1
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Liang Y, Wang W, Metzler R. Anomalous diffusion, non-Gaussianity, and nonergodicity for subordinated fractional Brownian motion with a drift. Phys Rev E 2023; 108:024143. [PMID: 37723819 DOI: 10.1103/physreve.108.024143] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/20/2023] [Accepted: 08/11/2023] [Indexed: 09/20/2023]
Abstract
The stochastic motion of a particle with long-range correlated increments (the moving phase) which is intermittently interrupted by immobilizations (the trapping phase) in a disordered medium is considered in the presence of an external drift. In particular, we consider trapping events whose times follow a scale-free distribution with diverging mean trapping time. We construct this process in terms of fractional Brownian motion with constant forcing in which the trapping effect is introduced by the subordination technique, connecting "operational time" with observable "real time." We derive the statistical properties of this process such as non-Gaussianity and nonergodicity, for both ensemble and single-trajectory (time) averages. We demonstrate nice agreement with extensive simulations for the probability density function, skewness, kurtosis, as well as ensemble and time-averaged mean-squared displacements. We place a specific emphasis on the comparisons between the cases with and without drift.
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Affiliation(s)
- Yingjie Liang
- College of Mechanics and Materials, Hohai University, 211100 Nanjing, China
- University of Potsdam, Institute of Physics and Astronomy, 14476 Potsdam-Golm, Germany
| | - Wei Wang
- University of Potsdam, Institute of Physics and Astronomy, 14476 Potsdam-Golm, Germany
| | - Ralf Metzler
- University of Potsdam, Institute of Physics and Astronomy, 14476 Potsdam-Golm, Germany
- Asia Pacific Centre for Theoretical Physics, Pohang 37673, Republic of Korea
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2
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Garibo-I-Orts Ò, Firbas N, Sebastiá L, Conejero JA. Gramian angular fields for leveraging pretrained computer vision models with anomalous diffusion trajectories. Phys Rev E 2023; 107:034138. [PMID: 37072993 DOI: 10.1103/physreve.107.034138] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Accepted: 02/28/2023] [Indexed: 04/20/2023]
Abstract
Anomalous diffusion is present at all scales, from atomic to large ones. Some exemplary systems are ultracold atoms, telomeres in the nucleus of cells, moisture transport in cement-based materials, arthropods' free movement, and birds' migration patterns. The characterization of the diffusion gives critical information about the dynamics of these systems and provides an interdisciplinary framework with which to study diffusive transport. Thus, the problem of identifying underlying diffusive regimes and inferring the anomalous diffusion exponent α with high confidence is critical to physics, chemistry, biology, and ecology. Classification and analysis of raw trajectories combining machine learning techniques with statistics extracted from them have widely been studied in the Anomalous Diffusion Challenge [Muñoz-Gil et al., Nat. Commun. 12, 6253 (2021)2041-172310.1038/s41467-021-26320-w]. Here we present a new data-driven method for working with diffusive trajectories. This method utilizes Gramian angular fields (GAF) to encode one-dimensional trajectories as images (Gramian matrices), while preserving their spatiotemporal structure for input to computer-vision models. This allows us to leverage two well-established pretrained computer-vision models, ResNet and MobileNet, to characterize the underlying diffusive regime and infer the anomalous diffusion exponent α. Short raw trajectories of lengths between 10 and 50 are commonly encountered in single-particle tracking experiments and are the most difficult ones to characterize. We show that GAF images can outperform the current state-of-the-art while increasing accessibility to machine learning methods in an applied setting.
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Affiliation(s)
- Òscar Garibo-I-Orts
- GRID-Grupo de Investigacion en Ciencia de Datos Valencian International University-VIU, Carrer Pintor Sorolla 21, 46002 València, Spain
| | - Nicolas Firbas
- DBS-Department of Biological Sciences, National University of Singapore 16 Science Drive 4, Singapore 117558, Singapore
| | - Laura Sebastiá
- VRAIN-Valencian Research Institute for Artificial Intelligence Universitat Politècnica de València, Cami de Vera s/n, 46022 València, Spain
| | - J Alberto Conejero
- Instituto Universitario de Matemática Pura y Aplicada Universitat Politècnica de València, Cami de Vera s/n, 46022 València, Spain
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3
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Lanoiselée Y, Stanislavsky A, Calebiro D, Weron A. Temperature and friction fluctuations inside a harmonic potential. Phys Rev E 2022; 106:064127. [PMID: 36671112 DOI: 10.1103/physreve.106.064127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2022] [Accepted: 10/20/2022] [Indexed: 06/17/2023]
Abstract
In this article we study the trapped motion of a molecule undergoing diffusivity fluctuations inside a harmonic potential. For the same diffusing-diffusivity process, we investigate two possible interpretations. Depending on whether diffusivity fluctuations are interpreted as temperature or friction fluctuations, we show that they display drastically different statistical properties inside the harmonic potential. We compute the characteristic function of the process under both types of interpretations and analyze their limit behavior. Based on the integral representations of the processes we compute the mean-squared displacement and the normalized excess kurtosis. In the long-time limit, we show for friction fluctuations that the probability density function (PDF) always converges to a Gaussian whereas in the case of temperature fluctuations the stationary PDF can display either Gaussian distribution or generalized Laplace (Bessel) distribution depending on the ratio between diffusivity and positional correlation times.
