Anomalous and subanomalous diffusion in stochastic trapping transport

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Abstract
Stochastic transport in the trapping model in d dimensions is studied with use of the generalized coherent-medium approximation. The existence of a transition between two states with vanishing dc diffusion and finite dc diffusion is predicted. The transition is determined by the inverse moment of the jump rate, in agreement with the exact result. The frequency-dependent (ac) part of the diffusion constant is shown to have a different frequency dependence for various values of the parameter of the distribution of the jump rate. Anomalous diffusion is obtained when the first inverse moment does not exist. The anomalous diffusion exponent agrees with the value predicted by a renormalization-group method. When the first inverse moment exists but the second inverse moment does not, a subanomalous diffusion is predicted where the dc diffusion exists and the ac part of the diffusion constant shows a power-law dependence on the frequency with a nonintegral exponent less than unity or a logarithmic dependence. This gives rise to a diverging term in the mean-square displacement in addition to a t-linear term determined by the dc diffusion. The spectral exponent is obtained, which is related to the anomalous diffusion exponent for a class of distribution. It is shown that the real part of the ac diffusion constant in three dimensions can behave as ∼ω, which implies the imaginary part behaves as ∼ω lnω. This behavior of the diffusion constant qualitatively agrees with ionic conduction in a fast-ionic conductor. Analysis for the diffusion constant, the mean-square displacement, and the staying probability is carried out in general d dimensions and relations among various exponents are discussed.