1
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Zhan Z, Wang X. Ergodic criterion of a random diffusivity model. Phys Rev E 2024; 109:044115. [PMID: 38755829 DOI: 10.1103/physreve.109.044115] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2023] [Accepted: 03/06/2024] [Indexed: 05/18/2024]
Abstract
The random diffusivity, initially proposed to explain Brownian yet non-Gaussian diffusion, has garnered significant attention due to its capacity not only for elucidating the internal physical mechanism of non-Gaussian diffusion, but also for establishing an analytical framework to characterize particle motion in complex environments. In this paper, based on the correlation function C(t_{1},t_{2})=〈D(t_{1})D(t_{2})〉 of random diffusivity D(t), we quantitatively propose a general criterion of determining the ergodic property of the Langevin equation with the arbitrary random diffusivity D(t). Due to the critical role of correlation function C(t_{1},t_{2}), we derive the criterion for the two cases with stationary diffusivity or nonstationary diffusivity, respectively. By utilizing the quantitative criterion, we can directly judge the ergodic properties of the random diffusivity model based on the correlation function C(t_{1},t_{2}) of random diffusivity D(t). Several typical diffusivities, including the common square of the Brownian motion and of the (fractional) Ornstein-Uhlenbeck process, are found to contribute to different ergodic properties, which validates our proposed criterion built on the correlation function C(t_{1},t_{2}).
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Affiliation(s)
- Zhongshuai Zhan
- School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, People's Republic of China
| | - Xudong Wang
- School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, People's Republic of China
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2
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Zhao XB, Zhang X, Guo W. Diffusion of active Brownian particles under quenched disorder. PLoS One 2024; 19:e0298466. [PMID: 38437208 PMCID: PMC10911629 DOI: 10.1371/journal.pone.0298466] [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: 06/30/2023] [Accepted: 01/24/2024] [Indexed: 03/06/2024] Open
Abstract
The motion of a single active particle in one dimension with quenched disorder under the external force is investigated. Within the tailored parameter range, anomalous diffusion that displays weak ergodicity breaking is observed, i.e., non-ergodic subdiffusion and non-ergodic superdiffusion. This non-ergodic anomalous diffusion is analyzed through the time-dependent probability distributions of the particle's velocities and positions. Its origin is attributed to the relative weights of the locked state (predominant in the subdiffusion state) and running state (predominant in the superdiffusion state). These results may contribute to understanding the dynamical behavior of self-propelled particles in nature and the extraordinary response of nonlinear dynamics to the externally biased force.
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Affiliation(s)
- Xiong-Biao Zhao
- Key Laboratory of Artificial Microstructures in Yunnan Higher Education Institutions, School of Physical Science and Technology, Kunming University, Kunming, China
| | - Xiao Zhang
- Key Laboratory of Artificial Microstructures in Yunnan Higher Education Institutions, School of Physical Science and Technology, Kunming University, Kunming, China
| | - Wei Guo
- Key Laboratory of Artificial Microstructures in Yunnan Higher Education Institutions, School of Physical Science and Technology, Kunming University, Kunming, China
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3
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Li Y, Suleiman K, Xu Y. Anomalous diffusion, non-Gaussianity, nonergodicity, and confinement in stochastic-scaled Brownian motion with diffusing diffusivity dynamics. Phys Rev E 2024; 109:014139. [PMID: 38366530 DOI: 10.1103/physreve.109.014139] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2023] [Accepted: 12/07/2023] [Indexed: 02/18/2024]
Abstract
Scaled Brownian motions (SBMs) with power-law time-dependent diffusivity have been used to describe various types of anomalous diffusion yet Gaussian observed in granular gases kinetics, turbulent diffusion, and molecules mobility in cells, to name a few. However, some of these systems may exhibit non-Gaussian behavior which can be described by SBM with diffusing diffusivity (DD-SBM). Here, we numerically investigate both free and confined DD-SBM models characterized by fixed or stochastic scaling exponent of time-dependent diffusivity. The effects of distributed scaling exponent, random diffusivity, and confinement are considered. Different regimes of ultraslow diffusion, subdiffusion, normal diffusion, and superdiffusion are observed. In addition, weak ergodic and non-Gaussian behaviors are also detected. These results provide insights into diffusion in time-fluctuating diffusivity landscapes with potential applications to time-dependent temperature systems spreading in heterogeneous environments.
