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 $⟨x^2(t)⟩\sim t^{1/(2-\alpha )}$, where $\alpha$ is the power-law exponent of the position dependent diffusion coefficient $D(x)\sim |x{|}^{\alpha }$. Depending on the value of $\alpha$ 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., $⟨x^2(t)⟩\sim t^{(1+\nu )/(2-\alpha )}$, where $0<\nu <1$ is the fractal dimension of the backbones structure. The reduced probability distribution functions for both cases are obtained by help of the Fox $H$-functions.

References in corpus (4)

• Fractional diffusion on a fractal grid comb
• Isothermal Langevin dynamics in systems with power-law spatially-dependent friction
• Non-Markovian Levy diffusion in nonhomogeneous media
• Influence of external potentials on heterogeneous diffusion processes

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• Probability distribution functions of sub- and super-diffusive systems
• Efficiency of random search with space-dependent diffusivity
• From continuous-time random walks to controlled-diffusion reaction