1
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Vezzani A, Burioni R. Fast Rare Events in Exit Times Distributions of Jump Processes. PHYSICAL REVIEW LETTERS 2024; 132:187101. [PMID: 38759165 DOI: 10.1103/physrevlett.132.187101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2023] [Revised: 03/11/2024] [Accepted: 04/04/2024] [Indexed: 05/19/2024]
Abstract
Rare events in the first-passage distributions of jump processes are capable of triggering anomalous reactions or series of events. Estimating their probability is particularly important when the jump probabilities have broad-tailed distributions, and rare events are therefore not so rare. We formulate a general approach for estimating the contribution of fast rare events to the exit probabilities in the presence of fat-tailed distributions. Using this approach, we study three jump processes that are used to model a wide class of phenomena ranging from biology to transport in disordered systems, ecology, and finance: discrete time random walks, Lévy walks, and the Lévy-Lorentz gas. We determine the exact form of the scaling function for the probability distribution of fast rare events, in which the jump process exits from an interval in a very short time at a large distance opposite to the starting point. In particular, we show that events occurring on timescales orders of magnitude smaller than the typical timescale of the process can make a significant contribution to the exit probability. Our results are confirmed by extensive numerical simulations.
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Affiliation(s)
- Alessandro Vezzani
- Istituto dei Materiali per l'Elettronica ed il Magnetismo (IMEM-CNR), Parco Area delle Scienze, 37/A-43124 Parma, Italy; Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Parco Area delle Scienze, 7/A 43124 Parma, Italy; and INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy
| | - Raffaella Burioni
- Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma, Parco Area delle Scienze, 7/A 43124 Parma, Italy and INFN, Gruppo Collegato di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy
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2
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Klinger J, Voituriez R, Bénichou O. Leftward, rightward, and complete exit-time distributions of jump processes. Phys Rev E 2023; 107:054109. [PMID: 37329110 DOI: 10.1103/physreve.107.054109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/05/2022] [Accepted: 04/03/2023] [Indexed: 06/18/2023]
Abstract
First-passage properties of continuous stochastic processes confined in a one-dimensional interval are well described. However, for jump processes (discrete random walks), the characterization of the corresponding observables remains elusive, despite their relevance in various contexts. Here we derive exact asymptotic expressions for the leftward, rightward, and complete exit-time distributions from the interval [0,x] for symmetric jump processes starting from x_{0}=0, in the large x and large time limit. We show that both the leftward probability F_{[under 0]̲,x}(n) to exit through 0 at step n and rightward probability F_{0,[under x]̲}(n) to exit through x at step n exhibit a universal behavior dictated by the large-distance decay of the jump distribution parametrized by the Levy exponent μ. In particular, we exhaustively describe the n≪(x/a_{μ})^{μ} and n≫(x/a_{μ})^{μ} limits and obtain explicit results in both regimes. Our results finally provide exact asymptotics for exit-time distributions of jump processes in regimes where continuous limits do not apply.
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Affiliation(s)
- J Klinger
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
- Laboratoire Jean Perrin, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
| | - R Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
- Laboratoire Jean Perrin, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
| | - O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
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3
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Klinger J, Voituriez R, Bénichou O. Splitting Probabilities of Symmetric Jump Processes. PHYSICAL REVIEW LETTERS 2022; 129:140603. [PMID: 36240405 DOI: 10.1103/physrevlett.129.140603] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2022] [Accepted: 09/13/2022] [Indexed: 06/16/2023]
Abstract
We derive a universal, exact asymptotic form of the splitting probability for symmetric continuous jump processes, which quantifies the probability π_{0,[under x]_}(x_{0}) that the process crosses x before 0 starting from a given position x_{0}∈[0,x] in the regime x_{0}≪x. This analysis provides in particular a fully explicit determination of the transmission probability (x_{0}=0), in striking contrast with the trivial prediction π_{0,[under x]_}(0)=0 obtained by taking the continuous limit of the process, which reveals the importance of the microscopic properties of the dynamics. These results are illustrated with paradigmatic models of jump processes with applications to light scattering in heterogeneous media in realistic 3D slab geometries. In this context, our explicit predictions of the transmission probability, which can be directly measured experimentally, provide a quantitative characterization of the effective random process describing light scattering in the medium.