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Affiliation(s)
- Yongge Li
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Kheder Suleiman
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Yong Xu
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
- MOE Key Laboratory for Complexity Science in Aerospace, Northwestern Polytechnical University, Xi'an 710072, China
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4
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Seckler H, Szwabiński J, Metzler R. Machine-Learning Solutions for the Analysis of Single-Particle Diffusion Trajectories. J Phys Chem Lett 2023; 14:7910-7923. [PMID: 37646323 DOI: 10.1021/acs.jpclett.3c01351] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 09/01/2023]
Abstract
Single-particle traces of the diffusive motion of molecules, cells, or animals are by now routinely measured, similar to stochastic records of stock prices or weather data. Deciphering the stochastic mechanism behind the recorded dynamics is vital in understanding the observed systems. Typically, the task is to decipher the exact type of diffusion and/or to determine the system parameters. The tools used in this endeavor are currently being revolutionized by modern machine-learning techniques. In this Perspective we provide an overview of recently introduced methods in machine-learning for diffusive time series, most notably, those successfully competing in the anomalous diffusion challenge. As such methods are often criticized for their lack of interpretability, we focus on means to include uncertainty estimates and feature-based approaches, both improving interpretability and providing concrete insight into the learning process of the machine. We expand the discussion by examining predictions on different out-of-distribution data. We also comment on expected future developments.
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Affiliation(s)
- Henrik Seckler
- Institute of Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Janusz Szwabiński
- Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
| | - Ralf Metzler
- Institute of Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
- Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
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5
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Evstigneev M, Kaffashnia A. Diffusion coefficient scaling of a free Brownian particle with velocity-dependent damping. Phys Rev E 2023; 107:064129. [PMID: 37464597 DOI: 10.1103/physreve.107.064129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2023] [Accepted: 06/08/2023] [Indexed: 07/20/2023]
Abstract
An analytical expression for the diffusion coefficient D of a free Brownian particle with velocity-dependent damping γ(v) is derived from the Green-Kubo formula. A special case of damping that decreases monotonically with velocity is considered. At high temperature T, the diffusion coefficient is found to exhibit two scaling types: (i) for a power-law decrease of damping with the particle's kinetic energy, γ(v)∝1/v^{2α}, it scales as D∝T^{α+1}; (ii) for a Gaussian function γ(v), it diverges at temperatures above a critical value T_{c} and behaves as D∝1/sqrt[T_{c}-T] at T slightly below T_{c}. At T>T_{c}, the particle trajectory contains long flight events, which are not observed at T<T_{c} in case (ii) and at all temperatures in case (i).
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Affiliation(s)
- Mykhaylo Evstigneev
- Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John's, NL, Canada A1B 3X7
| | - Amir Kaffashnia
- Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John's, NL, Canada A1B 3X7
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6
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Montana F, Camporeale C, Porporato A, Rondoni L. Inertial and geometrical effects of self-propelled elliptical Brownian particles. Phys Rev E 2023; 107:054607. [PMID: 37328983 DOI: 10.1103/physreve.107.054607] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2023] [Accepted: 05/11/2023] [Indexed: 06/18/2023]
Abstract
Active particles that self-propel by transforming energy into mechanical motion represent a growing area of research in mathematics, physics, and chemistry. Here we investigate the dynamics of nonspherical inertial active particles moving in a harmonic potential, introducing geometric parameters which take into account the role of eccentricity for nonspherical particles. A comparison between the overdamped and underdamped models for elliptical particles is performed. The model of overdamped active Brownian motion has been used to describe most of the basic aspects of micrometer-sized particles moving in a liquid ("microswimmers"). We consider active particles by extending the active Brownian motion model to incorporate translation and rotation inertia and account for the role of eccentricity. We show how the overdamped and the underdamped models behave in the same way for small values of activity (Brownian case) if eccentricity is equal to zero, but increasing eccentricity leads the two dynamics to substantially depart from each other-in particular, the action of a torque induced by external forces, induced a marked difference close to the walls of the domain if eccentricity is high. Effects induced by inertia include an inertial delay time of the self-propulsion direction from the particle velocity, and the differences between the overdamped and underdamped systems are particularly evident in the first and second moments of the particle velocities. Comparison with the experimental results of vibrated granular particles shows good agreement and corroborates the notion that self-propelling massive particles moving in gaseous media are dominated by inertial effects.
