R, the radius to which spinodal decomposition occurs, as a function of the parameter ε of the governing equation. We give a description of the dominating regions on the surface of the ball by estimating certain densities of the distributions of the exit points. We observe, and can show rigorously, that the behavior of most solutions originating near the equilibrium is determined completely by the linearization for an unexpectedly long time. We explain the mechanism for this unexpectedly linear behavior, and show that for some exceptional solutions this cannot be observed. We also describe the dynamics of these exceptional solutions.
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 Title
 Monte Carlo Simulations for Spinodal Decomposition
 Journal

Journal of Statistical Physics
Volume 95, Issue 56 , pp 925948
 Cover Date
 19990601
 DOI
 10.1023/A:1004550416829
 Print ISSN
 00224715
 Online ISSN
 15729613
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Cahn–Hilliard equation
 spinodal decomposition
 phase separation
 exit distributions
 density estimation
 Monte Carlo method
 Galerkin approximation
 Industry Sectors
 Authors

 Evelyn Sander ^{(1)}
 Thomas Wanner ^{(2)}
 Author Affiliations

 1. Department of Mathematical Sciences, George Mason University, Fairfax, Virginia, 22030
 2. Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, Maryland, 21250