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Liu J, Jin Y, Bao JD, Chen X. Coexistence of ergodicity and nonergodicity in the aging two-state random walks. SOFT MATTER 2022; 18:8687-8699. [PMID: 36349834 DOI: 10.1039/d2sm01093c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/16/2023]
Abstract
The two-state stochastic phenomenon is observed in various systems and is attracting more interest, and it can be described by the two-state random walk (TSRW) model. The TSRW model is a typical two-state renewal process alternating between the continuous-time random walk state and the Lévy walk state, in both of which the sojourn time distributions follow a power law. In this paper, by discussing the statistical properties and calculating the ensemble averaged and time averaged mean squared displacement, the ergodic property and the ultimate diffusive behavior of the aging TSRW is studied. Results reveal that because of the two-state intermittent feature, ergodicity and nonergodicity can coexist in the aging TSRW, which behave as the time scalings of the time averages and ensemble averages not being identically equal. In addition, we find that the unique state occupation mechanism caused by the diverging mean of the sojourn times of one state, determines the aging TSRW's ultimate diffusive behavior at extremely large timescales, i.e., instead of the term with the larger diffusion exponent, the diffusion is surprisingly characterized by the term with the smaller one, which is distinctly different from previous conclusions and known results. At last, we note that the Lévy walk with rests model which also displays aging and ergodicity breaking, can be generalized by the TSRW model.
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Affiliation(s)
- Jian Liu
- Department of Physics, Beijing Technology and Business University, Beijing, 100048, China.
- Institute of Systems Science, Beijing Technology and Business University, Beijing, 100048, China
- Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Yuliang Jin
- Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
- Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325000, China
| | - Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing, 100875, China
| | - Xiaosong Chen
- School of Systems Science, Beijing Normal University, Beijing, 100875, China
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2
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Bao JD, Li Y, Marchesoni F. Consistent Hamiltonian models for space-momentum diffusion. Phys Rev E 2022; 105:L052105. [PMID: 35706220 DOI: 10.1103/physreve.105.l052105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2021] [Accepted: 05/15/2022] [Indexed: 06/15/2023]
Abstract
We develop a unified Hamiltonian approach to the diffusion of a particle coupled to a dissipative environment, an archetypal model widely invoked to interpret condensed phase phenomena, such as polymerization and cold-atom diffusion in optical lattices. By appropriate choices of the coupling functions, we reformulate phenomenological diffusion models by adding otherwise ignored space-momentum terms. We thus numerically predict a variety of diffusion regimes, from diffusion saturation to superballistic diffusion. With reference to ultracold atoms in optical lattices, we also show that time correlated external noises prevent superdiffusion from exceeding Richardson's law. Some of these results are unexpected and call for experimental validation.
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Affiliation(s)
- Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Yunyun Li
- Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
| | - Fabio Marchesoni
- Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
- Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy
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3
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Barkai E, Radons G, Akimoto T. Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory. J Chem Phys 2022; 156:044118. [DOI: 10.1063/5.0076552] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany and Institute of Mechatronics, 09126 Chemnitz, Germany
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
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4
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Liu J, Zhu P, Bao JD, Chen X. Strong anomalous diffusive behaviors of the two-state random walk process. Phys Rev E 2022; 105:014122. [PMID: 35193269 DOI: 10.1103/physreve.105.014122] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2021] [Accepted: 01/06/2022] [Indexed: 06/14/2023]
Abstract
The phenomenon of the two-state process is observed in various systems and is increasingly attracting attention, such that there is a need for a theoretical model of the process. In this paper, we present a prototypal two-state random walk (TSRW) model of a renewal process alternating between the continuous-time random walk (CTRW) state and Lévy walk (LW) state. The jump length distribution of the CTRW state is assumed to be Gaussian whereas the time distributions of the two states are both considered to follow a power law. The diffusive behavior is analyzed and discussed by calculating the mean squared displacement (MSD) analytically and numerically. The results reveal that it displays strong anomalous diffusive behaviors caused by random motions of both states, i.e., two anomalous diffusion terms coexist in the expression of the MSD, and the time distribution which has the heavier tail determines their forms. Moreover, because the two diffusion terms originate from different mechanisms, we find that the diffusion can be characterized by either the term with the largest diffusion exponent or the term with the largest diffusion coefficient at long timescales, which shows very different properties from the single-state process. In addition, the two-state nature of the process of the particle moving in a velocity field makes the TSRW model applicable to describe it. Results obtained from the two-state model reveal that the diffusion can even exhibit subdiffusive behavior, which is significantly different from known results obtained using the single-state model.
