This paper investigates the motion of a Brownian particle experiencing both a friction (biased) force and a randomly fluctuating force with a long-time-correlation function C(f)(t) approximately t(-B), 0 < beta < 1, 1 < beta < 2, and beta = 1, instead of a Dirac delta-function. The generalized Langevin equation and Fokker-Planck equation and corresponding solution are presented. It is shown that when 0 < beta < 1 or 1 < beta < 2, the diffusion motion of the Brownian particle is the anomalous diffusion that is related to fractal Brownian motion (FBM). But when beta = 1 the diffusion motion is anomalous diffusion with no connection to FBM. The effects of friction retardation result in a probability density function for finding the particle at displacement X at time t that depends on the initial value of velocity of the particle. The approach in this paper may provide a systematic method for the study of particles diffusing in fractal media.
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chinese ctr adv sci & technol,world lab,beijing 100080,peoples r china. acad sinica,int ctr mat phys,shenyang 110015,peoples r china.;wang, kg (reprint author), acad sinica,inst met res,shenyang 100015,peoples r china
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