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Affiliation(s)
- Yann Lanoiselée
- Institute of Metabolism and Systems Research, University of Birmingham, Birmingham B15 2TT, United Kingdom
- Centre of Membrane Proteins and Receptors (COMPARE), Universities of Nottingham and Birmingham, Birmingham B15 2TT, United Kingdom
| | | | - Davide Calebiro
- Institute of Metabolism and Systems Research, University of Birmingham, Birmingham B15 2TT, United Kingdom
- Centre of Membrane Proteins and Receptors (COMPARE), Universities of Nottingham and Birmingham, Birmingham B15 2TT, United Kingdom
| | - Aleksander Weron
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
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4
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Vinod D, Cherstvy AG, Metzler R, Sokolov IM. Time-averaging and nonergodicity of reset geometric Brownian motion with drift. Phys Rev E 2022; 106:034137. [PMID: 36266856 DOI: 10.1103/physreve.106.034137] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2022] [Accepted: 08/09/2022] [Indexed: 06/16/2023]
Abstract
How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Δ)≠TAMSD(Δ) and Variance(Δ)≠TAMSD(Δ) at short lag times Δ and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Δ/T≪1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.
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Affiliation(s)
- Deepak Vinod
- Institute for Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
| | - Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
| | - Igor M Sokolov
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
- IRIS Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany
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5
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Zhao H, Zhang C, Guo Y, Lin S. Least squares estimator for a class of subdiffusion processes. COMMUN STAT-THEOR M 2022. [DOI: 10.1080/03610926.2020.1838546] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Affiliation(s)
- Huiyan Zhao
- School of Applied Mathematics, Beijing Normal University, Zhuhai, Guangdong, P.R. China
| | - Chongqi Zhang
- School of Economics and Statistics, Guangzhou University, Guangzhou, Guangdong, P.R. China
| | - Yu Guo
- School of Applied Mathematics, Beijing Normal University, Zhuhai, Guangdong, P.R. China
| | - Sheng Lin
- School of Applied Mathematics, Beijing Normal University, Zhuhai, Guangdong, P.R. China
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6
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Muñoz-Gil G, Volpe G, Garcia-March MA, Aghion E, Argun A, Hong CB, Bland T, Bo S, Conejero JA, Firbas N, Garibo I Orts Ò, Gentili A, Huang Z, Jeon JH, Kabbech H, Kim Y, Kowalek P, Krapf D, Loch-Olszewska H, Lomholt MA, Masson JB, Meyer PG, Park S, Requena B, Smal I, Song T, Szwabiński J, Thapa S, Verdier H, Volpe G, Widera A, Lewenstein M, Metzler R, Manzo C. Objective comparison of methods to decode anomalous diffusion. Nat Commun 2021; 12:6253. [PMID: 34716305 PMCID: PMC8556353 DOI: 10.1038/s41467-021-26320-w] [Citation(s) in RCA: 57] [Impact Index Per Article: 19.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2021] [Accepted: 09/30/2021] [Indexed: 11/17/2022] Open
Abstract
Deviations from Brownian motion leading to anomalous diffusion are found in transport dynamics from quantum physics to life sciences. The characterization of anomalous diffusion from the measurement of an individual trajectory is a challenging task, which traditionally relies on calculating the trajectory mean squared displacement. However, this approach breaks down for cases of practical interest, e.g., short or noisy trajectories, heterogeneous behaviour, or non-ergodic processes. Recently, several new approaches have been proposed, mostly building on the ongoing machine-learning revolution. To perform an objective comparison of methods, we gathered the community and organized an open competition, the Anomalous Diffusion challenge (AnDi). Participating teams applied their algorithms to a commonly-defined dataset including diverse conditions. Although no single method performed best across all scenarios, machine-learning-based approaches achieved superior performance for all tasks. The discussion of the challenge results provides practical advice for users and a benchmark for developers.
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Affiliation(s)
- Gorka Muñoz-Gil
- ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860, Castelldefels (Barcelona), Spain
| | - Giovanni Volpe
- Department of Physics, University of Gothenburg, Origovägen 6B, SE-41296, Gothenburg, Sweden.
| | - Miguel Angel Garcia-March
- Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, Spain
| | - Erez Aghion
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, DE-01187, Dresden, Germany
| | - Aykut Argun
- Department of Physics, University of Gothenburg, Origovägen 6B, SE-41296, Gothenburg, Sweden
| | - Chang Beom Hong
- Department of Physics, Pohang University of Science and Technology, Pohang, 37673, Korea
| | - Tom Bland
- The Francis Crick Institute, 1 Midland Road, London, NW1 1AT, UK
| | - Stefano Bo
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, DE-01187, Dresden, Germany
| | - J Alberto Conejero
- Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, Spain
| | - Nicolás Firbas
- Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, Spain
| | - Òscar Garibo I Orts
- Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, Spain
| | - Alessia Gentili
- Department of Chemistry, University College London, 20 Gordon Street, London, WC1H 0AJ, UK
| | - Zihan Huang
- School of Physics and Electronics, Hunan University, Changsha, 410082, China
| | - Jae-Hyung Jeon
- Department of Physics, Pohang University of Science and Technology, Pohang, 37673, Korea
| | - Hélène Kabbech
- Department of Cell Biology, Erasmus University Medical Center, Dr. Molewaterplein 40, 3015, GD, Rotterdam, the Netherlands
| | - Yeongjin Kim
- Department of Physics, Pohang University of Science and Technology, Pohang, 37673, Korea
| | - Patrycja Kowalek
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wrocław, Poland
| | - Diego Krapf
- Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, Colorado, 80523, USA
| | - Hanna Loch-Olszewska
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wrocław, Poland
| | - Michael A Lomholt
- PhyLife, Department of Physics, Chemistry and Pharmacy, University of Southern Denmark, DK-5230, Odense M, Denmark
| | - Jean-Baptiste Masson
- Institut Pasteur, Université de Paris, USR 3756 (C3BI/DBC) & Neuroscience department CNRS UMR 3751, Decision and Bayesian Computation lab, F-75015, Paris, France
| | - Philipp G Meyer
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, DE-01187, Dresden, Germany
