Closure method for spatially averaged dynamics of particle chains

We study the closure problem for continuum balance equations that model the mesoscale dynamics of large ODE systems. The underlying microscale model consists of classical Newton equations of particle dynamics. As a mesoscale model we use the balance equations for spatial averages obtained earlier by a number of authors: Murdoch and Bedeaux, Hardy, Noll and others. The momentum balance equation contains a flux (stress), which is given by an exact function of particle positions and velocities. We propose a method for approximating this function by a sequence of operators applied to the average density and momentum. The resulting approximate mesoscopic models are systems in closed form. The closed form property allows one to work directly with the mesoscale equations without the need to calculate the underlying particle trajectories, which is useful for the modeling and simulation of large particle systems. The proposed closure method utilizes the theory of ill-posed problems, in particular iterative regularization methods for solving first order linear integral equations. The closed form approximations are obtained in two steps. First, we use Landweber regularization to (approximately) reconstruct the interpolants of the relevant microscale quantities from the average density and momentum. Second, these reconstructions are substituted into the exact formulas for stress. The developed general theory is then applied to non-linear oscillator chains. We conduct a detailed study of the simplest zero-order approximation, and show numerically that it works well as long as the fluctuations of velocity are nearly constant.