Commun. Math. Phys. 185, 1 – 36 (1997) Communications in Mathematical Physics c  Springer-Verlag 1997 Motion by Mean Curvature from the Ginzburg-Landau ∇φ Interface Model T. Funaki 1 , H. Spohn 2 ,⋆ 1 Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153 Japan. E-mail: funaki@ms.u-tokyo.ac.jp 2 Theoretische Physik, Ludwig-Maximilians-Universit¨ at, Theresienstr. 37, D-80333 M¨ unchen, Germany. E-mail: spohn@stat.physik.uni-muenchen.de Received: 1 February 1996 / Accepted: 2 July 1996 Abstract: We consider the scalar field φ t with a reversible stochastic dynamics which is defined by the standard Dirichlet form relative to the Gibbs measure with formal energy  d d xV (∇φ(x)). The potential V is even and strictly convex. We prove that under a suitable large scale limit the φ t -field becomes deterministic such that locally its normal velocity is proportional to its mean curvature, except for some anisotropy effects. As an essential input we prove that for every tilt there is a unique shift invariant, ergodic Gibbs measure for the ∇φ-field. 1. Introduction and Main Results There has been a considerable effort to prove that particle models with stochastic dynam- ics behave deterministically and are governed by a suitable partial differential equation on a sufficiently coarse scale [21,32]. Almost exclusively, the models are constructed in such a way that they have at least one local conservation law, like the number of particles. The locally conserved fields vary slowly in space and therefore also slowly in time. It is this slow motion which persists on a sufficiently coarse space-time scale and which is governed by a partial differential equation. Local degrees of freedom relax quickly under given constraints and have a statistics as defined by the corresponding Gibbs measure. As pointed out long ago, physically, slow motion arises also from broken symmetry. The prime example is the ferromagnetic Ising model at low temperatures and zero external field, which has then the two distinct shift invariant pure phases (ergodic Gibbs measures) µ + and µ − . We assume reversible spin-flip dynamics - no conservation law. Then in the pure phases the relaxation is (essentially) exponentially fast. If however we prepare µ + and µ − as spatially coexisting and glued together at a fairly sharp interface, then such a situation will persist (essentially) forever. Of course, in general the interface ⋆ H.S. is partially supported by DFG