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Stella AL, Chechkin A, Teza G. Anomalous Dynamical Scaling Determines Universal Critical Singularities. PHYSICAL REVIEW LETTERS 2023; 130:207104. [PMID: 37267558 DOI: 10.1103/physrevlett.130.207104] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2022] [Accepted: 04/19/2023] [Indexed: 06/04/2023]
Abstract
Anomalous diffusion phenomena occur on length scales spanning from intracellular to astrophysical ranges. A specific form of decay at a large argument of the probability density function of rescaled displacement (scaling function) is derived and shown to imply universal singularities in the normalized cumulant generator. Exact calculations for continuous time random walks provide paradigmatic examples connected with singularities of second order phase transitions. In the biased case scaling is restricted to displacements in the drift direction and singularities have no equilibrium analogue.
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Affiliation(s)
- Attilio L Stella
- Department of Physics and Astronomy, University of Padova, Via Marzolo 8, I-35131 Padova, Italy and INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy
| | - Aleksei Chechkin
- Institute of Physics and Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany, Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, University of Science and Technology, Wyspianskiego 27, 50-370 Wrocław, Poland, and Akhiezer Institute for Theoretical Physics National Science Center ''Kharkov Institute of Physics and Technology,'' 61108, Kharkiv, Ukraine
| | - Gianluca Teza
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
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Stella AL, Chechkin A, Teza G. Universal singularities of anomalous diffusion in the Richardson class. Phys Rev E 2023; 107:054118. [PMID: 37329006 DOI: 10.1103/physreve.107.054118] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2022] [Accepted: 04/21/2023] [Indexed: 06/18/2023]
Abstract
Inhomogeneous environments are rather ubiquitous in nature, often implying anomalies resulting in deviation from Gaussianity of diffusion processes. While sub- and superdiffusion are usually due to contrasting environmental features (hindering or favoring the motion, respectively), they are both observed in systems ranging from the micro- to the cosmological scale. Here we show how a model encompassing sub- and superdiffusion in an inhomogeneous environment exhibits a critical singularity in the normalized generator of the cumulants. The singularity originates directly and exclusively from the asymptotics of the non-Gaussian scaling function of displacement, and the independence from other details confers it a universal character. Our analysis, based on the method first applied by Stella et al. [Phys. Rev. Lett. 130, 207104 (2023)10.1103/PhysRevLett.130.207104], shows that the relation connecting the scaling function asymptotics to the diffusion exponent characteristic of processes in the Richardson class implies a nonstandard extensivity in time of the cumulant generator. Numerical tests fully confirm the results.
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Affiliation(s)
- Attilio L Stella
- Department of Physics and Astronomy, University of Padova, Via Marzolo 8, I-35131 Padova, Italy and INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy
| | - Aleksei Chechkin
- Institute of Physics and Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany; Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, University of Science and Technology, Wyspianskiego 27, 50-370 Wrocław, Poland; and Akhiezer Institute for Theoretical Physics, National Science Center "Kharkov Institute of Physics and Technology", 61108 Kharkiv, Ukraine
| | - Gianluca Teza
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
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Balankin AS. Effective degrees of freedom of a random walk on a fractal. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062146. [PMID: 26764671 DOI: 10.1103/physreve.92.062146] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2015] [Indexed: 06/05/2023]
Abstract
We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν-dimensional space F(ν) equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν) and fractal dimensionalities is deduced. The intrinsic time of random walk in F(ν) is inferred. The Laplacian operator in F(ν) is constructed. This allows us to map physical problems on fractals into the corresponding problems in F(ν). In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.
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Affiliation(s)
- Alexander S Balankin
- Grupo "Mecánica Fractal," ESIME, Instituto Politécnico Nacional, México D.F., 07738, Mexico
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Balankin AS, Mena B, Martínez-González CL, Matamoros DM. Random walk in chemical space of Cantor dust as a paradigm of superdiffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:052101. [PMID: 23214828 DOI: 10.1103/physreve.86.052101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2012] [Revised: 07/31/2012] [Indexed: 06/01/2023]
Abstract
We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.
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Affiliation(s)
- Alexander S Balankin
- Grupo Mecánica Fractal, Instituto Politécnico Nacional, México Distrito Federal 07738, Mexico
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Kondratenko PS, Matveev LV. Random advection in a fractal medium with finite correlation length. Phys Rev E 2007; 75:051102. [PMID: 17677017 DOI: 10.1103/physreve.75.051102] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2006] [Indexed: 11/07/2022]
Abstract
We investigate tracer advection in fractal media with finite correlation length. The process is superdiffusive at early times and transforms to a classical diffusion regime later. At large time the spatial tail of the concentration has a two-stage structure with a long-distant part corresponding to superdiffusive regime.
