Modelling anomalous diffusion in semi-infinite disordered systems and porous media
- For an effectively one-dimensional, semi-infinite disordered system connected to a reservoir of tracer particles kept at constant concentration, we provide the dynamics of the concentration profile. Technically, we start with the Montroll-Weiss equation of a continuous time random walk with a scale-free waiting time density. From this we pass to a formulation in terms of the fractional diffusion equation for the concentration profile C(x, t) in a semi-infinite space for the boundary condition C(0, t) = C-0, using a subordination approach. From this we deduce the tracer flux and the so-called breakthrough curve (BTC) at a given distance from the tracer source. In particular, BTCs are routinely measured in geophysical contexts but are also of interest in single-particle tracking experiments. For the "residual' BTCs, given by 1- P(x, t), we demonstrate a long-time power-law behaviour that can be compared conveniently to experimental measurements. For completeness we also derive expressions for the moments in thisFor an effectively one-dimensional, semi-infinite disordered system connected to a reservoir of tracer particles kept at constant concentration, we provide the dynamics of the concentration profile. Technically, we start with the Montroll-Weiss equation of a continuous time random walk with a scale-free waiting time density. From this we pass to a formulation in terms of the fractional diffusion equation for the concentration profile C(x, t) in a semi-infinite space for the boundary condition C(0, t) = C-0, using a subordination approach. From this we deduce the tracer flux and the so-called breakthrough curve (BTC) at a given distance from the tracer source. In particular, BTCs are routinely measured in geophysical contexts but are also of interest in single-particle tracking experiments. For the "residual' BTCs, given by 1- P(x, t), we demonstrate a long-time power-law behaviour that can be compared conveniently to experimental measurements. For completeness we also derive expressions for the moments in this constant-concentration boundary condition.…
Author details: | Ralf MetzlerORCiDGND, Ashish Rajyaguru, Brian Berkowitz |
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DOI: | https://doi.org/10.1088/1367-2630/aca70c |
ISSN: | 1367-2630 |
Title of parent work (English): | New journal of physics : the open-access journal for physics |
Publisher: | IOP Publ. Ltd. |
Place of publishing: | London |
Publication type: | Article |
Language: | English |
Date of first publication: | 2022/12/09 |
Publication year: | 2022 |
Release date: | 2024/09/23 |
Tag: | anomalous diffusion; breakthrough curves; constant boundary concentration; diffusion |
Volume: | 24 |
Issue: | 12 |
Number of pages: | 12 |
Funding institution: | German Research Foundation (DFG) [ME 1535/12-1]; European Union; [701647]; Swiss Society of Friends of the Weizmann Institute of Science;; Crystal Family Foundation; Estate of Claire Weiss; P. & A.; Guggenheim-Ascarelli Foundation |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie |
DDC classification: | 5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik |
Peer review: | Referiert |
Publishing method: | Open Access / Gold Open-Access |
DOAJ gelistet | |
License (German): | CC-BY - Namensnennung 4.0 International |