Enza Orlandi

Stability of the interface in a model of phase separation.

The paper is concerned with the asymptotic behaviour of the solutions to a nonlocal evolution equation which arises in models of phase separation. As in the Allen–Cahn equations, stationary spatially nonhomogeneous solutions exist, which represent the interface profile between stable phases. Local stability of these interface profiles is proved.

Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics

This is the first of three papers on the Glauber evolution of Ising spin systems with Kac potentials. We begin with the analysis of the mesoscopic limit, where space scales like the diverging range, gamma -1, of the interaction while time is kept finite: we prove that in this limit the magnetization density converges to the solution of a deterministic, nonlinear, nonlocal evolution equation. We also show that the long time behaviour of this equation describes correctly the evolution of the spin system till times which diverge as gamma to 0 but are small in units log gamma -1. In this time regime we can give a very precise description of the evolution and a sharp characterization of the spin trajectories. As an application of the general theory, we then prove that for ferromagnetic interactions, in the absence of external magnetic fields and below the critical temperature, on a suitable macroscopic limit, an interface between two stable phases moves by mean curvature. All the proofs are consequence of sharp estimates on special correlation functions, the v-functions, whose analysis is reminiscent of the cluster expansion in equilibrium statistical mechanics.

A particle model for spinodal decomposition

We study a one-dimensional lattice gas where particles jump stochastically obeying an exclusion rule and having a “small” drift toward regions of higher concentration. We prove convergence in the continuum limit to a nonlinear parabolic equation whenever the initial density profile satisfies suitable conditions which depend on the strengtha of the drift. There is a critical valueac ofa. Fora<ac, the density values are unrestricted, while fora⩾ac, they should all be to the right or to the left of a given interval ℐ(a). The diffusion coefficient of the limiting equation can be continued analytically to ℐ(a), and, in the interior of ℐ(a), it has negative values which should correspond to particle aggregation phenomena. We also show that the dynamics can be obtained as a limit of a Kawasaki evolution associated to a Kac potential. The coefficienta plays the role of the inverse temperatureβ. The critical value ofa coincides with the critical inverse temperature in the van der Waals limit and ℐ(a) with the spinodal region. It is finally seen that in a scaling intermediate between the microscopic and the hydrodynamic, the system evolves according to an integrodifferential equation. The instanton solutions of this equation, as studied by Dal Passo and De Mottoni, are then related to the phase transition region in the thermodynamic phase diagram; analogies with the Cahn-Hilliard equations are also discussed.

Interfaces and typical Gibbs configurations for one-dimensional Kac potentials

SummaryWe consider a one dimensional Ising spin system with a ferromagnetic Kac potential γJ(γ|r|),J having compact support. We study the system in the limit, γ»0, below the Lebowitz-Penrose critical temperature, where there are two distinct thermodynamic phases with different magnetizations. We prove that the empirical spin average in blocks of size δγ−1 (for any positive δ) converges, as γ»0, to one of the two thermodynamic magnetizations, uniformly in the intervals of size γ−p, for any given positivep≧1. We then show that the intervals where the magnetization is approximately constant have lengths of the order of exp(cγ−1),c>0, and that, when normalized, they converge to independent variables with exponential distribution. We show this by proving large deviation estimates and applying the Ventsel and Friedlin methods to Gibbs random fields. Finally, if the temperature is low enough, we characterize the interface, namely the typical magnetization pattern in the region connecting the two phases.

Motion by curvature by scaling nonlocal evolution equations

We prove convergence to a motion by mean curvature by scaling diffusively a nonlinear, nonlocal evolution equation. This equation was introduced earlier to describe the macroscopic behavior of a ferromagnetic spin system with Kac interaction which evolves with Glauber dynamics. The convergence is proven in any time interval in which the limiting motion is regular.

The Laplacian in Regions with Many Small Obstacles: Fluctuations Around the Limit Operator

We consider the Laplacianδm in ℝ3 (or in a bounded region of ℝ3) with Dirichlet boundary conditions on the surfaces of some identical (small) neighborhoods ofm randomly distributed points, in the limit whenm goes to infinity and their linear size decreases as 1/m. We give here a stronger form of the result showing the convergence of the above operator toδ − C(x), whereC(x) is the limit density of electrostatic capacity of the “obstacles.” In particular results on the rate of convergence and on the fluctuations ofδm around the limit operator are given.

Glauber evolution with Kac potentials: III. Spinodal decomposition

This is the last of a series of papers on the Glauber dynamics of spin systems in with Kac potentials. It deals with phase separation, studying the evolution of an initial state which is a Bernoulli measure with zero average while the temperature of the Glauber dynamics is below the critical value. The state with 0 magnetization is then thermodynamically unstable and we prove that it is so also dynamically. In fact the stable phases, that have magnetization , develop into non-trivial patterns after times proportional to , the range of the Kac interaction. We characterize the typical spin configurations, both during the separation and when this is completed. In particular, we study the magnetization pattern at the boundaries of the clusters and the development of the interfaces.

Travelling fronts in nonlocal models for phase separation in an external field

We consider the one-dimensional, nonlocal, evolution equation derived by De Masi et al. (1995) for Ising systems with Glauber dynamics, Kac potentials and magnetic field. We prove the existence of travelling fronts, their uniqueness modulo translations among the monotone profiles and their linear stability for all the admissible values of the magnetic field for which the underlying spin system exhibits a stable and metastable phase.

Typical Configurations for One-Dimensional Random Field Kac Model

In this paper we study the typical profiles of a random field Kac model. We give upper and lower bounds of the space scaie where the profiles are constant. The results hold almost surely with respect to the realizations of the random field. The analysis is based on a block-spin construction, deviation techniques for the local empirical order parameters and concentration inequalities for the realizations of the random magnetic field. For the upper bound, we exhibit a scale related to the law of the iterated logarithm, where the random field makes an almost sure fluctuation that obliges the system to break its rigidity. For the lower bound, we prove that on a smaller scale the fluctuations are not strong enough to allow this transition.

Lattice Gas Model in Random Medium and Open Boundaries: Hydrodynamic and Relaxation to the Steady State

We consider a lattice gas interacting by the exclusion rule in the presence of a random field given by i.i.d. bounded random variables in a bounded domain in contact with particles reservoir at different densities. We show, in dimensions d≥3, that the rescaled empirical density field almost surely, with respect to the random field, converges to the unique weak solution of a quasilinear parabolic equation having the diffusion matrix determined by the statistical properties of the external random field and boundary conditions determined by the density of the reservoir. Further we show that the rescaled empirical density field, in the stationary regime, almost surely with respect to the random field, converges to the solution of the associated stationary transport equation.