Random walks and diffusion

The usual solution of the diffusion equation gives inaccurate results for short diffusion times for displacements which are relatively large compared to the maximum range of the diffusion. A method is proposed for solving the diffusion equation by a random-walk approximation technique which utilizes the appropriate generating functions and the complex integration method of steepest descent. The power of the method is illustrated for a one-dimensional random walk (SRW) and for physical situations in which the elementary diffusion process is governed by a normal or a rectangular probability distribution. Formulas for higher approximations applicable to any probability distribution are also derived. The method, which can also be applied to two and three dimensions, is capable of yielding high accuracy for the diffusion solution over the entire diffusion domain.