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Affiliation(s)
- Jian Liu
- Department of Physics, Institute of Systems Science, Beijing Technology and Business University, Beijing 100048, China
| | - Ping Zhu
- Department of Physics, Institute of Systems Science, Beijing Technology and Business University, Beijing 100048, China
| | - Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | - Xiaosong Chen
- School of Systems Science, Beijing Normal University, Beijing 100875, China
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5
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Barkai E, Radons G, Akimoto T. Transitions in the Ergodicity of Subrecoil-Laser-Cooled Gases. PHYSICAL REVIEW LETTERS 2021; 127:140605. [PMID: 34652191 DOI: 10.1103/physrevlett.127.140605] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2021] [Accepted: 08/26/2021] [Indexed: 06/13/2023]
Abstract
With subrecoil-laser-cooled atoms, one may reach nanokelvin temperatures while the ergodic properties of these systems do not follow usual statistical laws. Instead, due to an ingenious trapping mechanism in momentum space, power-law-distributed sojourn times are found for the cooled particles. Here, we show how this gives rise to a statistical-mechanical framework based on infinite ergodic theory, which replaces ordinary ergodic statistical physics of a thermal gas of atoms. In particular, the energy of the system exhibits a sharp discontinuous transition in its ergodic properties. Physically, this is controlled by the fluorescence rate, but, more profoundly, it is a manifestation of a transition for any observable, from being an integrable to becoming a nonintegrable observable, with respect to the infinite (non-normalized) invariant density.
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Affiliation(s)
- Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany and Institute of Mechatronics, 09126 Chemnitz, Germany
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
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6
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Sharaf MK, El-Shewy EK, Zahran MA. Fractional anisotropic diffusion equation in cylindrical brush model. JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 2020. [DOI: 10.1080/16583655.2020.1824743] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/08/2023]
Affiliation(s)
- M. K. Sharaf
- Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura, Egypt
| | - E. K. El-Shewy
- Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura, Egypt
- Taibah University, Al-Madinah Al-Munawarrah, Saudi Arabia
| | - M. A. Zahran
- Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura, Egypt
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7
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Zhao MD, Gao XZ, Wang Q, Zhang GL, Wang K, Dai F, Wang D, Li Y, Tu C, Wang HT. Multifractal vector optical fields. OPTICS EXPRESS 2019; 27:20608-20620. [PMID: 31510151 DOI: 10.1364/oe.27.020608] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/01/2019] [Accepted: 06/16/2019] [Indexed: 06/10/2023]
Abstract
We introduce the concept of multifractal into vector optical fields (VOFs). We propose, design and generate new fractal VOFs-multifractal VOFs (MF-VOFs), in which multifractal structure and VOF act as the lattice and the base, respectively. We generate two kinds of MF-VOFs experimentally and explore their focusing behaviors. We also investigate the self-healing and information recovering abilities of MF-VOFs, comparing with those of single-fractal VOFs (SF-VOFs) when their lattices are composed of the same hierarchy of fractal geometries. The results show that MF-VOFs have better self-healing and information recovering abilities than that of traditional SF-VOFs, meaning that MF-VOFs have better ability to resist the information loss during the focusing and imaging processes. These properties may find potential applications in information transmission, optical communication, and so on.
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8
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Levernier N, Bénichou O, Guérin T, Voituriez R. Universal first-passage statistics in aging media. Phys Rev E 2018; 98:022125. [PMID: 30253583 DOI: 10.1103/physreve.98.022125] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2017] [Indexed: 11/07/2022]
Abstract
It has been known for a long time that the kinetics of diffusion-limited reactions can be quantified by the time needed for a diffusing molecule to reach a target: the first-passage time (FPT). So far the general determination of the mean first-passage time to a target in confinement has left aside aging media, such as glassy materials, cellular media, or cold atoms in optical lattices. Here we consider general non-Markovian scale-invariant diffusion processes, which model a broad class of transport processes of molecules in aging media, and demonstrate that all the moments of the FPT obey universal scalings with the confining volume with nontrivial exponents. Our analysis shows that a nonlinear scaling with the volume of the mean FPT, which quantities the mean reaction time, is the hallmark of aging and provides a general tool to quantify its impact on reaction kinetics in confinement.