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Affiliation(s)
- Federica Montana
- Department of Mathematical Sciences, Politecnico di Torino, Turin, Italy and INFN, Sezione di Torino, Turin, Italy
| | - Carlo Camporeale
- Department of Environmental, Land and Infrastructure Engineering, Politecnico di Torino, Turin, Italy
| | - Amilcare Porporato
- Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey, USA and High Meadows Environmental Institute, Princeton University, Princeton, New Jersey, USA
| | - Lamberto Rondoni
- Department of Mathematical Sciences, Politecnico di Torino, Turin, Italy and INFN, Sezione di Torino, Turin, Italy
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7
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Mayer Martins J, Wittkowski R. Inertial dynamics of an active Brownian particle. Phys Rev E 2022; 106:034616. [PMID: 36266913 DOI: 10.1103/physreve.106.034616] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2022] [Accepted: 08/23/2022] [Indexed: 06/16/2023]
Abstract
Active Brownian motion commonly assumes spherical overdamped particles. However, self-propelled particles are often neither symmetric nor overdamped yet underlie random fluctuations from their surroundings. Active Brownian motion has already been generalized to include asymmetric particles. Separately, recent findings have shown the importance of inertial effects for particles of macroscopic size or in low-friction environments. We aim to consolidate the previous findings into the general description of a self-propelled asymmetric particle with inertia. We derive the Langevin equation of such a particle as well as the corresponding Fokker-Planck equation. Furthermore, a formula is presented that allows reconstructing the hydrodynamic resistance matrix of the particle by measuring its trajectory. Numerical solutions of the Langevin equation show that, independently of the particle's shape, the noise-free trajectory at zero temperature starts with an inertial transition phase and converges to a circular helix. We discuss this universal convergence with respect to the helical motion that many microorganisms exhibit.
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Affiliation(s)
- Jonas Mayer Martins
- Institute of Theoretical Physics, Center for Soft Nanoscience, University of Münster, 48149 Münster, Germany
| | - Raphael Wittkowski
- Institute of Theoretical Physics, Center for Soft Nanoscience, University of Münster, 48149 Münster, Germany
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8
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Yamamoto T, Sussman DM, Shibata T, Manning ML. Non-monotonic fluidization generated by fluctuating edge tensions in confluent tissues. SOFT MATTER 2022; 18:2168-2175. [PMID: 35212696 DOI: 10.1039/d0sm01559h] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
In development and homeostasis, multi-cellular systems exhibit spatial and temporal heterogeneity in their biochemical and mechanical properties. Nevertheless, it remains unclear how spatiotemporally heterogeneous forces affect the dynamical and mechanical properties of confluent tissue. To address this question, we study the dynamical behavior of the two-dimensional cellular vertex model for epithelial monolayers in the presence of fluctuating cell-cell interfacial tensions, which is a biologically relevant source of mechanical spatiotemporal heterogeneity. In particular, we investigate the effects of the amplitude and persistence time of fluctuating tension on the tissue dynamics. We unexpectedly find that the long-time diffusion constant describing cell rearrangements depends non-monotonically on the persistence time, while it increases monotonically as the amplitude increases. Our analysis indicates that at low and intermediate persistence times tension fluctuations drive motion of vertices and promote cell rearrangements, while at the highest persistence times the tension in the network evolves so slowly that rearrangements become rare.
