Numerical Analysis Seminars: From particles to diffusion: a new limit passage via entropy
9 March 2012 14:00 in CG83
The diffusion equation is the scaling limit of Brownian motion, and can be reformulated as gradient flow of the entropy in the Wasserstein metric. The latter formulation is physically very appealing, since it reveals in a mathematically rigorous the entropy as driving force out of equilibrium. How can we directly obtain the entropic gradient flow as a scaling limit of particles undergoing Brownian motion? One passage, combining a Large Deviation principle from probability and Gamma-convergence from analysis, will be presented. Extensions to other particle systems and other gradient flows will be described, and the result will be embedded in previous results for the equilibrium setting. If time permits we will discuss Kramers' equation, an equation describing for example particle motion in a fluctuating environment, or plasma or stellar dynamics, and describe how ideas from optimal transport and gradient flows lead to a Hamiltonian-dissipative splitting scheme.
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