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Affiliation(s)
- J Klinger
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
- Laboratoire Jean Perrin, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
| | - R Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
- Laboratoire Jean Perrin, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
| | - O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
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4
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Cui Y, Feng D, Kang K, Qin S. Anomalous band center localization in the one-dimensional Anderson model with a disordered distribution of infinite variance. Phys Rev E 2022; 105:024131. [PMID: 35291121 DOI: 10.1103/physreve.105.024131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2021] [Accepted: 02/03/2022] [Indexed: 06/14/2023]
Abstract
We perform a detailed numerical study of the influence of distributions without a finite second moment on the Lyapunov exponent through the one-dimensional tight-binding Anderson model with diagonal disorder. Using the transfer matrix parametrization method and considering a specific distribution function, we calculate the Lyapunov exponent numerically and demonstrate its relation with the fractional lower order moments of the disorder probability density function. For the lower order of moments of disorder distribution with an infinite variance, we obtain the anomalous behavior near the band center.
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Affiliation(s)
- Yang Cui
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China and School of Physical Sciences, University of Chinese Academy of Sciences. No. 19A Yuquan Road, Beijing 100049, China
| | - Delong Feng
- School of Medical Information and Engineering, Southwest Medical University, No. 1, Section 1, Xianglin Road, Longmatan District, Luzhou City, Sichuan Province 646000, China
| | - Kai Kang
- Department of Biomedical Engineering, Chengde Medical University, Anyuan Road, Shuangqiao District, Chengde City, Hebei Province 067000, China
| | - Shaojing Qin
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
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5
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Smith NR. Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process. Phys Rev E 2022; 105:014120. [PMID: 35193315 DOI: 10.1103/physreve.105.014120] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2021] [Accepted: 01/06/2022] [Indexed: 06/14/2023]
Abstract
We study the full distribution of A=∫_{0}^{T}x^{n}(t)dt, n=1,2,⋯, where x(t) is an Ornstein-Uhlenbeck process. We find that for n>2 the long-time (T→∞) scaling form of the distribution is of the anomalous form P(A;T)∼e^{-T^{μ}f_{n}(ΔA/T^{ν})} where ΔA is the difference between A and its mean value, and the anomalous exponents are μ=2/(2n-2) and ν=n/(2n-2). The rate function f_{n}(y), which we calculate exactly, exhibits a first-order dynamical phase transition which separates between a homogeneous phase that describes the Gaussian distribution of typical fluctuations, and a "condensed" phase that describes the tails of the distribution. We also calculate the most likely realizations of A(t)=∫_{0}^{t}x^{n}(s)ds and the distribution of x(t) at an intermediate time t conditioned on a given value of A. Extensions and implications to other continuous-time systems are discussed.
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Affiliation(s)
- Naftali R Smith
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 Rue Lhomond, F-75231 Paris Cedex, France and Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 8499000, Israel
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6
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Ślęzak J, Burov S. From diffusion in compartmentalized media to non-Gaussian random walks. Sci Rep 2021; 11:5101. [PMID: 33658556 PMCID: PMC7930099 DOI: 10.1038/s41598-021-83364-0] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2020] [Accepted: 12/18/2020] [Indexed: 01/31/2023] Open
Abstract
In this work we establish a link between two different phenomena that were studied in a large and growing number of biological, composite and soft media: the diffusion in compartmentalized environment and the non-Gaussian diffusion that exhibits linear or power-law growth of the mean square displacement joined by the exponential shape of the positional probability density. We explore a microscopic model that gives rise to transient confinement, similar to the one observed for hop-diffusion on top of a cellular membrane. The compartmentalization of the media is achieved by introducing randomly placed, identical barriers. Using this model of a heterogeneous medium we derive a general class of random walks with simple jump rules that are dictated by the geometry of the compartments. Exponential decay of positional probability density is observed and we also quantify the significant decrease of the long time diffusion constant. Our results suggest that the observed exponential decay is a general feature of the transient regime in compartmentalized media.