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Sellers S, Barker JA. Generalized diffusion equation for anisotropic anomalous diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:061103. [PMID: 17280034 DOI: 10.1103/physreve.74.061103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2006] [Indexed: 05/13/2023]
Abstract
Motivated by studies of comblike structures, we present a generalization of the classical diffusion equation to model anisotropic, anomalous diffusion. We assume that the diffusive flux is given by a diffusion tensor acting on the gradient of the probability density, where each component of the diffusion tensor can have its own scaling law. We also assume scaling laws that have an explicit power-law dependence on space and time. Solutions of the proposed generalized diffusion equation are consistent with previously derived asymptotic results for the probability density on comblike structures.
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Affiliation(s)
- S Sellers
- Mechanical and Aerospace Engineering, Washington University, St Louis, Missouri 63119, USA
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Sahimi M, Imdakm AO. The effect of morphological disorder on hydrodynamic dispersion in flow through porous media. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/21/19/019] [Citation(s) in RCA: 99] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Sahimi M. Hydrodynamic dispersion near the percolation threshold: scaling and probability densities. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/20/18/013] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Roman HE, Alemany PA. Continuous-time random walks and the fractional diffusion equation. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/27/10/017] [Citation(s) in RCA: 68] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Giona M, Schwalm WA, Schwalm MK, Adrover A. Exact solution of linear transport equations in fractal media—II. Diffusion and convection. Chem Eng Sci 1996. [DOI: 10.1016/0009-2509(96)00308-9] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Exact solution of linear transport equations in fractal media—I. Renormalization analysis and general theory. Chem Eng Sci 1996. [DOI: 10.1016/0009-2509(96)00307-7] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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First-order kinetics in fractal catalysts: Renormalization analysis of the effectiveness factor. Chem Eng Sci 1996. [DOI: 10.1016/0009-2509(96)00084-x] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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Hattori T, Nakajima H. Transition density of diffusion on the Sierpinski gasket and extension of Flory's formula. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:1202-1205. [PMID: 9963527 DOI: 10.1103/physreve.52.1202] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
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Eisenberg E, Havlin S, Weiss GH. Fluctuations of the probability density of diffusing particles for different realizations of a random medium. PHYSICAL REVIEW LETTERS 1994; 72:2827-2830. [PMID: 10055995 DOI: 10.1103/physrevlett.72.2827] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Sahimi M. Fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media. Transp Porous Media 1993. [DOI: 10.1007/bf00613269] [Citation(s) in RCA: 79] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Kimmich R, Weber HW. NMR relaxation and the orientational structure factor. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:11788-11794. [PMID: 10005348 DOI: 10.1103/physrevb.47.11788] [Citation(s) in RCA: 63] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Kim Y. Flory approximants and self-avoiding walks on critical percolation clusters. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 45:6103-6106. [PMID: 9907710 DOI: 10.1103/physreva.45.6103] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Zumofen G, Klafter J, Blumen A. Interdomain gaps in transient A+B-->0 reactions on fractals. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 44:8394-8397. [PMID: 9905998 DOI: 10.1103/physreva.44.8394] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Halley JW, Johnson B, Vallés JL. Model for diffusion with interactions and trapping on realizations of the percolation model. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 42:4383-4387. [PMID: 9995967 DOI: 10.1103/physrevb.42.4383] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Sahimi M. Diffusion, adsorption, and reaction in pillared clays. I. Rod‐like molecules in a regular pore space. J Chem Phys 1990. [DOI: 10.1063/1.458544] [Citation(s) in RCA: 24] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Sahimi M. Statistical physics of linear and nonlinear, scalar vector transport processes in disordered media. ACTA ACUST UNITED AC 1988. [DOI: 10.1016/0920-5632(88)90041-2] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Kärger J, Pfeifer H, Vojta G. Time correlation during anomalous diffusion in fractal systems and signal attenuation in NMR field-gradient spectroscopy. PHYSICAL REVIEW. A, GENERAL PHYSICS 1988; 37:4514-4517. [PMID: 9899589 DOI: 10.1103/physreva.37.4514] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Kärger J, Vojta G. On the use of NMR pulsed field-gradient spectroscopy for the study of anomalous diffusion in fractal networks. Chem Phys Lett 1987. [DOI: 10.1016/0009-2614(87)85050-9] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Fourcade B, Tremblay A. Diffusion noise of fractal networks and percolation clusters. PHYSICAL REVIEW. B, CONDENSED MATTER 1986; 34:7802-7812. [PMID: 9939462 DOI: 10.1103/physrevb.34.7802] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Lakhtakia A, Messier R, Varadan VK, Varadan VV. Use of combinatorial algebra for diffusion on fractals. PHYSICAL REVIEW. A, GENERAL PHYSICS 1986; 34:2501-2504. [PMID: 9897547 DOI: 10.1103/physreva.34.2501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Hansen A, Nelkin M. Nyquist noise in a fractal resistor network. PHYSICAL REVIEW. B, CONDENSED MATTER 1986; 33:649-651. [PMID: 9937968 DOI: 10.1103/physrevb.33.649] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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