Beschreibung:
The diffusion equation in physical space–time for a Brownian particle driven by an external force field has been derived by Smoluchowski in the two particular cases where the external field is uniform or varies linearly with position (elastic force). In more general cases, correction terms must be added to the Smoluchowski equation. We show here how to use a multi-scale Chapman–Enskog expansion to obtain, in the hydrodynamic limit, the first corrective terms to the Smoluchowski equation, without any restriction on the friction coefficient, and for any sufficiently small position-dependent constant force field. We also compare our approach with the works of Wilemski, Titulaer, and van Kampen.