Nucleation in the one-dimensional stochastic Cahn-Hilliard model

Abstract

Despite their misleading label, rare events in stochastic systems are central to many applied phenomena. In this paper, we concentrate on one such situationphase separation through homogeneous nucleation in binary alloys as described by the stochastic partial differential equation model due to Cahn, Hilliard, and Cook. We show that in the limit of small noise intensity, nucleation can be explained by the stochastically driven exit from the domain of attraction of an asymptotically stable homogeneous equilibrium state for the associated deterministic model. Furthermore, we provide insight into the subsequent nucleation dynamics via the structure of the attractor of the model in the absence of noise.

Degenerate Nucleation in the Cahn--Hilliard--Cook Model

“…For more details, see for example [4,5,6]. For sufficiently small noise intensity, it was shown in [2] that solution paths of the stochastic Cahn-Hilliard-Cook model (4) which originate near the homogeneous stateū ≡ µ will exit its deterministic domain of attraction with probability one, and at the time of exit these paths will be close to the boundary spikes which were identified in [1]. The behavior explains the development of the minimal perturbations mentioned above.…”

“…However, the latter result is only valid for mass values in the spinodal region −1/ √ 3 < µ < 1/ √ 3, and little is known rigorously in the nucleation region. See [2] for more details.…”

“…Consider the contribution to the energy provided by an equilibrium containing boundary droplets, interior droplets, or corner droplets, where µ and λ are fixed. Make the simplifying assumption that the solution is essentially constant +1 inside the droplet and constant −1 outside the droplet, where those constants correspond to the zeroes of F in (2). In this case, the only contribution to the energy E ε [u] occurs in the transition layer between the interior and exterior of the droplet, when the energy depends almost entirely on ε 2 = 1/λ times the integral of the gradient.…”

“…In fact, there are eight curves, but the ones for equilibria with labels 21, 22a, and 22b overlap exactly, as do the two curves for labels 31 and 32. For a heuristic un-derstanding of this, we appeal to the system energy (2). Consider the contribution to the energy provided by an equilibrium containing boundary droplets, interior droplets, or corner droplets, where µ and λ are fixed.…”

“…In this paper, we study a stochastic Cahn-Hilliard model of nucleation. Previous studies have used space-time white noise in this equation, and in this case, large deviation results give rigorous bounds on the time to nucleation and pattern formation at nucleation [2] at least in the small noise regime. In contrast, we concentrate on how the nature of the noise affects the droplet size.…”