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The paper introduces a general framework for derivation of continuumequations governing meso-scale dynamics of large particle systems. The balanceequations for spatial averages such as density, linear momentum, and energywere previously derived by a number of authors. These equations are not inclosed form because the stress and the heat flux cannot be evaluated withoutthe knowledge of particle positions and velocities. We propose a closure methodfor approximating fluxes in terms of other meso-scale averages. The main ideais to rewrite the non-linear averages as linear convolutions that relate micro-and meso-scale dynamical functions. The convolutions can be approximatelyinverted using regularization methods developed for solving ill-posed problems.This yields closed form constitutive equations that can be evaluated withoutsolving the underlying ODEs. We test the method numerically on Fermi-Pasta-Ulamchains with two different potentials: the classical Lennard-Jones, and thepurely repulsive potential used in granular materials modeling. The initialconditions incorporate velocity fluctuations on scales that are smaller thanthe size of the averaging window. The results show very good agreement betweenthe exact stress and its closed form approximation.

Deconvolution closure for mesoscopic continuum models of particle systems