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Affiliation(s)
- Takaki Yamamoto
- Laboratory for Physical Biology, RIKEN Center for Biosystems Dynamics Research, Kobe 650-0047, Japan
- Nonequilibrium Physics of Living Matter RIKEN Hakubi Research Team, RIKEN Center for Biosystems Dynamics Research, 2-2-3 Minatojima-minamimachi, Chuo-ku, Kobe 650-0047, Japan.
| | | | - Tatsuo Shibata
- Laboratory for Physical Biology, RIKEN Center for Biosystems Dynamics Research, Kobe 650-0047, Japan
| | - M Lisa Manning
- Department of Physics, Syracuse University, Syracuse, New York 13244, USA.
- BioInspired Institute, Syracuse University, Syracuse, New York 13244, USA
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9
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Wang W, Metzler R, Cherstvy AG. Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models. Phys Chem Chem Phys 2022; 24:18482-18504. [DOI: 10.1039/d2cp01741e] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
How does a systematic time-dependence of the diffusion coefficient $D (t)$ affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we examine how the behavior of the...
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10
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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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11
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Fritsch B, Hutzler A, Wu M, Khadivianazar S, Vogl L, Jank MPM, März M, Spiecker E. Accessing local electron-beam induced temperature changes during in situ liquid-phase transmission electron microscopy. NANOSCALE ADVANCES 2021; 3:2466-2474. [PMID: 36134158 PMCID: PMC9419575 DOI: 10.1039/d0na01027h] [Citation(s) in RCA: 18] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2020] [Accepted: 02/18/2021] [Indexed: 05/26/2023]
Abstract
A significant electron-beam induced heating effect is demonstrated for liquid-phase transmission electron microscopy at low electron flux densities using Au nanoparticles as local nanothermometers. The obtained results are in agreement with theoretical considerations. Furthermore, the impact of beam-induced heating on radiolysis chemistry is estimated and the consequences of the effect are discussed.
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Affiliation(s)
- Birk Fritsch
- Electron Devices (LEB), Department of Electrical, Electronic and Communication Engineering, Friedrich-Alexander University Erlangen-Nürnberg (FAU) Cauerstraße 6 91058 Erlangen Germany
| | - Andreas Hutzler
- Electron Devices (LEB), Department of Electrical, Electronic and Communication Engineering, Friedrich-Alexander University Erlangen-Nürnberg (FAU) Cauerstraße 6 91058 Erlangen Germany
| | - Mingjian Wu
- Institute of Micro- and Nanostructure Research (IMN) & Center for Nanoanalysis and Electron Microscopy (CENEM), Interdisciplinary Center for Nanostructured Films (IZNF), Department of Materials Science and Engineering, Friedrich-Alexander University Erlangen-Nürnberg (FAU) Cauerstraße 3 91058 Erlangen Germany
| | - Saba Khadivianazar
- Electron Devices (LEB), Department of Electrical, Electronic and Communication Engineering, Friedrich-Alexander University Erlangen-Nürnberg (FAU) Cauerstraße 6 91058 Erlangen Germany
| | - Lilian Vogl
- Institute of Micro- and Nanostructure Research (IMN) & Center for Nanoanalysis and Electron Microscopy (CENEM), Interdisciplinary Center for Nanostructured Films (IZNF), Department of Materials Science and Engineering, Friedrich-Alexander University Erlangen-Nürnberg (FAU) Cauerstraße 3 91058 Erlangen Germany
| | - Michael P M Jank
- Fraunhofer Institute for Integrated Systems and Device Technology IISB Schottkystraße 10 91058 Erlangen Germany
| | - Martin März
- Electron Devices (LEB), Department of Electrical, Electronic and Communication Engineering, Friedrich-Alexander University Erlangen-Nürnberg (FAU) Cauerstraße 6 91058 Erlangen Germany
