Department of Mathematics Colloquium


Abstract 

A general framework for derivation of
continuum equations governing mesoscale dynamics of large particle
systems is introduced. The balance equations for spatial averages
such as density, linear momentum, and energy were previously derived by
a number of authors (Irving, Kirkwood, Hardy, Murdoch and Bedeaux).
These equations are not in closed form because the stress and the heat
flux cannot be evaluated without the knowledge of particle positions
and velocities. A new closure method for approximating fluxes in
terms of other mesoscale averages is proposed. The main idea is to
rewrite the nonlinear averages as linear convolutions that relate
micro and mesoscale dynamical functions. The convolutions can be
approximately inverted using regularization methods developed for
solving illposed problems. This yields closed form constitutive
equations that can be evaluated without solving the underlying
ODEs. We test the method numerically on FermiPastaUlam chains
with two different potentials: the classical LennardJones, and the
purely repulsive Hertz like potential used in granular materials
modeling. The initial conditions incorporate velocity fluctuations on
scales that are smaller than the size of the averaging window. The
results show very good agreement between the exact stress and its
closed form approximation.