| | - Seongyu Park
- Department of Physics, Pohang University of Science and Technology, Pohang, 37673, Korea
| | - Borja Requena
- ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860, Castelldefels (Barcelona), Spain
| | - Ihor Smal
- Department of Cell Biology, Erasmus University Medical Center, Dr. Molewaterplein 40, 3015, GD, Rotterdam, the Netherlands
| | - Taegeun Song
- Department of Physics, Pohang University of Science and Technology, Pohang, 37673, Korea
- Center for AI and Natural Sciences, Korea Institute for Advanced Study, Seoul, Korea
- Department of Data Information and Physics, Kongju National University, Kongju, 32588, Korea
| | - Janusz Szwabiński
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wrocław, Poland
| | - Samudrajit Thapa
- Institute of Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Str 24/25, D-14476, Potsdam-Golm, Germany
- Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv, 69978, Israel
- School of Mechanical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel
| | - Hippolyte Verdier
- Institut Pasteur, Université de Paris, USR 3756 (C3BI/DBC) & Neuroscience department CNRS UMR 3751, Decision and Bayesian Computation lab, F-75015, Paris, France
| | - Giorgio Volpe
- Department of Chemistry, University College London, 20 Gordon Street, London, WC1H 0AJ, UK
| | - Artur Widera
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, 67663, Kaiserslautern, Germany
| | - Maciej Lewenstein
- ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860, Castelldefels (Barcelona), Spain
- ICREA, Pg. Lluís Companys 23, 08010, Barcelona, Spain
| | - Ralf Metzler
- Institute of Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Str 24/25, D-14476, Potsdam-Golm, Germany
| | - Carlo Manzo
- ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860, Castelldefels (Barcelona), Spain.
- Facultat de Ciències i Tecnologia, Universitat de Vic - Universitat Central de Catalunya (UVic-UCC), C. de la Laura,13, 08500, Vic, Spain.
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7
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McClure JE, Berg S, Armstrong RT. Thermodynamics of fluctuations based on time-and-space averages. Phys Rev E 2021; 104:035106. [PMID: 34654200 DOI: 10.1103/physreve.104.035106] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2020] [Accepted: 06/23/2021] [Indexed: 11/07/2022]
Abstract
We develop nonequilibrium theory by using averages in time and space as a generalized way to upscale thermodynamics in nonergodic systems. The approach offers a classical perspective on the energy dynamics in fluctuating systems. The rate of entropy production is shown to be explicitly scale dependent when considered in this context. We show that while any stationary process can be represented as having zero entropy production, second law constraints due to the Clausius theorem are preserved due to the fact that heat and work are related based on conservation of energy. As a demonstration, we consider the energy dynamics for the Carnot cycle and for Maxwell's demon. We then consider nonstationary processes, applying time-and-space averages to characterize nonergodic effects in heterogeneous systems where energy barriers such as compositional gradients are present. We show that the derived theory can be used to understand the origins of anomalous diffusion phenomena in systems where Fick's law applies at small length scales, but not at large length scales. We further characterize fluctuations in capillary-dominated systems, which are nonstationary due to the irreversibility of cooperative events.
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Affiliation(s)
- James E McClure
- Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA
| | - Steffen Berg
- Shell Global Solutions International B.V., Grasweg 31, 1031HW Amsterdam, The Netherlands
| | - Ryan T Armstrong
- School of Minerals and Energy Resources Engineering, University of New South Wales, Sydney 2052, Australia
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8
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Cherstvy AG, Wang W, Metzler R, Sokolov IM. Inertia triggers nonergodicity of fractional Brownian motion. Phys Rev E 2021; 104:024115. [PMID: 34525594 DOI: 10.1103/physreve.104.024115] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2021] [Accepted: 07/29/2021] [Indexed: 11/07/2022]
Abstract
How related are the ergodic properties of the over- and underdamped Langevin equations driven by fractional Gaussian noise? We here find that for massive particles performing fractional Brownian motion (FBM) inertial effects not only destroy the stylized fact of the equivalence of the ensemble-averaged mean-squared displacement (MSD) to the time-averaged MSD (TAMSD) of overdamped or massless FBM, but also dramatically alter the values of the ergodicity-breaking parameter (EB). Our theoretical results for the behavior of EB for underdamped or massive FBM for varying particle mass m, Hurst exponent H, and trace length T are in excellent agreement with the findings of stochastic computer simulations. The current results can be of interest for the experimental community employing various single-particle-tracking techniques and aiming at assessing the degree of nonergodicity for the recorded time series (studying, e.g., the behavior of EB versus lag time). To infer FBM as a realizable model of anomalous diffusion for a set single-particle-tracking data when massive particles are being tracked, the EBs from the data should be compared to EBs of massive (rather than massless) FBM.
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Affiliation(s)
- Andrey G Cherstvy
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany.,Institute for Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
| | - Wei Wang
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
| | - Igor M Sokolov
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany.,IRIS Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany
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9
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Wang W, Cherstvy AG, Kantz H, Metzler R, Sokolov IM. Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes. Phys Rev E 2021; 104:024105. [PMID: 34525678 DOI: 10.1103/physreve.104.024105] [Citation(s) in RCA: 20] [Impact Index Per Article: 6.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/27/2021] [Accepted: 07/14/2021] [Indexed: 12/12/2022]
Abstract
How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x)=D_{0}|x|^{γ} and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity-breaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB∼(1/r)-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.
