Generalized diffusion equation of interacting brownian particles

Macroscopic behavior of a system of brownian particles interacting with each other through potential forces is described by a generalized diffusion equation (GDE) for the density of particles. The diffusion coefficient in the GDE is given by the generalized Stokes—Einstein relation and generally depends on the density. In the presence of long-range interactions, the GDE becomes non-local in space. When a Coulomb interaction exists, the GDE corresponds to an improvement of the Poisson—Boltzmann equation.