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Affiliation(s)
- Jakub Ślęzak
- Physics Department, Bar-Ilan University, Ramat Gan, 5290002 Israel
| | - Stanislav Burov
- Physics Department, Bar-Ilan University, Ramat Gan, 5290002 Israel
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7
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Stivanello S, Bet G, Bianchi A, Lenci M, Magnanini E. Limit theorems for Lévy flights on a 1D Lévy random medium. ELECTRON J PROBAB 2021. [DOI: 10.1214/21-ejp626] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | - Gianmarco Bet
- Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze, Italy
| | - Alessandra Bianchi
- Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze, Italy
| | - Marco Lenci
- Dipartimento di Matematica, Università di Bologna, Italy
| | - Elena Magnanini
- Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze, Italy
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8
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Radice M, Onofri M, Artuso R, Pozzoli G. Statistics of occupation times and connection to local properties of nonhomogeneous random walks. Phys Rev E 2020; 101:042103. [PMID: 32422811 DOI: 10.1103/physreve.101.042103] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2020] [Accepted: 03/16/2020] [Indexed: 11/07/2022]
Abstract
We consider the statistics of occupation times, the number of visits at the origin, and the survival probability for a wide class of stochastic processes, which can be classified as renewal processes. We show that the distribution of these observables can be characterized by a single exponent, that is connected to a local property of the probability density function of the process, viz., the probability of occupying the origin at time t, P(t). We test our results for two different models of lattice random walks with spatially inhomogeneous transition probabilities, one of which of non-Markovian nature, and find good agreement with theory. We also show that the distributions depend only on the occupation probability of the origin by comparing them for the two systems: When P(t) shows the same long-time behavior, each observable follows indeed the same distribution.
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Affiliation(s)
- Mattia Radice
- Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Università degli studi dell'Insubria, Via Valleggio 11, I-22100 Como, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
| | - Manuele Onofri
- Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Università degli studi dell'Insubria, Via Valleggio 11, I-22100 Como, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
| | - Roberto Artuso
- Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Università degli studi dell'Insubria, Via Valleggio 11, I-22100 Como, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
| | - Gaia Pozzoli
- Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Università degli studi dell'Insubria, Via Valleggio 11, I-22100 Como, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
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9
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Vezzani A, Barkai E, Burioni R. Rare events in generalized Lévy Walks and the Big Jump principle. Sci Rep 2020; 10:2732. [PMID: 32066775 PMCID: PMC7026067 DOI: 10.1038/s41598-020-59187-w] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2019] [Accepted: 01/13/2020] [Indexed: 11/22/2022] Open
Abstract
The prediction and control of rare events is an important task in disciplines that range from physics and biology, to economics and social science. The Big Jump principle deals with a peculiar aspect of the mechanism that drives rare events. According to the principle, in heavy-tailed processes a rare huge fluctuation is caused by a single event and not by the usual coherent accumulation of small deviations. We consider generalized Lévy walks, a class of stochastic processes with power law distributed step durations and with complex microscopic dynamics in the single stretch. We derive the bulk of the probability distribution and using the big jump principle, the exact form of the tails that describes rare events. We show that the tails of the distribution present non-universal and non-analytic behaviors, which depend crucially on the dynamics of the single step. The big jump estimate also provides a physical explanation of the processes driving the rare events, opening new possibilities for their correct prediction.
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Affiliation(s)
- Alessandro Vezzani
- IMEM, CNR Parco Area delle Scienze 37/A, 43124, Parma, Italy
- Department of Mathematics, Physics and Computer Science, University of Parma, viale G.P. Usberti 7/A, 43124, Parma, Italy
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 52900, Israel
| | - Raffaella Burioni
- Department of Mathematics, Physics and Computer Science, University of Parma, viale G.P. Usberti 7/A, 43124, Parma, Italy.
- INFN, Gruppo Collegato di Parma, viale G.P. Usberti 7/A, 43124, Parma, Italy.
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10
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Bianchi A, Lenci M, Pène F. Continuous-time random walk between Lévy-spaced targets in the real line. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2019.03.010] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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11
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Vezzani A, Barkai E, Burioni R. Single-big-jump principle in physical modeling. Phys Rev E 2019; 100:012108. [PMID: 31499929 DOI: 10.1103/physreve.100.012108] [Citation(s) in RCA: 26] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2018] [Indexed: 11/07/2022]
Abstract
The big-jump principle is a well-established mathematical result for sums of independent and identically distributed random variables extracted from a fat-tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: Instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big-jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory, and quenched disorder. Doing so we are able to predict rare events using the extended big-jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure, and in a class of Lévy walks with memory. We argue that the generalized big-jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy-tailed distributions, ranging from contamination spreading to active transport in the cell.
