Diffusion in the Presence of a Force Field

In this chapter, we derive the diffusion equation for the case when an external or internal force on each of the diffusing particles is applied. This equation is referred to in the literature as the Smoluchowski equation, which is fundamental in the study of stochastic dynamics, as it can be applied to a number of problems related to physics and chemistry. An important result is that the backward Smoluchowski operator is the adjoint operator of the forward Smoluchowski operator. We also found that for particles diffusing in a potential, detailed balance guarantees the absence of net fluxes at equilibrium.