A functional analytic approach to LSI and the hydrodynamic limit

Math Physics Seminar
Wednesday, May 4, 2011 - 16:30
1 hour (actually 50 minutes)
Skiles 006
School of Math, Georgia Tech
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.