Multiscale derivation of an augmented Smoluchowski equation

Smoluchowski and Fokker–Planck equations for the stochastic dynamics of order parameters have been derived previously. The question of the validity of the truncated perturbation series and the initial data for which these equations exist remains unexplored. To address these questions, we take a simple example, a nanoparticle in a host medium. A perturbation parameter ε, the ratio of the mass of a typical atom to that of the nanoparticle, is introduced and the Liouville equation is solved to O(ε2). Via a general kinematic equation for the reduced probability W of the location of the center-of-mass of the nanoparticle, the O(ε2) solution of the Liouville equation yields an equation for W to O(ε3). An augmented Smoluchowski equation for W is obtained from the O(ε2) analysis of the Liouville equation for a particular class of initial data. However, for a less restricted assumption, analysis of the Liouville equation to higher order is required to obtain closure.