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Affiliation(s)
- Wei Wang
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Andrey G Cherstvy
- Institute for Physics & Astronomy University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany.,Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
| | - Holger Kantz
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
| | - Igor M Sokolov
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany.,IRIS Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany
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10
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Maraj K, Szarek D, Sikora G, Wyłomańska A. Time-averaged mean squared displacement ratio test for Gaussian processes with unknown diffusion coefficient. CHAOS (WOODBURY, N.Y.) 2021; 31:073120. [PMID: 34340341 DOI: 10.1063/5.0054119] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2021] [Accepted: 06/23/2021] [Indexed: 06/13/2023]
Abstract
The time-averaged mean squared displacement (TAMSD) is one of the most common statistics used for the analysis of anomalous diffusion processes. Anomalous diffusion is manifested by non-linear (mostly power-law) characteristics of the process in contrast to normal diffusion where linear characteristics are expected. One can distinguish between sub- and super-diffusive processes. We consider Gaussian anomalous diffusion models and propose a new approach used for their testing. This approach is based on the TAMSD ratio statistic for different time lags. Similar to the TAMSD, this statistic exhibits a specific behavior in the anomalous diffusion regime. Through its structure, it is independent of the diffusion coefficient, which, in general, does not influence anomalous diffusion behavior. Thus, the TAMSD ratio-based approach does not require preliminary knowledge of the diffusion coefficient's value, in contrast to the TAMSD-approach, where this value is crucial in the testing procedure. Based on the quadratic form representation of the TAMSD ratio, we calculate its main characteristics and propose a step-by-step testing procedure that can be applied for any Gaussian process. For the anomalous diffusion model used here, namely, the fractional Brownian motion, we demonstrate the effectiveness of the proposed methodology. We show that the new approach outperforms the TAMSD-based one, especially for small sample sizes. Finally, the methodology is applied to the real data from the financial market.
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Affiliation(s)
- Katarzyna Maraj
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland
| | - Dawid Szarek
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland
| | - Grzegorz Sikora
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland
| | - Agnieszka Wyłomańska
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland
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11
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Xu Y, Liu X, Li Y, Metzler R. Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes. Phys Rev E 2021; 102:062106. [PMID: 33466052 DOI: 10.1103/physreve.102.062106] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/09/2019] [Accepted: 10/22/2020] [Indexed: 01/03/2023]
Abstract
Heterogeneous diffusion processes (HDPs) with space-dependent diffusion coefficients D(x) are found in a number of real-world systems, such as for diffusion of macromolecules or submicron tracers in biological cells. Here, we examine HDPs in quenched-disorder systems with Gaussian colored noise (GCN) characterized by a diffusion coefficient with a power-law dependence on the particle position and with a spatially random scaling exponent. Typically, D(x) is considered to be centerd at the origin and the entire x axis is characterized by a single scaling exponent α. In this work we consider a spatially random scenario: in periodic intervals ("layers") in space D(x) is centerd to the midpoint of each interval. In each interval the scaling exponent α is randomly chosen from a Gaussian distribution. The effects of the variation of the scaling exponents, the periodicity of the domains ("layer thickness") of the diffusion coefficient in this stratified system, and the correlation time of the GCN are analyzed numerically in detail. We discuss the regimes of superdiffusion, subdiffusion, and normal diffusion realisable in this system. We observe and quantify the domains where nonergodic and non-Gaussian behaviors emerge in this system. Our results provide new insights into the understanding of weak ergodicity breaking for HDPs driven by colored noise, with potential applications in quenched layered systems, typical model systems for diffusion in biological cells and tissues, as well as for diffusion in geophysical systems.
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Affiliation(s)
- Yong Xu
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China.,MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi'an 710072, China
| | - Xuemei Liu
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Yongge Li
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China.,Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Ralf Metzler
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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12
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Lapolla A, Godec A. Single-file diffusion in a bi-stable potential: Signatures of memory in the barrier-crossing of a tagged-particle. J Chem Phys 2020; 153:194104. [PMID: 33218229 DOI: 10.1063/5.0025785] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate memory effects in barrier-crossing in the overdamped setting. We focus on the scenario where the hidden degrees of freedom relax on exactly the same time scale as the observable. As a prototypical model, we analyze tagged-particle diffusion in a single file confined to a bi-stable potential. We identify the signatures of memory and explain their origin. The emerging memory is a result of the projection of collective many-body eigenmodes onto the motion of a tagged-particle. We are interested in the "confining" (all background particles in front of the tagged-particle) and "pushing" (all background particles behind the tagged-particle) scenarios for which we find non-trivial and qualitatively different relaxation behaviors. Notably and somewhat unexpectedly, at a fixed particle number, we find that the higher the barrier, the stronger the memory effects are. The fact that the external potential alters the memory is important more generally and should be taken into account in applications of generalized Langevin equations. Our results can readily be tested experimentally and may be relevant for understanding transport in biological ion-channels.
