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Institute for Mathematics and Its Applications
David Kinderlehrer, Carnegie Mellon University
Nonlinear systems which are active across disparate length and time scales are among the most intriguing ones we encounter in nature. Moreover, many of these systems, although persistent for long times, reside in metastable states and their evolution is poorly understood. We are, thus, presented with difficult scientific challenges, both in the derivation of appropriate frameworks for modeling and in the effective use of large scale simulation techniques for their execution. We focus here on a mechanism we believe to be deeply intertwined with these properties. This is the competition between the thermodynamic energy and nearness in the appropriate sense for the distribution of microscopic variables, the `averages' that describe the evolution of the macroscopic system. The result is a new derivation of the Fokker-Planck Equation as the gradient flux or steepest descent of the ordinary thermodynamic energy (e.g., minus the entropy in the case of ordinary diffusion or Brownian motion.). This is joint work with Richard Jordan and Felix Otto.
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