Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

Abstract

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 finitedimensional subspace Y ε is high as long as the solution stays within distance r ε = O(ε R ) of the homogeneous state. The subspace Y 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.

Probabilistic and numerical validation of homology computations for nodal domains

“…On the other hand, for times t ≤ 70ε 2 the curves in both graphs are indistinguishable, despite the fact that they were obtained from a linear and a nonlinear model, respectively. Recent theoretical work has shown that in fact during the initial phase separation regime of the Cahn-Hilliard equation, the effects of the nonlinearity are suppressed for an unexpectedly long time [3,4,23,24,26]. These results have established rigorous lower bounds on the duration of the linear regime.…”

“…This stochastic partial differential equation has been proposed as a model for phase separation in metallic alloys and produces complicated patterns; see for example [3,4,5,7,26] and the references therein. As we mentioned in the Introduction, computational homology can be used to quantify these complicated structures [14], and the question of choosing the correct discretization size M for the homology computations is of utmost importance.…”

An Impact of Stochastic Dynamic Boundary Conditions on the Evolution of the Cahn-Hilliard System

“…The function f is the derivative of a logarithmic potential which is usually approximated by a polynomial with strictly positive dominant coefficient. Theoretical results on asymptotic dynamical behavior of the system (1.1), under either Neumann or periodic boundary conditions, can be founded, for example, in the survey [24] or the book [30], and [9,2,18].…”

“…Moreover, the environmental or surrounding fluctuations may also influence the system evolution and thus may be taken into account as well [2,9,15]. The present paper is concerned with the stochastic version of Eq.…”

“…It is given by dφ = ∆µdt + σ 1 dW (1) , µ = −∆φ + ε∂ t φ + f (φ), in G, ∂ ν µ = 0, on Γ, dφ = (∆ φ − λφ − ∂ ν φ − g(φ))dt + σ 2 dW (2) , on Γ, φ(0) = φ 0 , (1.3) where W (1) and W (2) are independent Wiener processes which will be explained in detail later, and the constants σ i > 0, i = 1, 2, for the noise intensities. The random fluctuation terms consisting of (W (1) , W (2) ) act in the domain but also on the boundary Γ and give a refined description of the underlying microscopic mechanism in the phase separation phenomena. To simplify the situation, we only consider the case with g = 0, and the potential f (u) is a polynomial of odd degree with a positive leading coefficient such as…”