Abstract
We study themacroscopic limit of an appropriately rescaledstochastic Ising model withlong range interactions evolving withGlauber dynamics as well as the correspondingmean field equation, which is nonlinear and nonlocal. In the limit we obtain an interface evolving with normal velocity ϑk, wherek isthe mean curvature and thetransport coefficient ϑ is identified by aneffective Green-Kubo type formula. The above assertions are valid for all positive times, the motion of the interface being interpreted in theviscosity sense after the onset of the geometric singularities.
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[AC] Allen, S., Cahn, J.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Act. Metall.27, 1089–1095 (1979)
[B] Bonaventura, L.: Motion by curvature in an interacting spin system. Preprint
[BSS] Barles, G., Soner, H.M., Souganidis, P.E.: Front Propagation and Phase Field Theory. Siam J. Cont. Opt.31, 439–469 (1993)
[C] Comets, F.: Nucleation for a long range magnetic model. Ann. Inst. H. Poincaré23, 135–178 (1987)
[CE] Comets, F., Eisele, T.: Asymptotic dynamics, noncritical and critical fluctuations for a geometric long range interacting model. Commun. Math. Phys.118, 531–568 (1988)
[CGG] Chen, Y.-G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Diff. Geom.33, 749–786 (1991)
[CIL] Crandall, M.G., Ishii, H., Lions, P.L.: User's guide to viscosity solutions of second order partial differential equations. Bull. AMS27, 1–67 (1992)
[DD] Dal Passo, R., De Mottoni, P.: The heat equation with a non-local density dependent advection term. Preprint
[DGP] De Masi, A., Gobron, T., Presutti, E.: Traveling fronts in nonlocal evolution equations. Preprint
[DOPT1] De Masi, A., Orlandi, E., Presutti, E., Triolo, L.: Glauber evolution with Kač potentials: I. Mesoscopic and macroscopic limits, interface dynamics. Preprint
[DOPT2] De Masi, A., Orlandi, E., Presutti, E., Triolo, L.: Glauber evolution with Kač potentials: II. Spinodal decomposition. Preprint
[DOPT3] De Masi, A., Orlandi, E., Presutti, E., Triolo, L.: Motion by curvature by scaling non local evolution equations. Preprint
[DOPT4] De Masi, A., Orlandi, E., Presutti, E., Triolo, L.: Stability of the interface in a model of phase separation. Proc. Royal Soc. Edinb., to appear
[DP] De Masi, A., Presutti, E.: Mathematical Methods for Hydrodynamic Limits. Lecture Notes in Mathematics, Berlin, Heidelberg, New York: Springer, 1991
[E] Evans, L.C.: The perturbed test function method for viscosity solutions of non-linear PDE. Proc. Royal Soc. Edinb.111A, 359–375 (1989)
[ES] Evans, L.C., Spruck, J.: Motion of level sets by mean curvature I. J. Diff. Geom.33, 635–681 (1991)
[ESS] Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math.XLV, 1097–1123 (1992)
[G] Gurtin, M.E.: Multiphase thermomechanics with interfacial structure. 1. Heat conduction and the capillarity balance law. Arch. Rat. Mech. Anal.104, 185–221 (1988)
[HL] Hemmer, P.C., Lebowitz, J.L.: Systems with weak long-range potentials. In: Phase transitions and critical phenomena, Vol 5b, Eds Domb, C., Green, M.S., London: Academic Press, 1976
[I] Ilmanen, T.: Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature. J. Diff. Geom.38, 417–461 (1993)
[IS] Ishii, H., Souganidis, P.E.: Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor. Tohuko Math. J., in Press
[J] Jerrard, R.: Fully nonlinear phase field equations and generalized mean curvature motion. CPDE to appear
[KS] Katsoulakis, M.A., Souganidis, P.E.: Interacting particle systems and generalized mean curvature evolution. Arch. Rat. Mech. Anal., to appear
[KUH] Kac, M., Uhlenbeck, G.E., Hemmer, P.C.: On the Van der Waals theory of vapor-liquid equilibrium. I. Discussion of an one-dimensional model. J. Math. Phys.4, 216–228 (1963)
[L] Liggett, T.: Interacting particle systems. Berlin, Heidelberg, New York: Springer, 1985
[LOP] Lebowitz, J.L., Orlandi, E., Presutti, E.: A particle model for spinodal decomposition. J. Stat. Phys.63, 933–975 (1991)
[LP] Lebowitz, J., Penrose, O.: Rigorous treatment of the Van der Waals Maxwell theory of the liquid vapour transition. J. Math. Phys.98, 98–113 (1966)
[LPV] Lions, P.L., Papanicolaou, G., Varadhan, S.R.S.: Homogenization of Hamilton-Jacobi equations. Preprint
[P] Penrose, O.: A mean field equation of motion for the dynamic Ising model. J. Stat. Phys.63, 975–986 (1991)
[So1] Soner, H.M.: Motion of a set by the curvature of its boundary. J. Diff. Eq.101, 313–372 (1993)
[So2] Soner, H.M.: Ginzburg-Landau equation and motion by mean curvature, I: Convergence. J. Geom. Anal., to appear
[Sp] Spohn, H.: Interface motion in models with stochastic dynamics. J. Stat. Phys.71, 1081–1132 (1993)
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Communicated by T. Spencer
Supported by ONR
Partially supported by NSF, ARO, ONR and the Alfred P. Sloan Foundation
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Katsoulakis, M.A., Souganidis, P.E. Generalized motion by mean curvature as a macroscopic limit of stochastic ising models with long range interactions and Glauber dynamics. Commun.Math. Phys. 169, 61–97 (1995). https://doi.org/10.1007/BF02101597
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DOI: https://doi.org/10.1007/BF02101597