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

Abstract

This is the fin1 of three papers on the Glauber evolution of king spin systems with Kac potentials. We begin with the analysis of the mesoscopic limit, where space scales like the diverging range. y-', of the interaction while time is kepi finite: ye 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 y + 0 but are small in units logy-'. In this time regime we can give a "ry precise description of the evolution and a sharp characterhion of the spin trajectories. As an application of the general theory, we then prove that for ferromagnetic interactions, in the absence of extema! magnetic fields and below the critical temperature, on a suitable macroscopic limit an interface between two stable phases moves by mean curvature. All ihe proofs are consequek of sharp estimates on special correlation functions, the v-functions, whose analysis is reminiscent of the cluster expansion in equilibrium statistical mechanics.

Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems

“…In addition, a discrete order parameter stochastic model, similar to the ones derived here from microscopics but without the inclusion of short range interactions, is constructed in [10], and following formal calculations in [17] is used to obtain a stochastic mesoscopic equations for diffusion and adsorption/desorption processes. Furthermore, in the context of a direct derivation of a mesoscopic model from a true microscopic one in the infinite particle limit, the coarse-grained stochastic processes proposed below are also intimately related to intermediate technical steps in the derivation of deterministic and stochastic mesoscopic models arising as mean field limits of Ising systems [4,7,8,16].…”

Normal Hyperbolicity and Continuity of Global Attractors for a Nonlocal Evolution Equations

“…We consider here (1) restricted to P 2 , with > 1. As shown in the previous work [7], this leads naturally to the consideration of the flow generated by (1) in 2 ( 1 ) where 1 is the unit sphere and * the convolution product in it. In what follows, we summarize the assumptions and results of [7].…”

“…Under hypothesis (H1) it was proved in [7] that the problem (1) is well posed in 2 ( 1 ) and its flow is 1 if we assume hypothesis (H2). Furthermore, assuming (H1) and (H2) the existence of a global compact attractor for the flow of (1) in the sense of [12] was also proved in [7].…”

“…There are several works in the literature dedicated to the analysis of (1) or its particular case when ≡ tanh (see [1][2][3][4][5][6][7][8]). …”

A stochastic phase-field model determined from molecular dynamics

“…To derive stochastic differential equations approximating kinetic Monte Carlo dynamics is a classical problem, studied e.g. in [7,20,28]. The work [16] derived coarse-grained stochastic differential equations from a kinetic Monte Carlo method with a technique related to the study here on molecular dynamics.…”