Continuous time random walk in a velocity field: role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixing
We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also ma...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2020-01-01
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| Series: | New Journal of Physics |
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| Online Access: | https://doi.org/10.1088/1367-2630/ab9ae2 |
| _version_ | 1797750390550167552 |
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| author | F Le Vot E Abad R Metzler S B Yuste |
| author_facet | F Le Vot E Abad R Metzler S B Yuste |
| author_sort | F Le Vot |
| collection | DOAJ |
| description | We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a Lévy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of Lévy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter–controlled reactions in real systems are discussed. |
| first_indexed | 2024-03-12T16:32:05Z |
| format | Article |
| id | doaj.art-050c5321453b4f0cad5899f022bffd0d |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:32:05Z |
| publishDate | 2020-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-050c5321453b4f0cad5899f022bffd0d2023-08-08T15:24:54ZengIOP PublishingNew Journal of Physics1367-26302020-01-0122707304810.1088/1367-2630/ab9ae2Continuous time random walk in a velocity field: role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixingF Le Vot0https://orcid.org/0000-0002-4316-182XE Abad1https://orcid.org/0000-0002-1765-409XR Metzler2https://orcid.org/0000-0002-6013-7020S B Yuste3https://orcid.org/0000-0001-8679-4195Departamento de Física and Instituto de Computación Científica Avanzada, Universidad de Extremadura , 06071 Badajoz, SpainDepartamento de Física Aplicada and Instituto de Computación Científica Avanzada, Centro Universitario de Mérida, Universidad de Extremadura , 06800 Mérida, SpainInstitute for Physics & Astronomy, University of Potsdam , Karl-Liebknecht-Str 24/25, 14476 Potsdam, GermanyDepartamento de Física and Instituto de Computación Científica Avanzada, Universidad de Extremadura , 06071 Badajoz, SpainWe consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a Lévy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of Lévy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter–controlled reactions in real systems are discussed.https://doi.org/10.1088/1367-2630/ab9ae2diffusionexpanding mediumcontinuous time random walk |
| spellingShingle | F Le Vot E Abad R Metzler S B Yuste Continuous time random walk in a velocity field: role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixing New Journal of Physics diffusion expanding medium continuous time random walk |
| title | Continuous time random walk in a velocity field: role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixing |
| title_full | Continuous time random walk in a velocity field: role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixing |
| title_fullStr | Continuous time random walk in a velocity field: role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixing |
| title_full_unstemmed | Continuous time random walk in a velocity field: role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixing |
| title_short | Continuous time random walk in a velocity field: role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixing |
| title_sort | continuous time random walk in a velocity field role of domain growth galilei invariant advection diffusion and kinetics of particle mixing |
| topic | diffusion expanding medium continuous time random walk |
| url | https://doi.org/10.1088/1367-2630/ab9ae2 |
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