The theory of Brownian motion was developed first by Einstein for a single spherical particle immersed in a fluid and subjected to an external potential. Einstein realized that on a sufficiently slow time scale the observed random displacements of the particle can be described by a generalized diffusion equation. The particle momentum can be ignored, since it changes on a much faster time scale. Its probability distribution rapidly becomes nearly maxwellian, corresponding to the temperature of the fluid. Einstein's simple equation for a single particle was generalized by Smoluchowski to the mutual diffusion of a pair of particles, interacting with a central potential. He also considered the possibility of reaction between the two partners. The modern theory of interacting Brownian particles is based on a generalized Smoluchowski equation for the probability distribution of the entire configuration of particles.
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