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Heterogeneous diffusion in comb and fractal grid structures

Abstract

We give an exact analytical results for diffusion with a power-law position dependent diffusion coefficient along the main channel (backbone) on a comb and grid comb structures. For the mean square displacement along the backbone of the comb we obtain behavior x2(t)t1/(2α)\langle x^2(t)\rangle\sim t^{1/(2-\alpha)}, where α\alpha is the power-law exponent of the position dependent diffusion coefficient D(x)xαD(x)\sim |x|^{\alpha}. Depending on the value of α\alpha we observe different regimes, from anomalous subdiffusion, superdiffusion, and hyperdiffusion. For the case of the fractal grid we observe the mean square displacement, which depends on the fractal dimension of the structure of the backbones, i.e., x2(t)t(1+ν)/(2α)\langle x^2(t)\rangle\sim t^{(1+\nu)/(2-\alpha)}, where 0<ν<10<\nu<1 is the fractal dimension of the backbones structure. The reduced probability distribution functions for both cases are obtained by help of the Fox HH-functions

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