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Affiliation(s)
- Alessio Lapolla
- Mathematical bioPhysics Group, Max Planck Institute for Biophysical Chemistry, Am Fassberg 11, 37077 Göttingen, Germany
| | - Aljaž Godec
- Mathematical bioPhysics Group, Max Planck Institute for Biophysical Chemistry, Am Fassberg 11, 37077 Göttingen, Germany
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13
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Gires PY, Thampi M, Weiss M. Quantifying active diffusion in an agitated fluid. Phys Chem Chem Phys 2020; 22:21678-21684. [PMID: 32966453 DOI: 10.1039/d0cp03629c] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
Mixing of reactants in microdroplets predominantly relies on diffusional motion due to small Reynolds numbers and the resulting absence of turbulent flows. Enhancing diffusion in microdroplets by an auxiliary noise source is therefore a topical problem. Here we report on how the diffusional motion of tracer beads is enhanced upon agitating the surrounding aqueous fluid with miniaturized magnetic stir bars that are compatible with microdroplets and microfluidic devices. Using single-particle tracking, we demonstrate via a broad palette of measures that local stirring of the fluid at different frequencies leads to an enhanced but apparently normal and homogenous diffusion process, i.e. diffusional steps follow the anticipated Gaussian distribution and no ballistic motion is observed whereas diffusion coefficients are significantly increased. The signature of stirring is, however, visible in the power-spectral density and in the velocity autocorrelation function of trajectories. Our data therefore demonstrate that diffusive mixing can be locally enhanced with miniaturized stir bars while only moderately affecting the ambient noise properties. The latter may also facilitate the controlled addition of nonequilibrium noise to complex fluids in future applications.
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Affiliation(s)
- Pierre-Yves Gires
- Experimental Physics I, University of Bayreuth, Universitätsstr. 30, D-95447 Bayreuth, Germany.
| | - Mithun Thampi
- Experimental Physics I, University of Bayreuth, Universitätsstr. 30, D-95447 Bayreuth, Germany.
| | - Matthias Weiss
- Experimental Physics I, University of Bayreuth, Universitätsstr. 30, D-95447 Bayreuth, Germany.
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14
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Wang W, Cherstvy AG, Liu X, Metzler R. Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise. Phys Rev E 2020; 102:012146. [PMID: 32794926 DOI: 10.1103/physreve.102.012146] [Citation(s) in RCA: 26] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2020] [Accepted: 06/22/2020] [Indexed: 01/09/2023]
Abstract
Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x)=D_{0}|x|^{α}. Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble- and time-averaged mean-squared displacements couple the scaling exponents α of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y∼|x|^{1/(2/(2-α))}/t^{H} coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), P_{HDP-FBM}(y)=e^{-y^{2}}/sqrt[π]. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.
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Affiliation(s)
- Wei Wang
- College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China.,Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Xianbin Liu
- College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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15
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Sun ZY, Yu X. Anomalous diffusion of discrete solitons driven by evolving disorder. Phys Rev E 2020; 101:062211. [PMID: 32688546 DOI: 10.1103/physreve.101.062211] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2020] [Accepted: 06/02/2020] [Indexed: 06/11/2023]
Abstract
Anomalous diffusion is simulated in this paper by studying the transport of discrete solitons in a lattice with evolving disorder. We find a Richardson-type diffusion for the small solitons and a regime of transient diffusion for larger solitons within the ensemble-averaged description. As a comparison, the time-averaged observables present a ballistic scaling for both cases. However, distribution of these observables changes remarkably with the soliton size. Our results suggest violation of ergodicity for the solitons' diffusive processes, which are expected to shed light on further understanding of the discreteness-disorder-nonlinearity interaction.
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Affiliation(s)
- Zhi-Yuan Sun
- Institute of Fluid Mechanics, Beihang University, Beijing 100191, China
- International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China
| | - Xin Yu
- Institute of Fluid Mechanics, Beihang University, Beijing 100191, China
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16
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Simonović I, Bošnjaković D, Petrović ZL, Stokes P, White RD, Dujko S. Third-order transport coefficient tensor of charged-particle swarms in electric and magnetic fields. Phys Rev E 2020; 101:023203. [PMID: 32168642 DOI: 10.1103/physreve.101.023203] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2019] [Accepted: 12/23/2019] [Indexed: 11/07/2022]
Abstract
Third-order transport coefficient tensor of charged-particle swarms in neutral gases in the presence of spatially uniform electric and magnetic fields is considered using a multiterm solution of Boltzmann's equation and Monte Carlo simulation technique. The structure of the third-order transport coefficient tensor and symmetries along its individual components in varying configurations of electric and magnetic fields are addressed using a group projector technique and through symmetry considerations of the Boltzmann equation. In addition, we focus upon the physical interpretation of the third-order transport coefficient tensor by considering the extended diffusion equation which incorporates the contribution of the third-order transport coefficients to the density profile of charged particles. Numerical calculations are carried out for electron and ion swarms for a range of model gases with the aim of establishing accurate benchmarks for third-order transport coefficients. The effects of ion to neutral-particle mass ratio are also examined. The errors of the two-term approximation for solving the Boltzmann equation and limitations of previous treatments of the high-order charged-particle transport properties are also highlighted.
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Affiliation(s)
- I Simonović
- Institute of Physics, University of Belgrade, PO Box 68, 11080 Belgrade, Serbia
| | - D Bošnjaković
- Institute of Physics, University of Belgrade, PO Box 68, 11080 Belgrade, Serbia
| | - Z Lj Petrović
- Serbian Academy of Sciences and Arts, Knez Mihailova 35, 11001 Belgrade, Serbia
| | - P Stokes
- College of Science and Engineering, James Cook University, 4810 Townsville, Australia
| | - R D White
- College of Science and Engineering, James Cook University, 4810 Townsville, Australia
| | - S Dujko
- Institute of Physics, University of Belgrade, PO Box 68, 11080 Belgrade, Serbia
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17
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Basak S, Sengupta S, Chattopadhyay K. Understanding biochemical processes in the presence of sub-diffusive behavior of biomolecules in solution and living cells. Biophys Rev 2019; 11:851-872. [PMID: 31444739 PMCID: PMC6957588 DOI: 10.1007/s12551-019-00580-9] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2019] [Accepted: 07/25/2019] [Indexed: 01/24/2023] Open
Abstract
In order to maintain cellular function, biomolecules like protein, DNA, and RNAs have to diffuse to the target spaces within the cell. Changes in the cytosolic microenvironment or in the nucleus during the fulfillment of these cellular processes affect their mobility, folding, and stability thereby impacting the transient or stable interactions with their adjacent neighbors in the organized and dynamic cellular interior. Using classical Brownian motion to elucidate the diffusion behavior of these biomolecules is hard considering their complex nature. The understanding of biomolecular diffusion inside cells still remains elusive due to the lack of a proper model that can be extrapolated to these cases. In this review, we have comprehensively addressed the progresses in this field, laying emphasis on the different aspects of anomalous diffusion in the different biochemical reactions in cell interior. These experiment-based models help to explain the diffusion behavior of biomolecules in the cytosolic and nuclear microenvironment. Moreover, since understanding of biochemical reactions within living cellular system is our main focus, we coupled the experimental observations with the concept of sub-diffusion from in vitro to in vivo condition. We believe that the pairing between the understanding of complex behavior and structure-function paradigm of biological molecules would take us forward by one step in order to solve the puzzle around diseases caused by cellular dysfunction.