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Affiliation(s)
- Alessandro Vezzani
- IMEM, CNR Parco Area delle Scienze 37/A 43124 Parma.,Dipartimento di Matematica, Fisica e Informatica Università degli Studi di Parma, viale G.P. Usberti 7/A, 43100 Parma, Italy
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 52900, Israel
| | - Raffaella Burioni
- Dipartimento di Matematica, Fisica e Informatica Università degli Studi di Parma, viale G.P. Usberti 7/A, 43100 Parma, Italy.,INFN, Gruppo Collegato di Parma, viale G.P. Usberti 7/A, 43100 Parma, Italy
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12
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Zarfaty L, Peletskyi A, Fouxon I, Denisov S, Barkai E. Dispersion of particles in an infinite-horizon Lorentz gas. Phys Rev E 2018; 98:010101. [PMID: 30110737 DOI: 10.1103/physreve.98.010101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/25/2017] [Indexed: 06/08/2023]
Abstract
We consider a two-dimensional Lorentz gas with infinite horizon. This paradigmatic model consists of pointlike particles undergoing elastic collisions with fixed scatterers arranged on a periodic lattice. It was rigorously shown that when t→∞, the distribution of particles is Gaussian. However, the convergence to this limit is ultraslow, hence it is practically unattainable. Here, we obtain an analytical solution for the Lorentz gas' kinetics on physically relevant timescales, and find that the density in its far tails decays as a universal power law of exponent -3. We also show that the arrangement of scatterers is imprinted in the shape of the distribution.
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Affiliation(s)
- Lior Zarfaty
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Alexander Peletskyi
- Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg Germany
- Sumy State University, Rimsky-Korsakov Street 2, 40007 Sumy, Ukraine
| | - Itzhak Fouxon
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Sergey Denisov
- Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg Germany
- Department of Applied Mathematics, Lobachevsky State University of Nizhny Novgorod, Gagarina Avenue 23, Nizhny Novgorod 603950, Russia
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
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13
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Méndez-Bermúdez JA, Aguilar-Sánchez R. Information-Length Scaling in a Generalized One-Dimensional Lloyd's Model. ENTROPY 2018; 20:e20040300. [PMID: 33265391 PMCID: PMC7512818 DOI: 10.3390/e20040300] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/27/2017] [Revised: 03/29/2018] [Accepted: 04/08/2018] [Indexed: 01/28/2023]
Abstract
We perform a detailed numerical study of the localization properties of the eigenfunctions of one-dimensional (1D) tight-binding wires with on-site disorder characterized by long-tailed distributions: For large ϵ , P ( ϵ ) ∼ 1 / ϵ 1 + α with α ∈ ( 0 , 2 ] ; where ϵ are the on-site random energies. Our model serves as a generalization of 1D Lloyd's model, which corresponds to α = 1 . In particular, we demonstrate that the information length β of the eigenfunctions follows the scaling law β = γ x / ( 1 + γ x ) , with x = ξ / L and γ ≡ γ ( α ) . Here, ξ is the eigenfunction localization length (that we extract from the scaling of Landauer's conductance) and L is the wire length. We also report that for α = 2 the properties of the 1D Anderson model are effectively reproduced.
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Affiliation(s)
- J. A. Méndez-Bermúdez
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Puebla 72570, Mexico
- Correspondence: ; Tel.: +52-222-229-5610
| | - R. Aguilar-Sánchez
- Facultad de Ciencias Químicas, Benemérita Universidad Autónoma de Puebla, Puebla 72570, Mexico
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14
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Amanatidis I, Kleftogiannis I, Falceto F, Gopar VA. Coherent wave transmission in quasi-one-dimensional systems with Lévy disorder. Phys Rev E 2018; 96:062141. [PMID: 29347281 DOI: 10.1103/physreve.96.062141] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2017] [Indexed: 11/07/2022]
Abstract
We study the random fluctuations of the transmission in disordered quasi-one-dimensional systems such as disordered waveguides and/or quantum wires whose random configurations of disorder are characterized by density distributions with a long tail known as Lévy distributions. The presence of Lévy disorder leads to large fluctuations of the transmission and anomalous localization, in relation to the standard exponential localization (Anderson localization). We calculate the complete distribution of the transmission fluctuations for a different number of transmission channels in the presence and absence of time-reversal symmetry. Significant differences in the transmission statistics between disordered systems with Anderson and anomalous localizations are revealed. The theoretical predictions are independently confirmed by tight-binding numerical simulations.