- Fraunhofer Institute for Integrated Systems and Device Technology IISB Schottkystraße 10 91058 Erlangen Germany
| | - Erdmann Spiecker
- Institute of Micro- and Nanostructure Research (IMN) & Center for Nanoanalysis and Electron Microscopy (CENEM), Interdisciplinary Center for Nanostructured Films (IZNF), Department of Materials Science and Engineering, Friedrich-Alexander University Erlangen-Nürnberg (FAU) Cauerstraße 3 91058 Erlangen Germany
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12
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Sprenger AR, Jahanshahi S, Ivlev AV, Löwen H. Time-dependent inertia of self-propelled particles: The Langevin rocket. Phys Rev E 2021; 103:042601. [PMID: 34005997 DOI: 10.1103/physreve.103.042601] [Citation(s) in RCA: 22] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2021] [Accepted: 03/16/2021] [Indexed: 06/12/2023]
Abstract
Many self-propelled objects are large enough to exhibit inertial effects but still suffer from environmental fluctuations. The corresponding basic equations of motion are governed by active Langevin dynamics, which involve inertia, friction, and stochastic noise for both the translational and orientational degrees of freedom coupled via the self-propulsion along the particle orientation. In this paper, we generalize the active Langevin model to time-dependent parameters and explicitly discuss the effect of time-dependent inertia for achiral and chiral particles. Realizations of this situation are manifold, ranging from minirockets (which are self-propelled by burning their own mass), to dust particles in plasma (which lose mass by evaporating material), to walkers with expiring activity. Here we present analytical solutions for several dynamical correlation functions, such as mean-square displacement and orientational and velocity autocorrelation functions. If the parameters exhibit a slow power law in time, we obtain anomalous superdiffusion with a nontrivial dynamical exponent. Finally, we constitute the "Langevin rocket" model by including orientational fluctuations in the traditional Tsiolkovsky rocket equation. We calculate the mean reach of the Langevin rocket and discuss different mass ejection strategies to maximize it. Our results can be tested in experiments on macroscopic robotic or living particles or in self-propelled mesoscopic objects moving in media of low viscosity, such as complex plasma.
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Affiliation(s)
- Alexander R Sprenger
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
| | - Soudeh Jahanshahi
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
| | - Alexei V Ivlev
- Max-Planck-Institut für Extraterrestrische Physik, 85748 Garching, Germany
| | - Hartmut Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
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13
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Sicard F, Koskin V, Annibale A, Rosta E. Position-Dependent Diffusion from Biased Simulations and Markov State Model Analysis. J Chem Theory Comput 2021; 17:2022-2033. [DOI: 10.1021/acs.jctc.0c01151] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/28/2022]
Affiliation(s)
- François Sicard
- Department of Chemistry, King’s College London, SE1 1DB London, U.K
- Department of Physics and Astronomy, University College London, WC1E 6BT London, U.K
| | - Vladimir Koskin
- Department of Chemistry, King’s College London, SE1 1DB London, U.K
- Department of Physics and Astronomy, University College London, WC1E 6BT London, U.K
| | - Alessia Annibale
- Department of Mathematics, King’s College London, SE11 6NJ London, U.K
| | - Edina Rosta
- Department of Chemistry, King’s College London, SE1 1DB London, U.K
- Department of Physics and Astronomy, University College London, WC1E 6BT London, U.K
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14
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Abstract
We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.
