Kinetic equation for spatially averaged molecular dynamics

We obtain a kinetic description of spatially averaged dynamics of particle systems. Spatial averaging is one of the three types of averaging relevant within the Irwing-Kirkwood procedure (IKP), a general method for deriving macroscopic equations from molecular models. The other two types, ensemble averaging and time averaging, have been extensively studied, while spatial averaging is relatively less understood. We show that spatially averaged density, linear momentum and kinetic energy can be obtained from a single generating function. A kinetic equation for the generating function is obtained. The deconvolution closure used in the derivation becomes more accurate with increasing density. This yields a kinetic description suitable for fluids and amorphous solids. In addition, we consider extensions of the theory which include time averaging and coarsening in the velocity space. Since averaging is essentially a low-pass filter which damps high frequency oscillations, adding time- and velocity-variable averaging allows us to filter out all oscillations below a specified threshold. Unlike traditional kinetic theory, the proposed equation applies to single realizations of atomistic dynamics.