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Affiliation(s)
- Ilias Amanatidis
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Republic of Korea
| | | | - Fernando Falceto
- Departamento de Física Teórica and BIFI, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
| | - Víctor A Gopar
- Departamento de Física Teórica and BIFI, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
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15
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Méndez-Bermúdez JA, Martínez-Mendoza AJ, Gopar VA, Varga I. Lloyd-model generalization: Conductance fluctuations in one-dimensional disordered systems. Phys Rev E 2016; 93:012135. [PMID: 26871052 DOI: 10.1103/physreve.93.012135] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2015] [Indexed: 11/07/2022]
Abstract
We perform a detailed numerical study of the conductance G through one-dimensional (1D) tight-binding wires with on-site disorder. The random configurations of the on-site energies ε of the tight-binding Hamiltonian are characterized by long-tailed distributions: For large ε, P(ε)∼1/ε^{1+α} with α∈(0,2). Our model serves as a generalization of the 1D Lloyd model, which corresponds to α=1. First, we verify that the ensemble average 〈-lnG〉 is proportional to the length of the wire L for all values of α, providing the localization length ξ from 〈-lnG〉=2L/ξ. Then, we show that the probability distribution function P(G) is fully determined by the exponent α and 〈-lnG〉. In contrast to 1D wires with standard white-noise disorder, our wire model exhibits bimodal distributions of the conductance with peaks at G=0 and 1. In addition, we show that P(lnG) is proportional to G^{β}, for G→0, with β≤α/2, in agreement with previous studies.
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Affiliation(s)
- J A Méndez-Bermúdez
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
| | - A J Martínez-Mendoza
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico.,Elméleti Fizika Tanszék, Fizikai Intézet, Budapesti Műszaki és Gazdaságtudományi Egyetem, H-1521 Budapest, Hungary
| | - V A Gopar
- Departamento de Física Teórica, Facultad de Ciencias, and BIFI, Universidad de Zaragoza, Pedro Cerbuna 12, E-50009, Zaragoza, Spain
| | - I Varga
- Elméleti Fizika Tanszék, Fizikai Intézet, Budapesti Műszaki és Gazdaságtudományi Egyetem, H-1521 Budapest, Hungary
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16
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Spanner M, Höfling F, Kapfer SC, Mecke KR, Schröder-Turk GE, Franosch T. Splitting of the Universality Class of Anomalous Transport in Crowded Media. PHYSICAL REVIEW LETTERS 2016; 116:060601. [PMID: 26918973 DOI: 10.1103/physrevlett.116.060601] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2015] [Indexed: 06/05/2023]
Abstract
We investigate the emergence of subdiffusive transport by obstruction in continuum models for molecular crowding. While the underlying percolation transition for the accessible space displays universal behavior, the dynamic properties depend in a subtle nonuniversal way on the transport through narrow channels. At the same time, the different universality classes are robust with respect to introducing correlations in the obstacle matrix as we demonstrate for quenched hard-sphere liquids as underlying structures. Our results confirm that the microscopic dynamics can dominate the relaxational behavior even at long times, in striking contrast to glassy dynamics.
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Affiliation(s)
- Markus Spanner
- Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
| | - Felix Höfling
- Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
- Max-Planck-Institut für Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany, and IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Sebastian C Kapfer
- Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
| | - Klaus R Mecke
- Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
| | - Gerd E Schröder-Turk
- Murdoch University, School of Engineering and IT, Mathematics and Statistics, Murdoch, Western Australia 6150, Australia
| | - Thomas Franosch
- Institut für Theoretische Physik, Leopold-Franzens-Universität Innsbruck, Technikerstraße 21A, A-6020 Innsbruck, Austria
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17
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Salari L, Rondoni L, Giberti C, Klages R. A simple non-chaotic map generating subdiffusive, diffusive, and superdiffusive dynamics. CHAOS (WOODBURY, N.Y.) 2015; 25:073113. [PMID: 26232964 DOI: 10.1063/1.4926621] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here, we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto the whole real line which preserves distances except at a countable set of points. This property, which leads to vanishing Lyapunov exponents, is designed to mimic diffusion in non-chaotic polygonal billiards that give rise to normal and anomalous diffusion in a fully deterministic setting. As these billiards are typically too complicated to be analyzed from first principles, simplified models are needed to identify the minimal ingredients generating the different transport regimes. For our model, which we call the slicer map, we calculate all its moments in position analytically under variation of a single control parameter. We show that the slicer map exhibits a transition from subdiffusion over normal diffusion to superdiffusion under parameter variation. Our results may help to understand the delicate parameter dependence of the type of diffusion generated by polygonal billiards. We argue that in different parameter regions the transport properties of our simple model match to different classes of known stochastic processes. This may shed light on difficulties to match diffusion in polygonal billiards to a single anomalous stochastic process.