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Affiliation(s)
- Sujit Basak
- Biochemistry and Molecular Pharmacology, University of Massachusetts Medical School, 364 Plantation Street, Worcester, MA, 01605, USA.
| | - Sombuddha Sengupta
- Protein Folding and Dynamics Lab, Structural Biology and Bioinformatics, CSIR-Indian Institute of Chemical Biology (CSIR-IICB), 4 Raja S.C Mullick Road, Jadavpur, Kolkata, West Bengal, 700032, India
| | - Krishnananda Chattopadhyay
- Protein Folding and Dynamics Lab, Structural Biology and Bioinformatics, CSIR-Indian Institute of Chemical Biology (CSIR-IICB), 4 Raja S.C Mullick Road, Jadavpur, Kolkata, West Bengal, 700032, India
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18
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Guerrero BV, Chakraborty B, Zuriguel I, Garcimartín A. Nonergodicity in silo unclogging: Broken and unbroken arches. Phys Rev E 2019; 100:032901. [PMID: 31639941 DOI: 10.1103/physreve.100.032901] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2019] [Indexed: 11/07/2022]
Abstract
We report an experiment on the unclogging dynamics in a two-dimensional silo submitted to a sustained gentle vibration. We find that arches present a jerking motion where rearrangements in the positions of their beads are interspersed with quiescent periods. This behavior occurs for both arches that break down and those that withstand the external perturbation: Arches evolve until they either collapse or get trapped in a stable configuration. This evolution is described in terms of a scalar variable characterizing the arch shape that can be modeled as a continuous-time random walk. By studying the diffusivity of this variable, we show that the unclogging is a weakly nonergodic process. Remarkably, arches that do not collapse explore different configurations before settling in one of them and break ergodicity much in the same way than arches that break down.
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Affiliation(s)
- B V Guerrero
- Dep. Física y Mat. Apl., Fac. Ciencias, Universidad de Navarra, 31080 Pamplona, Spain
| | - B Chakraborty
- Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02454, USA
| | - I Zuriguel
- Dep. Física y Mat. Apl., Fac. Ciencias, Universidad de Navarra, 31080 Pamplona, Spain
| | - A Garcimartín
- Dep. Física y Mat. Apl., Fac. Ciencias, Universidad de Navarra, 31080 Pamplona, Spain
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19
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Vojta T, Skinner S, Metzler R. Probability density of the fractional Langevin equation with reflecting walls. Phys Rev E 2019; 100:042142. [PMID: 31770994 DOI: 10.1103/physreve.100.042142] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2019] [Indexed: 06/10/2023]
Abstract
We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while antipersistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for nonthermal fractional Brownian motion with reflecting walls, and we discuss broader implications.
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Affiliation(s)
- Thomas Vojta
- Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA
| | - Sarah Skinner
- Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA
| | - Ralf Metzler
- Institute of Physics and Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany
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20
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Malsagov A, Mandjes M. Approximations for reflected fractional Brownian motion. Phys Rev E 2019; 100:032120. [PMID: 31640037 DOI: 10.1103/physreve.100.032120] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2019] [Indexed: 06/10/2023]
Abstract
Fractional Brownian motion is a widely used stochastic process that is particularly suited to model anomalous diffusion. We focus on capturing the mean and variance of fractional Brownian motion reflected at level 0. As explicit expressions or numerical techniques are not available, we base our analysis on Monte Carlo simulation. Our main findings concern closed-form approximations of the mean and variance, with a near-perfect fit.
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Affiliation(s)
- Artagan Malsagov
- Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 105, 1098 XH Amsterdam, The Netherlands
| | - Michel Mandjes
- Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 105, 1098 XH Amsterdam, The Netherlands
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21
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22
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Holmes WR. Subdiffusive Dynamics Lead to Depleted Particle Densities near Cellular Borders. Biophys J 2019; 116:1538-1546. [PMID: 30954212 DOI: 10.1016/j.bpj.2019.02.021] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2018] [Revised: 02/07/2019] [Accepted: 02/08/2019] [Indexed: 01/17/2023] Open
Abstract
It has long been known that the complex cellular environment leads to anomalous motion of intracellular particles. At a gross level, this is characterized by mean-squared displacements that deviate from the standard linear profile. Statistical analysis of particle trajectories has helped further elucidate how different characteristics of the cellular environment can introduce different types of anomalousness. A significant majority of this literature has, however, focused on characterizing the properties of trajectories that do not interact with cell borders (e.g., cell membrane or nucleus). Numerous biological processes ranging from protein activation to exocytosis, however, require particles to be near a membrane. This study investigates the consequences of a canonical type of subdiffusive motion, fractional Brownian motion, and its physical analog, generalized Langevin equation dynamics, on the spatial localization of particles near reflecting boundaries. Results show that this type of subdiffusive motion leads to the formation of significant zones of depleted particle density near boundaries and that this effect is independent of the specific model details encoding those dynamics. Rather, these depletion layers are a natural and robust consequence of the anticorrelated nature of motion increments that is at the core of fractional Brownian motion (or alternatively generalized Langevin equation) dynamics. If such depletion zones are present, it would be of profound importance given the wide array of signaling and transport processes that occur near membranes. If not, that would suggest our understanding of this type of anomalous motion may be flawed. Either way, this result points to the need to further investigate the consequences of anomalous particle motions near cell borders from both theoretical and experimental perspectives.
