Brownian motion in a field of force and the diffusion theory of chemical reactions

Kramers 1 ) has derived a diffusion equation in phase space, describing the motion of a particle subject to an external force and to the shuttling action of the Brownian forces caused by a surrounding medium in temperature equilibrium. In this paper a solution of Kramers' equation is obtained by systematic development of the particle density with respect to the Smoluchowski diffusion current operator. It is shown that Smoluchowski's diffusion equation in coordinate space is a first approximation to Kramers' equation, valid if all but the first power of the Smoluchowski current operator may be neglected. The region of validity of the Smoluchowski equation is determined for a particle in a potential well.