Study of Diffusion-controlled Processes. Solution of the Smoluchowski Equation with a Step Potential

Study of Diffusion-controlled Processes. Solution of the Smoluchowski Equation with a Step Potential

The Smoluchowski equation with a step potential is solved in one-dimensional case and three-dimensional case with spherical symmetry. Exact analytic expressions for the solution and the remaining probability are obtained in one-dimensional case for the reflecting boundary condition and the long time behavior of the remaining probability is compared with the earlier work. In three-dimensional case, only the long time behavior is evaluated. More general case with the radiation boundary condition is also investigated and the results are shown to approach correct limits of the reflecting boundary condition.