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Affiliation(s)
- N Levernier
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS, UPMC, 4 Place Jussieu, 75005 Paris, France
| | - O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS, UPMC, 4 Place Jussieu, 75005 Paris, France
| | - T Guérin
- Laboratoire Ondes et Matière d'Aquitaine, University of Bordeaux, Unité Mixte de Recherche No. 5798, CNRS, 33400 Talence, France
| | - R Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS, UPMC, 4 Place Jussieu, 75005 Paris, France.,Laboratoire Jean Perrin, CNRS, UPMC, 4 Place Jussieu, 75005 Paris, France
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9
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Afek G, Coslovsky J, Courvoisier A, Livneh O, Davidson N. Observing Power-Law Dynamics of Position-Velocity Correlation in Anomalous Diffusion. PHYSICAL REVIEW LETTERS 2017; 119:060602. [PMID: 28949641 DOI: 10.1103/physrevlett.119.060602] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2017] [Indexed: 06/07/2023]
Abstract
In this Letter, we present a measurement of the phase-space density distribution (PSDD) of ultracold ^{87}Rb atoms performing 1D anomalous diffusion. The PSDD is imaged using a direct tomographic method based on Raman velocity selection. It reveals that the position-velocity correlation function C_{xv}(t) builds up on a time scale related to the initial conditions of the ensemble and then decays asymptotically as a power law. We show that the decay follows a simple scaling theory involving the power-law asymptotic dynamics of position and velocity. The generality of this scaling theory is confirmed using Monte Carlo simulations of two distinct models of anomalous diffusion.
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Affiliation(s)
- Gadi Afek
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Jonathan Coslovsky
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Arnaud Courvoisier
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Oz Livneh
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Nir Davidson
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
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10
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Aghion E, Kessler DA, Barkai E. Large Fluctuations for Spatial Diffusion of Cold Atoms. PHYSICAL REVIEW LETTERS 2017; 118:260601. [PMID: 28707920 DOI: 10.1103/physrevlett.118.260601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2017] [Indexed: 06/07/2023]
Abstract
We use a new approach to study the large fluctuations of a heavy-tailed system, where the standard large-deviations principle does not apply. Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example, spreading packets of particles. Mathematically, it concerns the exponential falloff of the density of thin-tailed systems. Here we investigate the spatial density P_{t}(x) of laser-cooled atoms, where at intermediate length scales the shape is fat tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed non-normalized infinite-covariant density approach. The small and large fluctuations give rise to a bifractal nature of the spreading packet. We derive general relations which extend our theory to a class of systems with multifractal moments.
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Affiliation(s)
- Erez Aghion
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - David A Kessler
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
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11
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Dechant A, Shafier ST, Kessler DA, Barkai E. Heavy-tailed phase-space distributions beyond Boltzmann-Gibbs: Confined laser-cooled atoms in a nonthermal state. Phys Rev E 2016; 94:022151. [PMID: 27627290 DOI: 10.1103/physreve.94.022151] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2016] [Indexed: 06/06/2023]
Abstract
The Boltzmann-Gibbs density, a central result of equilibrium statistical mechanics, relates the energy of a system in contact with a thermal bath to its equilibrium statistics. This relation is lost for nonthermal systems such as cold atoms in optical lattices, where the heat bath is replaced with the laser beams of the lattice. We investigate in detail the stationary phase-space probability for Sisyphus cooling under harmonic confinement. In particular, we elucidate whether the total energy of the system still describes its stationary state statistics. We find that this is true for the center part of the phase-space density for deep lattices, where the Boltzmann-Gibbs density provides an approximate description. The relation between energy and statistics also persists for strong confinement and in the limit of high energies, where the system becomes underdamped. However, the phase-space density now exhibits heavy power-law tails. In all three cases we find expressions for the leading-order phase-space density and corrections which break the equivalence of probability and energy and violate energy equipartition. The nonequilibrium nature of the steady state is corroborated by explicit violations of detailed balance. We complement these analytical results with numerical simulations to map out the intricate structure of the phase-space density.