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Affiliation(s)
- William R Holmes
- Department of Pysics and Astronomy, Vanderbilt University, Nashville, Tennessee; Department of Mathematics, Vanderbilt University, Nashville, Tennessee; Quantitative Systems Biology Center, Vanderbilt University, Nashville, Tennessee.
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23
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Thapa S, Lukat N, Selhuber-Unkel C, Cherstvy AG, Metzler R. Transient superdiffusion of polydisperse vacuoles in highly motile amoeboid cells. J Chem Phys 2019; 150:144901. [PMID: 30981236 DOI: 10.1063/1.5086269] [Citation(s) in RCA: 21] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023] Open
Affiliation(s)
- Samudrajit Thapa
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Nils Lukat
- Institute of Materials Science, Christian-Albrechts-Universität zu Kiel, 24143 Kiel, Germany
| | | | - Andrey G. Cherstvy
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Ralf Metzler
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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24
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Cherstvy AG, Thapa S, Wagner CE, Metzler R. Non-Gaussian, non-ergodic, and non-Fickian diffusion of tracers in mucin hydrogels. SOFT MATTER 2019; 15:2526-2551. [PMID: 30734041 DOI: 10.1039/c8sm02096e] [Citation(s) in RCA: 45] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
Native mucus is polymer-based soft-matter material of paramount biological importance. How non-Gaussian and non-ergodic is the diffusive spreading of pathogens in mucus? We study the passive, thermally driven motion of micron-sized tracers in hydrogels of mucins, the main polymeric component of mucus. We report the results of the Bayesian analysis for ranking several diffusion models for a set of tracer trajectories [C. E. Wagner et al., Biomacromolecules, 2017, 18, 3654]. The models with "diffusing diffusivity", fractional and standard Brownian motion are used. The likelihood functions and evidences of each model are computed, ranking the significance of each model for individual traces. We find that viscoelastic anomalous diffusion is often most probable, followed by Brownian motion, while the model with a diffusing diffusion coefficient is only realised rarely. Our analysis also clarifies the distribution of time-averaged displacements, correlations of scaling exponents and diffusion coefficients, and the degree of non-Gaussianity of displacements at varying pH levels. Weak ergodicity breaking is also quantified. We conclude that-consistent with the original study-diffusion of tracers in the mucin gels is most non-Gaussian and non-ergodic at low pH that corresponds to the most heterogeneous networks. Using the Bayesian approach with the nested-sampling algorithm, together with the quantitative analysis of multiple statistical measures, we report new insights into possible physical mechanisms of diffusion in mucin gels.
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Affiliation(s)
- Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany.
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25
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Thapa S, Lomholt MA, Krog J, Cherstvy AG, Metzler R. Bayesian analysis of single-particle tracking data using the nested-sampling algorithm: maximum-likelihood model selection applied to stochastic-diffusivity data. Phys Chem Chem Phys 2018; 20:29018-29037. [PMID: 30255886 DOI: 10.1039/c8cp04043e] [Citation(s) in RCA: 47] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We employ Bayesian statistics using the nested-sampling algorithm to compare and rank multiple models of ergodic diffusion (including anomalous diffusion) as well as to assess their optimal parameters for in silico-generated and real time-series. We focus on the recently-introduced model of Brownian motion with "diffusing diffusivity"-giving rise to widely-observed non-Gaussian displacement statistics-and its comparison to Brownian and fractional Brownian motion, also for the time-series with some measurement noise. We conduct this model-assessment analysis using Bayesian statistics and the nested-sampling algorithm on the level of individual particle trajectories. We evaluate relative model probabilities and compute best-parameter sets for each diffusion model, comparing the estimated parameters to the true ones. We test the performance of the nested-sampling algorithm and its predictive power both for computer-generated (idealised) trajectories as well as for real single-particle-tracking trajectories. Our approach delivers new important insight into the objective selection of the most suitable stochastic model for a given time-series. We also present first model-ranking results in application to experimental data of tracer diffusion in polymer-based hydrogels.
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Affiliation(s)
- Samudrajit Thapa
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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26
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Hou R, Cherstvy AG, Metzler R, Akimoto T. Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing. Phys Chem Chem Phys 2018; 20:20827-20848. [PMID: 30066003 DOI: 10.1039/c8cp01863d] [Citation(s) in RCA: 29] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023]
Abstract
We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.
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Affiliation(s)
- Ru Hou
- School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China.
