We are most concerned with the dynamical mechanisms for anomalous diffusion. We study the motion of a Brownian particle (or diffusing particle), when it is experienced a frictional force and random fluctuation force F(t) with long-time correlation ～ t~(-β), 0 < β < 1, β = 1, 1 < β < 2, and the second fluctuation-dissipation theorem is assumed to be valid. The motion of the Brownian particle takes over much interesting behaviour. We find that when 0 < β 1, 1 < β < 2, and the velocity autocorrelation function has the property ∫_0~∞Cv(t)dt ≠ E(0 < E < ∞), the diffusion motion of the particle is anomalous, and related to fractal Brownian motion. The drift motion of the particle is also anomalous. When β = 1, the velocity autocorrelation function C_v(t) ～ 1/t, the motion of the particle is anomalous diffusion with ～ tlnt, but not related to fractal Brownian motion. It is worthy mentioning that we have derived the Fokker-Planck equation for anomalous diffusion for the first time. In other hand, we investigate a particle diffusing in a disordered materials with Gaussian distribution of potential well depths. The distribution of transition rates is found to be lognormal. Using scaling theory, we find a new class of anomalous diffusion. This type of anomalous diffusion is also derived by us, in addition to continuous-time random walks. The probability for finding the particle at site X at time t and the probability coming back to the origin are derived.
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