Front Fluctuations in One Dimensional Stochastic Phase Field Equations

Abstract

Abstract. We consider a conservative system of stochastic PDE's, namely a weakly coupled, one dimensional phase field model with additive noise. We study the fluctuations of the front proving that, in a suitable scaling limit, the front evolves according to a non-Markov process, solution of a linear stochastic equation with long memory drift. Part I. Introduction 1 General setting, model, and resultsThe term "sharp interface limit" denotes a scaling procedure aimed at the derivation of interfaces as geometric objects, e.g. surfaces of codimension one with bounded variation, that is, enough regular for the area measure to be well defined. Of course this makes only sense in the context of systems which undergo phase transitions and of states where different phases coexist. In the limit the other degrees of freedom are lost and we are left with the interface alone. Rigorous proofs are hard, yet a great variety of models has been successfully worked out. The mathematics involved is correspondingly rich, e.g. the theory of Γ-convergence (to study the sharp interface limit of Ginzburg-Landau like free energy functionals in relation with the equilibrium shape of the interface, as in the Wulff problem) and correspondingly the theory of Gibbsian large deviations (to study the same problems at the more microscopic level of statistical mechanics); singular limit in PDE's, like in the Allen-Cahn, Cahn-Hilliard and phase field equations, and correspondingly, at the microscopic level, hydrodynamic limits of spin or particle systems.This paper deals with fluctuations. Here again the questions are, first, whether in a sharp interface limit the system is described by a [fluctuating] interface with closed equations of motion and, secondly, the nature of such equations. The problem greatly simplifies in one space dimension where the limit interface is represented by a point which separates the two phases (one to its left and the other one *

Front Motion in the One-Dimensional Stochastic Cahn--Hilliard Equation

“…The motion of interfaces for Cahn-Hilliard equation was only studied in an unpublished note by S. Brassesco in 2003, where she studied a solution with a single interface on R. When properly rescaled, the interface is driven by nonMarkovian dynamics (cf. [12] for a similar result). In [46], the authors present a numerical study of the late stages of spinodal decomposition with noise.…”

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

“…Let us stress that the consideration of SDEs with coefficients depending on the entire past path of a process ("non-Markov case") is particularly important in certain applications, including the modelling of the dynamics of polymers moving in a random medium, see, e.g. [8,13,19].…”

Stochastic Phase Field Equations: Existence and Uniqueness

“…We also mention that the stochastic system (1.1) is very similar to the so-called, in the physical literature on critical phenomena, model C of Hohenberg and Halperin [5]. The need of an existence and uniqueness result for the stochastic system (1.1) in [1] is the main motivation for the present paper. Moreover, for λ small, m and h are weakly coupled so that we may refer to (1.1) as a weakly conservative system.…”

“…Referring to [1] for a more exhaustive discussion on the stochastic phase field equations, we next state precisely our results. For α > 0 and γ ∈ (0, 1], let us define the following norms on C(R):…”

“…Henri Poincaré 1 2 ∆)f are continuous on (0, T ) × R, f (0) = 0 and |f | 2n+2 and |∇f | 2n+2 are in L 1 (µ T ), then, for any n = 0, 1, . Ann.…”