中国物理B ›› 2023, Vol. 32 ›› Issue (1): 10202-010202.doi: 10.1088/1674-1056/ac5c39

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Anomalous diffusion in branched elliptical structure

Kheder Suleiman1,2, Xuelan Zhang(张雪岚)2, Erhui Wang(王二辉)1, Shengna Liu(刘圣娜)1, and Liancun Zheng(郑连存)2,†   

  1. 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
  • 收稿日期:2022-01-16 修回日期:2022-03-01 接受日期:2022-03-10 出版日期:2022-12-08 发布日期:2022-12-08
  • 通讯作者: Liancun Zheng E-mail:liancunzheng@ustb.edu.cn
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 11772046 and 81870345).

Anomalous diffusion in branched elliptical structure

Kheder Suleiman1,2, Xuelan Zhang(张雪岚)2, Erhui Wang(王二辉)1, Shengna Liu(刘圣娜)1, and Liancun Zheng(郑连存)2,†   

  1. 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
  • Received:2022-01-16 Revised:2022-03-01 Accepted:2022-03-10 Online:2022-12-08 Published:2022-12-08
  • Contact: Liancun Zheng E-mail:liancunzheng@ustb.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 11772046 and 81870345).

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摘要： 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 $\rho \left( \theta \right)=\frac{b}{\sqrt {1-e^{2}\cos^{2}\theta } }$ with $0<e<1$, $\theta \in \left[ 0,2\pi \right]$, and $b$ as a constant, can be obtained through stretched radius $r$ such that $\varUpsilon =r \rho \left( \theta \right)$ with $r\in \left[ 0,1 \right]$. 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.

关键词: 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 $\rho \left( \theta \right)=\frac{b}{\sqrt {1-e^{2}\cos^{2}\theta } }$ with $0<e<1$, $\theta \in \left[ 0,2\pi \right]$, and $b$ as a constant, can be obtained through stretched radius $r$ such that $\varUpsilon =r \rho \left( \theta \right)$ with $r\in \left[ 0,1 \right]$. 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.

Key words: anomalous diffusion, Fokker-Planck equation, branched elliptical structure

中图分类号:  (Stochastic analysis)

  • 02.50.Fz
02.50.Ey (Stochastic processes) 05.40.-a (Fluctuation phenomena, random processes, noise, and Brownian motion) 05.60.-k (Transport processes)