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.