GLAUBER EVOLUTION WITH KAC POTENTIALS .1. MESOSCOPIC AND MACROSCOPIC LIMITS, INTERFACE DYNAMICS

This is the first of three papers on the Glauber evolution of Ising sp in systems with Kac potentials. We begin with the analysis of the meso scopic limit, where space scales like the diverging range, gamma-1, of the interaction while time is kept finite: we prove that in this limi t the magnetization density converges to the solution of a determinist ic, 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 --> 0 but are small in units log gamma-1. In this time regime we can give a very precise d escription of the evolution and a sharp characterization of the spin t rajectories. As an application of the general theory, we then prove th at 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 correlat ion functions, the v-functions, whose analysis is reminiscent of the c luster expansion in equilibrium statistical mechanics.