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Affiliation(s)
- Lucia Salari
- Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24 I-10129 Torino, Italy
| | - Lamberto Rondoni
- Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24 I-10129 Torino, Italy
| | - Claudio Giberti
- Dipartimento di Scienze e Metodi dell'Ingegneria, Universita' di Modena e Reggio E., Via G. Amendola 2 - Pad. Morselli, I-42122 Reggio E., Italy
| | - Rainer Klages
- School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
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18
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Zakeri SS, Lepri S, Wiersma DS. Localization in one-dimensional chains with Lévy-type disorder. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032112. [PMID: 25871059 DOI: 10.1103/physreve.91.032112] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/08/2014] [Indexed: 06/04/2023]
Abstract
We study Anderson localization of the classical lattice waves in a chain with mass impurities distributed randomly through a power-law relation s-(1+α) with s as the distance between two successive impurities and α>0. This model of disorder is long-range correlated and is inspired by the peculiar structure of the complex optical systems known as Lévy glasses. Using theoretical arguments and numerics, we show that in the regime in which the average distance between impurities is finite with infinite variance, the small-frequency behavior of the localization length is ξ{ω)∼ω-α. The physical interpretation of this result is that, for small frequencies and long wavelengths, the waves feel an effective disorder whose fluctuations are scale dependent. Numerical simulations show that an initially localized wave-packet attains, at large times, a characteristic inverse power-law front with an α-dependent exponent which can be estimated analytically.
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Affiliation(s)
- Sepideh S Zakeri
- European Laboratory for Non-linear Spectroscopy (LENS), University of Florence, Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy
| | - Stefano Lepri
- Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via Giovanni Sansone 1, I-50019 Sesto Fiorentino, Italy
| | - Diederik S Wiersma
- European Laboratory for Non-linear Spectroscopy (LENS), University of Florence, Via Nello Carrara 1, I-50019 Sesto Fiorentino, Italy
- Consiglio Nazionale delle Ricerche, Istituto Nazionale di Ottica, Largo Fermi 6, I-50125 Firenze, Italy
- Universitá di Firenze, Dipartimento di Fisica e Astronomia, via Giovanni Sansone 1, I-50019 Sesto Fiorentino, Italy
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19
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Fernández-Marín AA, Méndez-Bermúdez JA, Carbonell J, Cervera F, Sánchez-Dehesa J, Gopar VA. Beyond Anderson localization in 1D: anomalous localization of microwaves in random waveguides. PHYSICAL REVIEW LETTERS 2014; 113:233901. [PMID: 25526129 DOI: 10.1103/physrevlett.113.233901] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2014] [Indexed: 06/04/2023]
Abstract
Experimental evidence demonstrating that anomalous localization of waves can be induced in a controllable manner is reported. A microwave waveguide with dielectric slabs randomly placed is used to confirm the presence of anomalous localization. If the random spacing between slabs follows a distribution with a power-law tail (Lévy-type distribution), unconventional properties in the microwave-transmission fluctuations take place revealing the presence of anomalous localization. We study both theoretically and experimentally the complete distribution of the transmission through random waveguides characterized by α=1/2 ("Lévy waveguides") and α=3/4, α being the exponent of the power-law tail of the Lévy-type distribution. As we show, the transmission distributions are determined by only two parameters, both of them experimentally accessible. Effects of anomalous localization on the transmission are compared with those from the standard Anderson localization.
