Wulff Droplets and the Metastable Relaxation of Kinetic Ising Models

Abstract:We consider the kinetic Ising models (Glauber dynamics) corresponding to the infinite volume Ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h. Minimal conditions on the flip rates are assumed, so that all the common choices are being considered. We study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature T and the evolution is started from a distribution which is stochastically lower than the (−)-phase. We show that as h↘ 0 the relaxation time blows up as exp(λc(T)/h), with lgr;c(T) =w(T)2/(12 T m*(T)). Here m*(T) is the spontaneous magnetization and w(T) is the integrated surface tension of the Wulff body of unit volume. Moreover, for 0 < λ < λc, the state of the process at time exp(λ/h) is shown to be close, when h is small, to the (−)-phase. The difference between this state and the (−)-phase can be described in terms of an asymptotic expansion in powers of the external field. This expansion can be interpreted as describing a set of ?∞ continuations in h of the family of Gibbs distributions with the negative magnetic fields into the region of positive fields.

[1]  F. Martinelli.  On the two-dimensional dynamical Ising model in the phase coexistence region , 1994 .

[2]  Metastable lifetimes in a kinetic Ising model: Dependence on field and system size. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  R. Kotecḱy,et al.  Droplet dynamics for asymmetric Ising model , 1993 .

[4]  Michael Aizenman,et al.  Metastability effects in bootstrap percolation , 1988 .

[5]  C. Pfister.  Large deviations and phase separation in the two-dimensional Ising model , 1991 .

[6]  Roberto H. Schonmann,et al.  Metropolis Dynamics Relaxation via Nucleation and Growth , 1997 .

[7]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[8]  O. Penrose,et al.  Towards a rigorous molecular theory of metastability , 1979 .

[9]  R. Schonmann.  Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region , 1994 .

[10]  D. Ioffe.  Exact large deviation bounds up toTc for the Ising model in two dimensions , 1995 .

[11]  R. Schonmann.  Theorems and Conjectures on the Droplet-Driven Relaxation of Stochastic Ising Models , 1994 .

[12]  S. N. Isakov.  Nonanalytic features of the first order phase transition in the Ising model , 1984 .

[13]  Roberto H. Schonmann,et al.  Complete analyticity for 2D Ising completed , 1995 .

[14]  A. Martin-Löf.  Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature , 1973 .

[15]  K. Binder,et al.  Investigation of metastable states and nucleation in the kinetic Ising model , 1974 .

[16]  D. Ioffe.  Large deviations for the 2D ising model: A lower bound without cluster expansions , 1994 .

[17]  S. Shlosman,et al.  Constrained variational problem with applications to the Ising model , 1996 .

[18]  James D. Gunton,et al.  Introduction to the Theory of Metastable and Unstable States , 1983 .