A New Algorithm for the Numerical Solution of Diffusion Problems Related to the Smoluchowski Equation By Bernhard Nickel

Diffusion-influenced reactions can often be described with simple kinetic models, whose basic features are a spherically symmetric potential, a distance-dependent relative diffusion coefficient, and a distance-dependent first-order rate coefficient. A new algorithm for the solution of the corresponding Smoluchowski equation has been developed. Its peculiarities are: (1) A logarithmic increase of the radius; (2) the systematic use of numerical fundamental solutions w of the Smoluchowski equation; (3) the use of polynomials of up to the 8 degree for the definition of the first and second partial derivatives of w with respect to the radius; (4) successive doubling of the total diffusion time. The power of the algorithm is illustrated by examples. In particular its usefulness for the combination of a short-range potential with a large radial range is demonstrated. Some aspects of the algorithm are explained in the context of one-dimensional diffusion. Diffusion in a harmonic potential (Ornstein-Uhlenbeck process) and in a double-minimum potential is treated in detail. It is shown that a detailed balance will in general not lead to the best approximation of the time-dependence of a distribution.