For anomalous diffusion the dependence of the mean square displacement < r~2(t) > on the diffusion time t usually differs from the linear dependence expected for the normal (i.e. Fickian) diffusion < r~2(t) > = 2dDt. (1) Eq.1 is known as the Einstein's equation with D and d denoting the self-diffusion coefficient and the dimensionality of the diffusion space, respectively. Proportionality between the mean square displacement and the diffusion time in Eq.1 may be easily rationalised when describing the diffusion as a random walk. In this case it is assumed that during a certain elementary time interval τ each particle makes a diffusion step of length l in arbitrary direction. Assuming further that the particle displacements during different elementary time intervals (i≠j) are uncorrelated we obtain < r~2(t = nτ) > = < ∑ from i=1 to n of r_i r_i > + < ∑ from i≠j=1 to n of r_i r_j > = n (l)~2 ∝ t, (2) where r_i is the displacement vector in the i-th interval of time. Under the conditions of anomalous diffusion, displacements in different time intervals are usually correlated, so that the second sum in Eq.2 does not vanish as in the case of normal diffusion and, consequently, the mean square displacement cannot be expected to increase proportionally to the diffusion time t.
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