Anomalous diffusion in branched elliptical structure

SuleimanKheder; ZhangXuelan; WangErhui; LiuShengna; ZhengLiancun

doi:10.1088/1674-1056/ac5c39

Anomalous diffusion in branched elliptical structure

doi: 10.1088/1674-1056/ac5c39

1.
School of Energy and Environmental Engineering, University of Science and Technology Beijing,Beijing 100083,China
2.
School of Mathematics and Physics, University of Science and Technology Beijing,Beijing 100083,China

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出版历程
- 收稿日期: 2022-01-16
- 网络出版日期: 2023-05-16
- 刊出日期: 2023-01-01

摘要

- anomalous diffusion /
- Fokker-Planck equation /
- branched elliptical structure
Abstract: Diffusion in narrow curved channels with dead-ends as in extracellular space in the biological tissues, e.g., brain, tumors, muscles, etc. is a geometrically induced complex diffusion and is relevant to different kinds of biological, physical, and chemical systems. In this paper, we study the effects of geometry and confinement on the diffusion process in an elliptical comb-like structure and analyze its statistical properties. The ellipse domain whose boundary has the polar equation $ρ (θ) = \frac{b}{\sqrt{1 - e^{2} \cos^{2} θ}}$ with 0 < e < 1, θ ∈ [0,2 π], and b as a constant, can be obtained through stretched radius r such that Υ = r ρ ( θ) with r ∈ [0,1]. We suppose that, for fixed radius r = R, an elliptical motion takes place and is interspersed with a radial motion inward and outward of the ellipse. The probability distribution function (PDF) in the structure and the marginal PDF and mean square displacement (MSD) along the backbone are obtained numerically. The results show that a transient sub-diffusion behavior dominates the process for a time followed by a saturating state. The sub-diffusion regime and saturation threshold are affected by the length of the elliptical channel lateral branch and its curvature.

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1 Schematic picture of an elliptical comb-like structure, where the radii are continuously distributed over the ellipse of radius R = ρ( θ) with θ ∈ [0,2 π].

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2 The comparison of the exact solution via Eq.(16) with numerical solution for b = 1, e = 0.5, $D = \tilde{D} = 1$ , τ = 10 ⁻⁴, h _r = 10 ⁻², and h _θ = π/20 at T = 1.

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3 Time evolution of PDF in the backbone of classical comb, along x direction, for $D = \tilde{D} = 1$ , L = 5, and (a) R = 1.5, (b) R = 6.

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4 Time evolution of PDF in the backbone of classical comb, along x direction, for $D = \tilde{D} = 1$ , R = 3, and (a) L = 5, (b) L = 10.

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5 Time evolution of probability distribution in the structure for different value of the parameter b at T = 1: (a) $D = \tilde{D} = 1$ , e = 0.5, and b = 2, (b) $D = \tilde{D} = 1$ , e = 0.5, and b = 3, (c) $D = \tilde{D} = 1$ , e = 0.5, and b = 4, (d) $D = \tilde{D} = 1$ , e = 0.5, and b = 5.

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6 Time evolution of PDF in the backbone of elliptical comb, along direction θ, for $D = \tilde{D} = 1$ , e = 0.5, and (a) b = 2, (b) b = 8.

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7 Time evolution of PDF in the backbone of elliptical comb, along direction θ, for $D = \tilde{D} = 1$ , b = 4, and (a) e = 0.3, (b) e = 0.8.

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8 Temporal evolution of MSD in the backbone of classical comb, along x direction for $D = \tilde{D} = 1$ , R = {1.5, 3, 6}, and (a) L = 5, (b) L = 10.

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9 Temporal evolution of MSD in the backbone of classical comb, along x direction for $D = \tilde{D} = 1$ , L = {5, 7.5, 10}, and (a) R = 1.5, (b) R = 6.

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10 Temporal evolution of MSD in the backbone of classical comb, along θ direction for $D = \tilde{D} = 1$ , b = {2, 4, 5, 6, 8, 10}, and (a) e = 0, (b) e = 0.2, (c) e = 0.5, (d) e = 0.8.

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11 MSD along θ direction for short time for $D = \tilde{D} = 1$ , e = {0, 0.3, 0.5, 0.8}, and (a) b = 4, (b) b = 7.

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12 MSD along elliptical channel without constrained geometries for short time for $D = \tilde{D} = 1$ , and (a) e = 0, 0.5, 0.8, and b = 10, (b) e = 0.5 and b = 3, 6, 8.

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