Numerical analysis of the Smoluchowski equation using the split operator method

We apply the split operator method of Feit, Fleck and Steiger to the numerical solution of the Smoluchowski equation. We study the time evolution of the first few moments and of the equilibrium time-correlation function. For bistable potentials, the time evolution of the probability density obtained by us is compared with the one predicted by the scaling theory of Suzuki. We also analyze the dependence of the total equilibrium relaxation time on the noise intensity, D, for small and large values of D. The relaxation from an initial condition away from the unstable point is also discussed. We find that the one-well population is well described by a single exponential law with decay times which grow extremely large as the value of D is decreased.