Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation
- Cite this article as:
- Blömker, D., Maier-Paape, S. & Wanner, T. Commun. Math. Phys. (2001) 223: 553. doi:10.1007/PL00005585
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This paper gives theoretical results on spinodal decomposition for the stochastic Cahn–Hilliard–Cook equation, which is a Cahn–Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size.
In more detail, our results can be summarized as follows. The Cahn–Hilliard–Cook equation depends on a small positive parameter ε which models atomic scale interaction length. We quantify the behavior of solutions as ε→ 0. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace ?ε is high as long as the solution stays within distance rε=O(εR) of the homogeneous state. The subspace ?ε is an affine space corresponding to the highly unstable directions for the linearized deterministic equation. The exponent R depends on both the strength and the regularity of the noise.