Journal of Statistical Physics

, Volume 95, Issue 5–6, pp 925–948

Monte Carlo Simulations for Spinodal Decomposition

  • Evelyn Sander
  • Thomas Wanner


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 u0μ in the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R in a Hilbert space centered at u0. There are two mathematical explanations for spinodal decomposition, due to Grant and to Maier-Paape 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 Maier-Paape 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.

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


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Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • Evelyn Sander
    • 1
  • Thomas Wanner
    • 2
  1. 1.Department of Mathematical SciencesGeorge Mason UniversityFairfax
  2. 2.Department of Mathematics and StatisticsUniversity of Maryland, Baltimore CountyBaltimore

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