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27
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Mardoukhi Y, Jeon JH, Chechkin AV, Metzler R. Fluctuations of random walks in critical random environments. Phys Chem Chem Phys 2018; 20:20427-20438. [PMID: 30043029 DOI: 10.1039/c8cp03212b] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Percolation networks have been widely used in the description of porous media but are now found to be relevant to understand the motion of particles in cellular membranes or the nucleus of biological cells. Random walks on the infinite cluster at criticality of a percolation network are asymptotically ergodic. On any finite size cluster of the network stationarity is reached at finite times, depending on the cluster's size. Despite of this we here demonstrate by combination of analytical calculations and simulations that at criticality the disorder and cluster size average of the ensemble of clusters leads to a non-vanishing variance of the time averaged mean squared displacement, regardless of the measurement time. Fluctuations of this relevant experimental quantity due to the disorder average of such ensembles are thus persistent and non-negligible. The relevance of our results for single particle tracking analysis in complex and biological systems is discussed.
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Affiliation(s)
- Yousof Mardoukhi
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany.
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28
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Cherstvy AG, Thapa S, Mardoukhi Y, Chechkin AV, Metzler R. Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity. Phys Rev E 2018; 98:022134. [PMID: 30253569 DOI: 10.1103/physreve.98.022134] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2018] [Indexed: 06/08/2023]
Abstract
How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and out-of-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.
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Affiliation(s)
- Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Samudrajit Thapa
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Yousof Mardoukhi
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Aleksei V Chechkin
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
- Institute for Theoretical Physics, Kharkov Institute of Physics and Technology, 61108 Kharkov, Ukraine
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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29
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Wada AHO, Vojta T. Fractional Brownian motion with a reflecting wall. Phys Rev E 2018; 97:020102. [PMID: 29548098 DOI: 10.1103/physreve.97.020102] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2017] [Indexed: 11/07/2022]
Abstract
Fractional Brownian motion, a stochastic process with long-time correlations between its increments, is a prototypical model for anomalous diffusion. We analyze fractional Brownian motion in the presence of a reflecting wall by means of Monte Carlo simulations. Whereas the mean-square displacement of the particle shows the expected anomalous diffusion behavior 〈x^{2}〉∼t^{α}, the interplay between the geometric confinement and the long-time memory leads to a highly non-Gaussian probability density function with a power-law singularity at the barrier. In the superdiffusive case α>1, the particles accumulate at the barrier leading to a divergence of the probability density. For subdiffusion α<1, in contrast, the probability density is depleted close to the barrier. We discuss implications of these findings, in particular, for applications that are dominated by rare events.
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Affiliation(s)
- Alexander H O Wada
- Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA.,Instituto de Física, Universidade de São Paulo, Rua do Matão, 1371, 05508-090 São Paulo, São Paulo, Brazil
| | - Thomas Vojta
- Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA.,Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA
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Loch-Olszewska H, Szwabiński J. Detection of ε-ergodicity breaking in experimental data—A study of the dynamical functional sensibility. J Chem Phys 2018; 148:204105. [DOI: 10.1063/1.5025941] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Affiliation(s)
- Hanna Loch-Olszewska
- Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
| | - Janusz Szwabiński
- Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
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31
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Third-order transport coefficients for localised and delocalised charged-particle transport. Sci Rep 2018; 8:2226. [PMID: 29396503 PMCID: PMC5797156 DOI: 10.1038/s41598-018-19711-5] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2017] [Accepted: 12/18/2017] [Indexed: 11/10/2022] Open
Abstract
We derive third-order transport coefficients of skewness for a phase-space kinetic model that considers the processes of scattering collisions, trapping, detrapping and recombination losses. The resulting expression for the skewness tensor provides an extension to Fick’s law which is in turn applied to yield a corresponding generalised advection-diffusion-skewness equation. A physical interpretation of trap-induced skewness is presented and used to describe an observed negative skewness due to traps. A relationship between skewness, diffusion, mobility and temperature is formed by analogy with Einstein’s relation. Fractional transport is explored and its effects on the flux transport coefficients are also outlined.
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32
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Sadhu T, Delorme M, Wiese KJ. Generalized Arcsine Laws for Fractional Brownian Motion. PHYSICAL REVIEW LETTERS 2018; 120:040603. [PMID: 29437446 DOI: 10.1103/physrevlett.120.040603] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2017] [Revised: 12/09/2017] [Indexed: 06/08/2023]
Abstract
The three arcsine laws for Brownian motion are a cornerstone of extreme-value statistics. For a Brownian B_{t} starting from the origin, and evolving during time T, one considers the following three observables: (i) the duration t_{+} the process is positive, (ii) the time t_{last} the process last visits the origin, and (iii) the time t_{max} when it achieves its maximum (or minimum). All three observables have the same cumulative probability distribution expressed as an arcsine function, thus the name arcsine laws. We show how these laws change for fractional Brownian motion X_{t}, a non-Markovian Gaussian process indexed by the Hurst exponent H. It generalizes standard Brownian motion (i.e., H=1/2). We obtain the three probabilities using a perturbative expansion in ϵ=H-1/2. While all three probabilities are different, this distinction can only be made at second order in ϵ. Our results are confirmed to high precision by extensive numerical simulations.
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Affiliation(s)
- Tridib Sadhu
- Tata Institute of Fundamental Research, Mumbai 400005, India
| | - Mathieu Delorme
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, PSL Research University, Sorbonne Universités, UPMC, 24 rue Lhomond, 75005 Paris, France
| | - Kay Jörg Wiese
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, PSL Research University, Sorbonne Universités, UPMC, 24 rue Lhomond, 75005 Paris, France
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Cherstvy AG, Nagel O, Beta C, Metzler R. Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells. Phys Chem Chem Phys 2018; 20:23034-23054. [DOI: 10.1039/c8cp04254c] [Citation(s) in RCA: 51] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?
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Affiliation(s)
- Andrey G. Cherstvy
- Institute for Physics & Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
| | - Oliver Nagel
- Institute for Physics & Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
| | - Carsten Beta
- Institute for Physics & Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
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