An analysis of smoluchowski equation for sedimentation

Summary

As an example of the theoretical analysis of the Smoluchowski equation (S eq.); the collective motion of Brownian particles in sedimentation is studied here exactly. First, using the standard procedures we show that the general solution of the S eq., which is satisfied with some given boundary and initial conditions, can be expressed as a series expansion by a complete set of orthogonal functions. Next, letting the region tend to infinity, we reduce the solution in series form to an asymptotic solution in integral form. Further we prove that execution of the integration just results in the formula described in Chandrasekhar’s review. Nowadays, the Brownian motion under external force-field has become a big target of active research. The analysis given hero is partially original and useful as a tool to solve the problems related to this motion such as the sedimentation (1,2), the surface kinetics of absorption or desorption (3,4), and the diffusion phenomena (5), etc.(*).