A random motion of Winterbottom-like shape

Tadahisa Funaki (Univ. Tokyo)

We are interested in deriving a motion of crystals from certain microscopic models. The derivation of a crystal shape, called Wulff shape, from the Ising model goes back to Dobrushin, Kotecký and Shlosman [1] under the equilibrium situation. The natural choice of the corresponding dynamics would be the Kawasaki dynamics (a system of random walks on a lattice with exclusion rule). However, at present, mathematically rigorous results for the derivation of a crystalline motion from such dynamics are not known (cf. [2]) except the zero temperature limit for a system of interacting Brownian particles, which is a continuous analogue of Kawasaki dynamics, see [3]. Our goal is to discuss this kind of problem in the context of the so-called ∇φinterface model. The following topics were discussed in my lecture after introducing the ∇φ-interface model with self potentials and the scaling from microscopic to macroscopic levels in this model.

(1) (Static theory) For the corresponding Gibbs measures, large deviation principles are obtained under several situations and the law of large numbers is deduced from them. The limits are characterized by variational principles, which determine Wulff shape, Winterbottom shape, free boundary problem of Alt-Caffarelli’s type and others depending on the situations we are concerned.

(2) (Dynamic theory) The corresponding dynamics are defined through the Langevin equation of non-conservative or conservative types. The hydrodynamic limit, the equilibrium fluctuation and the dynamic entropic repulsion under the situation with hard walls are discussed.

(3) (Motion of Winterbottom-like shape) If the macroscopic equilibrium shapes (i.e. large deviation minimizers) are not unique, one can expect to observe a random motion of the equilibrium shapes under the properly chosen time scale beyond the hydrodynamic one. This is still unsolved in a desirable way, for instance, for the ∇φ-interface dynamics of conservative type on a lattice torus. Instead, we consider related stochastic partial differential equations with smooth noise under a proper time scale, from which a diffusive motion of the equilibrium shapes is derived.

Recent results concerning the ∇φ-interface model are reviewed in the lecture notes [4].

References

[1] R.L. Dobrushin, R. Kotecký and S. Shlosman, Wulff Construction: a Global Shape from Local Interaction, AMS translation series, 104 [2] G. Favrin, E. Marinari and F. Martinelli, Droplet motion for the

conservative 2D Ising lattice gas dynamics below the critical temperature, J. Phys. A 34 (2001), pp. 5901–5910.

[3] T. Funaki, Zero temperature limit for interacting Brownian particles, I. Motion of a single body, Ann. Probab., 32 (2004), pp. 1201–1227.

[4] T. Funaki, Stochastic Interface Models, In: Lectures on Probability Theory and Statistics, Ecole d’Eté de Probabilités de Saint-Flour XXXIII - 2003 (ed. J. Picard), Lect. Notes Math., 1869 (2005), Springer.