Slow Motion, Rare Events, and Sharp Bounds in High-Dimensional Systems

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Nonlinear, random, and multi-scale phenomena dominate the physical world. This project investigates: (1) dynamic metastability, (2) rare events and action minimization, (3) logarithmic Sobolev inequalities, and (4) scaling bounds in thermal convection. Via the analysis of specific model problems such as heat transport in thermal convection or stochastically induced phase transformation, the project seeks to develop new tools and identify general trends in the areas of coarsening/decay rates and random nucleation. Dynamic metastability refers to systems that appear to be stable, although in fact they are far from equilibrium and will change dramatically after a long period of time. There are many problems in which the physically observed coarsening timescales are controversial, for instance certain epitaxial growth experiments; a better understanding of the connection between the statics and dynamics of complex energy landscapes can lend insight into these problems. Rare events, on the other hand, are in some sense the opposite extreme: How and at what rate does thermal noise drive systems away from energy minimizers? This work applies to problems in nanoscale magnetic memory devices, as well as diverse problems in chemistry and biology. Logarithmic Sobolev inequalities describe the trend to equilibrium in spin systems, and may be useful in the analysis of certain multi-scale numerical algorithms. Finally, thermal convection processes are important in a wide range of areas, including engineering, meteorology, oceanography, and astrophysics. The project investigates the scaling law for the Nusselt number (a nondimensional measure of heat transport), attempting to reconcile differences between the rate predicted analytically and the rate observed numerically.