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Abstract
This review addresses issues of various drift–diffusion and inhomogeneous advection problems with and without resetting on comblike structures. Both a Brownian diffusion search with drift and an inhomogeneous advection search on the comb structures are analyzed. The analytical results are verified by numerical simulations in terms of coupled Langevin equations for the comb structure. The subordination approach is one of the main technical methods used here, and we demonstrated how it can be effective in the study of various random search problems with and without resetting.
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Bénichou O, Illien P, Oshanin G, Sarracino A, Voituriez R. Diffusion and Subdiffusion of Interacting Particles on Comblike Structures. PHYSICAL REVIEW LETTERS 2015; 115:220601. [PMID: 26650285 DOI: 10.1103/physrevlett.115.220601] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2015] [Indexed: 06/05/2023]
Abstract
We study the dynamics of a tracer particle (TP) on a comb lattice populated by randomly moving hard-core particles in the dense limit. We first consider the case where the TP is constrained to move on the backbone of the comb only. In the limit of high density of the particles, we present exact analytical results for the cumulants of the TP position, showing a subdiffusive behavior ∼t^{3/4}. At longer times, a second regime is observed where standard diffusion is recovered, with a surprising nonanalytical dependence of the diffusion coefficient on the particle density. When the TP is allowed to visit the teeth of the comb, based on a mean-field-like continuous time random walk description, we unveil a rich and complex scenario with several successive subdiffusive regimes, resulting from the coupling between the geometrical constraints of the comb lattice and particle interactions. In this case, remarkably, the presence of hard-core interactions asymptotically speeds up the TP motion along the backbone of the structure.
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Affiliation(s)
- O Bénichou
- Sorbonne Universités, UPMC Université Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005 Paris, France
| | - P Illien
- Sorbonne Universités, UPMC Université Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005 Paris, France
| | - G Oshanin
- Sorbonne Universités, UPMC Université Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005 Paris, France
| | - A Sarracino
- Sorbonne Universités, UPMC Université Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005 Paris, France
| | - R Voituriez
- Sorbonne Universités, UPMC Université Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005 Paris, France
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Sandev T, Iomin A, Kantz H. Fractional diffusion on a fractal grid comb. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032108. [PMID: 25871055 DOI: 10.1103/physreve.91.032108] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/20/2014] [Indexed: 06/04/2023]
Abstract
A grid comb model is a generalization of the well known comb model, and it consists of N backbones. For N=1 the system reduces to the comb model where subdiffusion takes place with the transport exponent 1/2. We present an exact analytical evaluation of the transport exponent of anomalous diffusion for finite and infinite number of backbones. We show that for an arbitrarily large but finite number of backbones the transport exponent does not change. Contrary to that, for an infinite number of backbones, the transport exponent depends on the fractal dimension of the backbone structure.
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Affiliation(s)
- Trifce Sandev
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany and Radiation Safety Directorate, Partizanski odredi 143, P.O. Box 22, 1020 Skopje, Macedonia
| | - Alexander Iomin
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany and Department of Physics, Technion, Haifa 32000, Israel
| | - Holger Kantz
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
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