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15
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Xu XB, Tang T, Wang ZH, Xu XN, Fang GY, Gu M. Nonequilibrium pattern formation in circularly confined two-dimensional systems with competing interactions. Phys Rev E 2021; 103:012604. [PMID: 33601588 DOI: 10.1103/physreve.103.012604] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2020] [Accepted: 12/19/2020] [Indexed: 11/07/2022]
Abstract
We numerically investigate the nonequilibrium behaviors of classic particles with competing interactions confined in a two-dimensional logarithmic trap. We reveal a quench-induced surprising dynamics exhibiting rich dynamic patterns depending upon confinement strength and trap size, which is attributed to the time-dependent competition between interparticle repulsions and attractions under a circular confinement. Moreover, in the collectively diffusive motions of the particles, we find that the emergence of dynamic structure transformation coincides with a diffusive mode transition from superdiffusion to subdiffusion. These findings are likely useful in understanding the pattern selection and evolution in various chemical and biological systems in addition to modulated systems, and add a new route to tailoring the morphology of pattern-forming systems.
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Affiliation(s)
- X B Xu
- Collaborative Innovation Center of Advanced Microstructures, School of Physics, Nanjing University, Nanjing 210093, People's Republic of China
| | - T Tang
- Collaborative Innovation Center of Advanced Microstructures, School of Physics, Nanjing University, Nanjing 210093, People's Republic of China
| | - Z H Wang
- Collaborative Innovation Center of Advanced Microstructures, School of Physics, Nanjing University, Nanjing 210093, People's Republic of China
| | - X N Xu
- Collaborative Innovation Center of Advanced Microstructures, School of Physics, Nanjing University, Nanjing 210093, People's Republic of China
| | - G Y Fang
- Collaborative Innovation Center of Advanced Microstructures, School of Physics, Nanjing University, Nanjing 210093, People's Republic of China
| | - M Gu
- Collaborative Innovation Center of Advanced Microstructures, School of Physics, Nanjing University, Nanjing 210093, People's Republic of China
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16
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Yin H, Liu S, Wen X. Optimal transition paths of phenotypic switching in a non-Markovian self-regulation gene expression. Phys Rev E 2021; 103:022409. [PMID: 33736096 DOI: 10.1103/physreve.103.022409] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2020] [Accepted: 01/06/2021] [Indexed: 11/07/2022]
Abstract
Gene expression is a complex biochemical process involving multiple reaction steps, creating molecular memory because the probability of waiting time between consecutive reaction steps no longer follows exponential distributions. What effect the molecular memory has on metastable states in gene expression remains not fully understood. Here, we study transition paths of switching between bistable states for a non-Markovian model of gene expression equipped with a self-regulation. Employing the large deviation theory for this model, we analyze the optimal transition paths of switching between bistable states in gene expression, interestingly finding that dynamic behaviors in gene expression along the optimal transition paths significantly depend on the molecular memory. Moreover, we discover that the molecular memory can prolong the time of switching between bistable states in gene expression along the optimal transition paths. Our results imply that the molecular memory may be an unneglectable factor to affect switching between metastable states in gene expression.
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Affiliation(s)
- Hongwei Yin
- School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou 221111, People's Republic of China
- School of Science, Nanchang University, Nanchang 330031, People's Republic of China
| | - Shuqin Liu
- School of Science, Nanchang University, Nanchang 330031, People's Republic of China
| | - Xiaoqing Wen
- School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou 221111, People's Republic of China
- School of Science, Nanchang University, Nanchang 330031, People's Republic of China
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17
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Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration. MATHEMATICS 2020. [DOI: 10.3390/math8071123] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/17/2022]
Abstract
We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and the conditional variance were explicitly evaluated. Several time-inhomogeneous processes were studied in detail in view of their potential applications in population growth models and in queuing systems. A time-inhomogeneous linear birth–death processes with finite state-space was also taken into account. Special attention was devoted to the cases of periodic immigration intensity functions that play an important role in the description of the evolution of dynamic systems influenced by seasonal immigration or other regular environmental cycles. Various numerical computations were performed for periodic immigration intensity functions.