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Affiliation(s)
- A A Fernández-Marín
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
| | - J A Méndez-Bermúdez
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
| | - J Carbonell
- Wave Phenomena Group, Universitat Politècnica de València, Camino de vera s.n. (Edificio 7F), ES-46022 Valencia, Spain
| | - F Cervera
- Wave Phenomena Group, Universitat Politècnica de València, Camino de vera s.n. (Edificio 7F), ES-46022 Valencia, Spain
| | - J Sánchez-Dehesa
- Wave Phenomena Group, Universitat Politècnica de València, Camino de vera s.n. (Edificio 7F), ES-46022 Valencia, Spain
| | - V A Gopar
- Departamento de Física Teórica, Facultad de Ciencias, and Instituto de Biocomputación y Física de Sistemas Complejos, Universidad de Zaragoza, Pedro Cerbuna 12, E-50009 Zaragoza, Spain
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20
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Rebenshtok A, Denisov S, Hänggi P, Barkai E. Infinite densities for Lévy walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062135. [PMID: 25615072 DOI: 10.1103/physreve.90.062135] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/20/2014] [Indexed: 06/04/2023]
Abstract
Motion of particles in many systems exhibits a mixture between periods of random diffusive-like events and ballistic-like motion. In many cases, such systems exhibit strong anomalous diffusion, where low-order moments 〈|x(t)|(q)〉 with q below a critical value q(c) exhibit diffusive scaling while for q>q(c) a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent α, and the diffusion coefficient K(α). We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.
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Affiliation(s)
- A Rebenshtok
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 52900, Israel
| | - S Denisov
- Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135, Augsburg, Germany and Department for Bioinformatics, Lobachevsky State University, Gagarin Avenue 23, 603950 Nizhny Novgorod, Russia and Sumy State University, Rimsky-Korsakov Street 2, 40007 Sumy, Ukraine
| | - P Hänggi
- Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135, Augsburg, Germany and Nanosystems Initiative Munich, Schellingstr, 4, D-80799 München, Germany
| | - E Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 52900, Israel
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21
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Burioni R, Ubaldi E, Vezzani A. Superdiffusion and transport in two-dimensional systems with Lévy-like quenched disorder. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:022135. [PMID: 25353450 DOI: 10.1103/physreve.89.022135] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/28/2013] [Indexed: 06/04/2023]
Abstract
We present an extensive analysis of transport properties in superdiffusive twodimensional quenched random media, obtained by packing disks with radii distributed according to a Lévy law. We consider transport and scaling properties in samples packed with two different procedures, at fixed filling fraction and at self-similar packing, and we clarify the role of the two procedures in the superdiffusive effects. Using the behavior of the filling fraction in finite size systems as the main geometrical parameter, we define an effective Lévy exponent that correctly estimates the finite size effects. The effective Lévy exponent rules the dynamical scaling of the main transport properties and identifies the region where superdiffusive effects can be detected.
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Affiliation(s)
- Raffaella Burioni
- Dipartimento di Fisica e Scienza della Terra, Università di Parma, viale G.P. Usberti 7/A, 43124 Parma, Italy and INFN, Gruppo Collegato di Parma, viale G.P. Usberti 7/A, 43124 Parma, Italy
| | - Enrico Ubaldi
- Dipartimento di Fisica e Scienza della Terra, Università di Parma, viale G.P. Usberti 7/A, 43124 Parma, Italy
| | - Alessandro Vezzani
- Dipartimento di Fisica e Scienza della Terra, Università di Parma, viale G.P. Usberti 7/A, 43124 Parma, Italy and Centro S3, CNR-Istituto di Nanoscienze, Via Campi 213A, 41125 Modena, Italy
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22
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Metzler R, Jeon JH, Cherstvy AG, Barkai E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys Chem Chem Phys 2014; 16:24128-64. [DOI: 10.1039/c4cp03465a] [Citation(s) in RCA: 1046] [Impact Index Per Article: 104.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Abstract
This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.
