Diffusion in an External Potential

Abstract

We write down the Langevin equation for a tagged particle in the presence of an external or applied force derived from a time-independent potential. The general form of the SDE↔FPE correspondence is introduced. Using this result, we obtain the Fokker-Planck equation for the phase space density (the Kramers equation). The case of a harmonically bound particle (the Brownian oscillator) is discussed in detail. In the long-time or high-friction regime, the Kramers equation leads to a modified diffusion equation, the Smoluchowski equation, for the PDF of the position of the tagged particle. Kramers’ escape rate formula for thermally-activated diffusion over a potential barrier is derived. We also consider the case of a constant external field, applicable to the problem of sedimentation, in one of the exercises at the end of the chapter.