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Math Seminar: "Nonlocal Continuum Models of Particle Systems"
4:10pm Neill 5W
Abstract: The main question addressed in the talk is how to obtain continuum equations for spatial averages from the ODEs of classical particle dynamics. Balance equations for the average density, linear momentum, and energy were derived by Irving and Kirkwood, Noll, Hardy, Murdoch and others. These equations are exact, but not in closed form since fluxes are given as functions of particle positions and velocities. Evaluating the exact fluxes requires solving the full ODE system which can be prohibitively expensive. We present a closure approximation that leads to continuum models in the true sense of the word. The fluxes in these models are given by operators acting on the average density and velocity. The closure construction is based on the use of regularized deconvolution. In the discrete setting, characterized by a finite non-improvable resolution, the error associated with deconvolution closure can be further reduced by incorporating a priori knowledge of empirical statistics of fluctuations. At the end of the talk we briefly discuss connections with large eddy simulation and quasi-continuum method. Results of numerical experiments and partial error estimates are presented as well.