Spinodal decomposition for the Cahn-Hilliard-Cook equation

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 epsilon which models atomic scale interaction length. We quantify the behavior of solutions as epsilon --> 0. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace Y-epsilon is high as long as the solution stays within distance r(epsilon) = O(epsilon (R)) of the homogeneous state. The subspace Y-epsilon 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.