Coarsening in stochastically perturbed Ginzburg -Landau -type equations and statistical studies of the Potts model

• Main Author: Fatkullin, Ibrahim
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-2002
• Subjects: Mathematics
• ISBN: 9780493731124; 0493731121

This work is devoted to the statistical description of certain stochastic partial differential equations (PDEs) which exhibit the so-called phase separation dynamics. In particular we consider the randomly perturbed scalar Ginzburg-Landau (or Allen-Cahn) equation (1.1) and the Cahn-Hilliard equation (1.2). Deterministic dynamics of these systems in certain asymptotic limits is quickly attracted to the so-called slow manifold—a set of functions assuming a discrete set of values (phases) almost everywhere, and further on proceeds restricted to this set. We develop a formalism which allows the analysis of the asymptotic dynamics in both the deterministic and stochastic settings via a restriction of the full gradient flow to the slow manifold. The second problem that we study is the influence of small stochastic perturbations on the reduced dynamics of the aforementioned systems. It turns out that in the proper asymptotic limit the deterministic dynamics of one-dimensional systems is totally dominated by the noise, and reduces to the dynamics of an ensemble of particles which undergo Brownian motions and interact by collision. We also discuss the nucleation phenomenon which consists of the creation of new domains induced by the large fluctuations of the stochastic forcing. We complete the PDE aspect of this work by summarizing these ideas and establishing the connection with the so-called Potts model with voters dynamics, to which we devote the rest of our studies. Our interest lies in the understanding of the coarsening phenomenon. It appears in connection with the question of the general structure of solutions in spatially extended systems, i.e., in the situation when there exists a large number of domains (connected regions of a certain phase), and concerns the distribution of domain sizes and their elimination in the process of evolution. The Potts model with voters dynamics is a stochastic process on lattice spin systems. It describes the switching of the spin values at random times to the values of their immediate neighbors. The continuous limit of the Potts model is directly related to the reduced dynamics of the Ginzburg-Landau equation and therefore the analysis of the statistical properties of the former provides understanding of the coarsening phenomena in original PDEs. We begin with the introduction of some basic facts concerning one-dimensional spin lattices and comparison of several different ways to obtain their probabilistic description, e.g., by means of correlation functions and domain-length densities. Next we derive evolution equations for these quantities induced by the voters dynamics and analyze their continuous limits. We find that an infinite hierarchy of coupled equations is necessary to provide the complete description and discuss possible decouplings and closures.