A new algorithm for the numerical solution of diffusion problems related to the Smoluchowski equation

Diffusion-influenced reactions can often be described with simple kinetic m odels, whose basic features are a spherically symmetric potential, a distan ce-dependent relative diffusion coefficient, and a distance-dependent first -order rate coefficient. A new algorithm for the solution of the correspond ing Smoluchowski equation has been developed. Its peculiarities are: (1) A logarithmic increase of the radius; (2) the systematic use of numerical fun damental solutions w of the Smoluchowski equation: (3) the use of polynomia ls of up to the 8(th) degree for the definition of the first and second par tial derivatives of w with respect to the radius; (4) successive doubling o f the total diffusion time. The power of the algorithm is illustrated by ex amples. In particular its usefulness for the combination of a short-range p otential with a large radial range is demonstrated. Some aspects of the alg orithm are explained in the context of one-dimensional diffusion. Diffusion in a harmonic potential (Ornstein-Uhlenbeck process) and in a double-minim um 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.