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Affiliation(s)
- Andreas Dechant
- Department of Physics and Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel
- Department of Physics #1, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
| | - Shalom Tzvi Shafier
- Department of Physics and Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel
| | - David A Kessler
- Department of Physics and Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel
| | - Eli Barkai
- Department of Physics and Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel
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12
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Dechant A, Kessler DA, Barkai E. Deviations from Boltzmann-Gibbs Statistics in Confined Optical Lattices. PHYSICAL REVIEW LETTERS 2015; 115:173006. [PMID: 26551114 DOI: 10.1103/physrevlett.115.173006] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2014] [Indexed: 06/05/2023]
Abstract
We investigate the semiclassical phase-space probability distribution P(x,p) of cold atoms in a Sisyphus cooling lattice with an additional harmonic confinement. We pose the question of whether this nonequilibrium steady state satisfies the equivalence of energy and probability. This equivalence is the foundation of Boltzmann-Gibbs and generalized thermostatic statistics, and a prerequisite for the description in terms of a temperature. At large energies, P(x,p) depends only on the Hamiltonian H(x,p) and the answer to the question is yes. In distinction to the Boltzmann-Gibbs state, the large-energy tails are power laws P(x,p)∝H(x,p)(-1/D), where D is related to the depth of the optical lattice. At intermediate energies, however, P(x,p) cannot be expressed as a function of the Hamiltonian and the equivalence between energy and probability breaks down. As a consequence the average potential and kinetic energy differ and no well-defined temperature can be assigned. The Boltzmann-Gibbs state is regained only in the limit of deep optical lattices. For strong confinement relative to the damping, we derive an explicit expression for the stationary phase-space distribution.
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Affiliation(s)
- Andreas Dechant
- Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel
- Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
| | - David A Kessler
- Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel
| | - Eli Barkai
- Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel
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13
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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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14
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Rebenshtok A, Denisov S, Hänggi P, Barkai E. Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem. PHYSICAL REVIEW LETTERS 2014; 112:110601. [PMID: 24702341 DOI: 10.1103/physrevlett.112.110601] [Citation(s) in RCA: 43] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2013] [Indexed: 06/03/2023]
Abstract
Strong anomalous diffusion, where ⟨|x(t)|(q)⟩ ∼ tqν(q) with a nonlinear spectrum ν(q) ≠ const, is wide spread and has been found in various nonlinear dynamical systems and experiments on active transport in living cells. Using a stochastic approach we show how this phenomenon is related to infinite covariant densities; i.e., the asymptotic states of these systems are described by non-normalizable distribution functions. Our work shows that the concept of infinite covariant densities plays an important role in the statistical description of open systems exhibiting multifractal anomalous diffusion, as it is complementary to the central limit theorem.
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Affiliation(s)
- Adi Rebenshtok
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Sergey Denisov
- Sumy State University, Rimsky-Korsakov Street 2, 40007 Sumy, Ukraine and 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
| | - Peter Hänggi
- Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
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15
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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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16
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Dykman MI, Schwartz IB. Large rare fluctuations in systems with delayed dissipation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:031145. [PMID: 23030904 DOI: 10.1103/physreve.86.031145] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2012] [Revised: 07/04/2012] [Indexed: 06/01/2023]
Abstract
We study the probability distribution and the escape rate in systems with delayed dissipation that comes from the coupling to a thermal bath. To logarithmic accuracy in the fluctuation intensity, the problem is reduced to a variational problem. It describes the most probable fluctuational paths, which are given by acausal equations due to the delay. In thermal equilibrium, the most probable path passing through a remote state has time-reversal symmetry, even though one cannot uniquely define a path that starts from a state with given system coordinate and momentum. The corrections to the distribution and the escape activation energy for small delay and small noise correlation time are obtained in explicit form.
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Affiliation(s)
- M I Dykman
- Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
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