Georgia Institute of Technology College of Sciences

The logarithmic Sobolev inequality (LSI) is a powerful tool for studying convergence to equilibrium in spin systems. The Bakry-Emery criterion implies LSI in the case of a convex Hamiltonian. What can be said in the nonconvex case? We present two recent sufficient conditions for LSI. The first is a Bakry-Emery-type criterion that requires only LSI (not convexity) for the single-site conditional measures. The second is a two-scale condition: An LSI on the microscopic scale (conditional measures) and an LSI on the macroscopic scale (marginal measure) are combined to prove a global LSI. We extend the two-scale method to derive an abstract theorem for convergence to the hydrodynamic limit which we then apply to the example of Guo-Papanicolaou-Varadhan. We also survey some new results.This work is joint with Grunewald, Otto, and Villani.

Colloids are mixtures of molecules  well-studied in material science that are not well-understood mathematically.  Physicists model colloids as a system of two types of tiles (type A and type B) embedded on a region of the plane, where no two tiles can overlap.  It is conjectured that at high density, the type A tiles tend to separate out and form large "clusters".   To verify this conjecture, we need methods for counting these configurations directly or efficient algorithms for sampling.  Local sampling algorithms are known to be inefficient. However, we provide the first rigorous analysis of a global "DK Algorithm" introduced by Dress and Krauth.  We also examine the clustering effect directly via a combinatorial argument. We prove for a certain class of colloid models that at high density the configurations are likely to exhibit clustering, whereas at low density the tiles are all well-distributed. Joint work with Sarah Miracle and Dana Randall.