Sharp-interface limit of a Ginzburg-Landau functional with a random external field

We add a random bulk term, modeling the interaction with the impurities of the medium, to a standard functional in the gradient theory of phase transitions consisting of a gradient term with a double-well potential. For the resulting functional we study the asymptotic properties of minimizers and minimal energy under a rescaling in space, i.e., on the macroscopic scale. By bounding the energy from below by a coarse-grained, discrete functional, we show that for a suitable strength of the random field the random energy functional has two types of random global minimizers, corresponding to two phases. Then we derive the macroscopic cost of low energy “excited” states that correspond to a bubble of one phase surrounded by the opposite phase

Phase Segregation Dynamics in Particle Systems with Long Range Interactions I: Macroscopic Limits

We present and discuss the derivation of a nonlinear non-local integro-differential equation for the macroscopic time evolution of the conserved order parameter of a binary alloy undergoing phase segregation. Our model is a d-dimensional lattice gas evolving via Kawasaki exchange dynamics, i.e. a (Poisson) nearest-neighbor exchange process, reversible with respect to the Gibbs measure for a Hamiltonian which includes both short range (local) and long range (nonlocal) interactions. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (part II), we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.Comment: amstex with macros (included in the file), tex twice, 20 page

Transient Nucleation near the Mean-Field Spinodal

Nucleation is considered near the pseudospinodal in a one-dimensional $\phi^4$ model with a non-conserved order parameter and long-range interactions. For a sufficiently large system or a system with slow relaxation to metastable equilibrium, there is a non-negligible probability of nucleation occurring before reaching metastable equilibrium. This process is referred to as transient nucleation. The critical droplet is defined to be the configuration of maximum likelihood that is dynamically balanced between the metastable and stable wells. Time-dependent droplet profiles and nucleation rates are derived, and theoretical results are compared to computer simulation. The analysis reveals a distribution of nucleation times with a distinct peak characteristic of a nonstationary nucleation rate. Under the quench conditions employed, transient critical droplets are more compact than the droplets found in metastable equilibrium simulations and theoretical predictions.Comment: 7 Pages, 5 Figure

Capillary Wave Theory of Adsorbed Liquid Films and the Structure of the Liquid-Vapor Interface

In this paper we try to work out in detail the implications of a microscopic theory for capillary waves under the assumption that the density is given along lines normal to the interface. Our study provides interface Hamiltonians for adsorbed films in a variety of systems, and shows that the corrections to the classical capillary wave spectrum are of the same order as the surface tension. This implies that it is possible, at least in principle, to measure them in x-ray surface scattering experiments. Interestingly, our study also sheds some light on the nature of the liquid-vapor interface in the absence of external fields and allows us to reconcile the Fisk-Widom scaling hypothesis with capillary wave theory.Comment: Revised version, 51 pages, 4 figure

Driven Diffusive Systems: An Introduction and Recent Developments

Nonequilibrium steady states in driven diffusive systems exhibit many features which are surprising or counterintuitive, given our experience with equilibrium systems. We introduce the prototype model and review its unusual behavior in different temperature regimes, from both a simulational and analytic view point. We then present some recent work, focusing on the phase diagrams of driven bi-layer systems and two-species lattice gases. Several unresolved puzzles are posed.Comment: 25 pages, 5 figures, to appear in Physics Reports vol. 299, June 199