THE DIFFUSION EQUATION FOR A CLASSICAL MECHANICAL SYSTEM IN A NONLINEAR FORCE-FIELD - AN IMPROVED TREATMENT

The diffusion of a uni-dimensional classical mechanical system in a stochastic force field with white spectrum is analysed by splitting the velocity into a stochastic diffusive component plus a deterministic part. Making use of the expansion of the full response function in terms of an approximate expression, the conditional average of the stochastic component results in terms of a single functional including the response function and the autocorrelation of the random force. This result is subsequently applied to an arbitrary system in the limit of high friction, so as to obtain a diffusion equation in configuration space (Smoluchowski equation).