/pdf/topological-microstructure-analysis-using-persistence-1ud3my9skq.pdf

Topological Microstructure Analysis Using Persistence Landscapes

We study the stochastic mass-conserving Allen--Cahn equation posed on a smoothly bounded domain of $\mathbb{R}^2$ with additive, spatially smooth, space-time noise. This equation describes the stochastic motion of a small almost semicircular droplet attached to domain's boundary and moving toward a point of locally maximum curvature. We apply Ito calculus to derive the stochastic dynamics of the center of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen, and Fusco [Calc. Var., 11 (2000), pp. 233--305] for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption of a sufficiently small noise strength, we establish stochastic stability of a neighborhood of the manifold of boundary droplet states in the $L^2$- and $H^1$-norms, which means that with overwhelming probability the solution stays close to the manifold for very l...

/pdf/motion-of-a-droplet-for-the-stochastic-mass-conserving-allen-43o6z6wa7e.pdf

Motion of a droplet for the stochastic mass-conserving allen-cahn equation ∗

The diblock copolymer model is a fourth-order parabolic partial differential equation which models phase separation with fine structure. The equation is a gradient flow with respect to an extension of the standard van der Waals free energy functional which involves nonlocal interactions. Thus, the long-term dynamical behavior of the diblock copolymer model is described by its finite-dimensional attractor. However, even on one-dimensional domains, the full structure of the underlying equilibrium set is not fully understood. In this paper, we develop a rigorous computational approach for establishing the existence of equilibrium solutions of the diblock copolymer model. We consider both individual solutions, as well as pieces of solution branches in a parameter-dependent situation. The results are presented for the case of one-dimensional domains, and can easily be implemented using standard interval arithmetic packages.

https://www.aimsciences.org/article/exportPdf?id=c5f67c6a-cc6c-48bd-890f-894fff69651e

Computer-assisted equilibrium validation for the diblock copolymer model

Despite their misleading label, rare events in stochastic systems
are central to many applied phenomena. In this paper, we concentrate
on one such situation - phase 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.

https://www.aimsciences.org/article/exportPdf?id=1674d889-dc35-40e9-8698-452c68b73bae

Nucleation in the one-dimensional stochastic Cahn-Hilliard model

We develop general methods for rigorously computing continuous branches of bifurcation points of equilibria, specifically focusing on fold points and on pitchfork bifurcations which are forced through \begin{document}${\mathbb{Z}}_2$\end{document} symmetries in the equation. We apply these methods to secondary bifurcation points of the one-dimensional diblock copolymer model.

Rigorous continuation of bifurcation points in the diblock copolymer equation

Cahn-Morral systems serve as models for several phase separation phenomena in multicomponent alloys. In this paper we study the dynamical aspects of nucleation in a stochastic version of these models using numerical simulations, concentrating on ternary, i.e., three-component, alloys on two- dimensional square domains. We perform numerical studies and give a statistical classification for the distribution of droplet types as the component structure of the alloy is varied. We relate these statistics to the low-energy equilibria of the deterministic equation.

/pdf/the-dynamics-of-nucleation-in-stochastic-cahn-morral-systems-1658phxhjv.pdf

Joseph J. Price

Papers

The Dynamics of Nucleation in Stochastic Cahn–Morral Systems