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Affiliation(s)
- Ralf Metzler
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
- Physics Department
- Tampere University of Technology
| | - Jae-Hyung Jeon
- Physics Department
- Tampere University of Technology
- Tampere, Finland
- Korean Institute for Advanced Study (KIAS)
- Seoul, Republic of Korea
| | - Andrey G. Cherstvy
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
| | - Eli Barkai
- Physics Department and Institute of Nanotechnology and Advanced Materials
- Bar-Ilan University
- Ramat Gan, Israel
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23
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Svensson T, Vynck K, Grisi M, Savo R, Burresi M, Wiersma DS. Holey random walks: optics of heterogeneous turbid composites. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022120. [PMID: 23496473 DOI: 10.1103/physreve.87.022120] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2012] [Indexed: 06/01/2023]
Abstract
We present a probabilistic theory of random walks in turbid media with nonscattering regions. It is shown that important characteristics such as diffusion constants, average step lengths, crossing statistics, and void spacings can be analytically predicted. The theory is validated using Monte Carlo simulations of light transport in heterogeneous systems in the form of random sphere packings and good agreement is found. The role of step correlations is discussed and differences between unbounded and bounded systems are investigated. Our results are relevant to the optics of heterogeneous systems in general and represent an important step forward in the understanding of media with strong (fractal) heterogeneity in particular.
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Affiliation(s)
- Tomas Svensson
- European Laboratory for Non-Linear Spectroscopy, University of Florence, Via Nello Carrara 1, 50019 Sesto Fiorentino, Firenze, Italy.
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24
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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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25
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Burioni R, di Santo S, Lepri S, Vezzani A. Scattering lengths and universality in superdiffusive Lévy materials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:031125. [PMID: 23030884 DOI: 10.1103/physreve.86.031125] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2012] [Revised: 07/18/2012] [Indexed: 06/01/2023]
Abstract
We study the effects of scattering lengths on Lévy walks in quenched, one-dimensional random and fractal quasilattices, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling properties of the random-walk probability distribution, we show that the effect of the varying scattering length can be reabsorbed in the multiplicative coefficient of the scaling length. This leads to a superscaling behavior, where the dy-namical exponents and also the scaling functions do not depend on the value of the scattering length. Within the scaling framework, we obtain an exact expression for the multiplicative coefficient as a function of the scattering length both in the annealed and in the quenched random and fractal cases. Our analytic results are compared with numerical simulations, with excellent agreement, and are supposed to hold also in higher dimensions.
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Affiliation(s)
- Raffaella Burioni
- Dipartimento di Fisica, Università degli Studi di Parma, viale G.P. Usberti 7/A, 43100 Parma, Italy
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26
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Groth CW, Akhmerov AR, Beenakker CWJ. Transmission probability through a Lévy glass and comparison with a Lévy walk. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:021138. [PMID: 22463183 DOI: 10.1103/physreve.85.021138] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2011] [Revised: 01/14/2012] [Indexed: 05/31/2023]
Abstract
Recent experiments on the propagation of light over a distance L through a random packing of spheres with a power-law distribution of radii (a so-called Lévy glass) have found that the transmission probability T∝1/L(γ) scales superdiffusively (γ<1). The data has been interpreted in terms of a Lévy walk. We present computer simulations to demonstrate that diffusive scaling (γ≈1) can coexist with a divergent second moment of the step size distribution [p(s)∝1/s(1+α) with α<2]. This finding is in accord with analytical predictions for the effect of step size correlations, but deviates from what one would expect for a Lévy walk of independent steps.
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Affiliation(s)
- C W Groth
- Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, NL-2300 RA Leiden, The Netherlands
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27
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Buonsante P, Burioni R, Vezzani A. Transport and scaling in quenched two- and three-dimensional Lévy quasicrystals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:021105. [PMID: 21928947 DOI: 10.1103/physreve.84.021105] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2011] [Indexed: 05/31/2023]
Abstract
We consider correlated Lévy walks on a class of two- and three-dimensional deterministic self-similar structures, with correlation between steps induced by the geometrical distribution of regions, featuring different diffusion properties. We introduce a geometric parameter α, playing a role analogous to the exponent characterizing the step-length distribution in random systems. By a single-long-jump approximation, we analytically determine the long-time asymptotic behavior of the moments of the probability distribution as a function of α and of the dynamic exponent z associated with the scaling length of the process. We show that our scaling analysis also applies to experimentally relevant quantities such as escape-time and transmission probabilities. Extensive numerical simulations corroborate our results which, in general, are different from those pertaining to uncorrelated Lévy-walk models.
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Affiliation(s)
- P Buonsante
- Dipartimento di Fisica, Università degli Studi di Parma, Viale Usberti 7/a, I-43124 Parma, Italy
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