CORRELATION-EFFECTS, GENERALIZED BROWNIAN-MOTION AND ANOMALOUS DIFFUSION

We investigate the diffusion motion of a Brownian particle which is acted upon by both a friction force with memory effect and a noise. The noise is expressed as f(X, t) is similar to X-sigma F(t), sigma > 0, where X and t are the displacement and time, respectively, and F(t) has the long-time correlation effect [F(0) F(t)] is similar to t(-beta), 0 < beta < 1, beta = 1, 1 < beta < 2. The generalized Langevin equation, the corresponding Fokker-Planck equation and its solution at large time are established. A variety of anomalous diffusion patterns are proposed. The correlation effects of noise may bring about that the effective diffusion coefficient is dependent on both the displacement and time, and the probability density for finding the Brownian particle is a non-Gaussian distribution. O'Shaughnessy and Procaccia's results on the diffusion on fractals can be derived from our model.