4:00 pm Monday, March 2, 2009
Colloquium: Particles and Fluids by Peter Constantin in RLM 6.104
Imagine many simple particles, each described by few degrees of freedom, like for instance the ball-and-stick models of molecules. Suppose that they are put in a bucket and the bucket is shaken vigorously a few times. Do the particles arrange themselves in any structured way? What if the bucket is filled with fluid? If the particles are rod-like, (and the interactions are electrostatic), then they will align. This was discovered by Onsager in the late forties, but proved mathematically only recently. I will describe a mathematical framework, that leads to an "Onsager equation" on metric spaces. I'll explain also, why such generality might be justified. The corresponding kinetic theory does not yet exist in such generality, but it corresponds to gradient flows on metric spaces. There exists a "good" kinetic theory on smooth Riemannian manifolds. When the particles are suspended in a fluid, then the effect that they have on the fluid is difficult to deduce from first principles. This is a manifestation of the fundamnetal problem of up-scaling -- deducing macroscopic properties from microscopic structure. I will explain a simple rule that gives familiar results for rod-like particles, and that is quite general. The mathematical analysis of the coupled system, Navier-Stokes equation for the fluid, Nonlinear Fokker-Planck equation for the kinetics of particles, is not trivial. I'll describe some new results obtained recently. Submitted by
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