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18
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Bodrova AS, Sokolov IM. Continuous-time random walks under power-law resetting. Phys Rev E 2020; 101:062117. [PMID: 32688585 DOI: 10.1103/physreve.101.062117] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2019] [Accepted: 05/20/2020] [Indexed: 06/11/2023]
Abstract
We study continuous-time random walks (CTRW) with power-law distribution of waiting times under resetting which brings the walker back to the origin, with a power-law distribution of times between the resetting events. Two situations are considered. Under complete resetting, the CTRW after the resetting event starts anew, with a new waiting time, independent of the prehistory. Under incomplete resetting, the resetting of the coordinate does not influence the waiting time until the next jump. We focus on the behavior of the mean-squared displacement (MSD) of the walker from its initial position, on the conditions under which the probability density functions of the walker's displacement show universal behavior, and on this universal behavior itself. We show, that the behavior of the MSD is the same as in the scaled Brownian motion (SBM), being the mean-field model of the CTRW. The intermediate asymptotics of the probability density functions (PDF) for CTRW under complete resetting (provided they exist) are also the same as in the corresponding case for SBM. For incomplete resetting, however, the behavior of the PDF for CTRW and SBM is vastly different.
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Affiliation(s)
- Anna S Bodrova
- Department of Physics, Humboldt University, Newtonstrasse 15, D-12489 Berlin, Germany
| | - Igor M Sokolov
- Department of Physics, Humboldt University, Newtonstrasse 15, D-12489 Berlin, Germany
- IRIS Adlershof, Zum Großen Windkanal 6, D-12489 Berlin, Germany
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19
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Bodrova AS, Chechkin AV, Sokolov IM. Nonrenewal resetting of scaled Brownian motion. Phys Rev E 2019; 100:012119. [PMID: 31499839 DOI: 10.1103/physreve.100.012119] [Citation(s) in RCA: 30] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2018] [Indexed: 11/07/2022]
Abstract
We investigate an intermittent stochastic process in which diffusive motion with a time-dependent diffusion coefficient, D(t)∼t^{α-1}, α>0 (scaled Brownian motion), is stochastically reset to its initial position and starts anew. The resetting follows a renewal process with either an exponential or a power-law distribution of the waiting times between successive renewals. The resetting events, however, do not affect the time dependence of the diffusion coefficient, so that the whole process appears to be a nonrenewal one. We discuss the mean squared displacement of a particle and the probability density function of its positions in this process. We show that scaled Brownian motion with resetting demonstrates rich behavior whose properties essentially depend on the interplay of the parameters of the resetting process and the particle's displacement infree motion. The motion of particles can remain almost unaffected by resetting but can also get slowed down or even be completely suppressed. Especially interesting are the nonstationary situations in which the mean squared displacement stagnates but the distribution of positions does not tend to any steady state. This behavior is compared to the situation [discussed in the companion paper; A. S. Bodrova et al., Phys. Rev. E 100, 012120 (2019)10.1103/PhysRevE.100.012120] in which the memory of the value of the diffusion coefficient at a resetting time is erased, so that the whole process is a fully renewal one. We show that the properties of the probability densities in such processes (erasing or retaining the memory on the diffusion coefficient) are vastly different.
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Affiliation(s)
- Anna S Bodrova
- Department of Physics, Humboldt University, Newtonstrasse 15, 12489 Berlin, Germany.,Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, Moscow 123458, Russia.,Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow 119991, Russia
| | - Aleksei V Chechkin
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany.,Akhiezer Institute for Theoretical Physics, Kharkov Institute of Physics and Technology, 61108 Kharkov, Ukraine
| | - Igor M Sokolov
- Department of Physics, Humboldt University, Newtonstrasse 15, 12489 Berlin, Germany
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20
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Sikdar S, Ghosh M, Adak A, Chakrabarti J. Structural and dynamic responses of calcium ion binding loop residues in metallo-proteins. Biophys Chem 2019; 252:106207. [PMID: 31252378 DOI: 10.1016/j.bpc.2019.106207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/25/2019] [Revised: 06/11/2019] [Accepted: 06/11/2019] [Indexed: 10/26/2022]
Abstract
Conformational changes in bio-molecular systems are fundamental to several biological processes. It is important to study changes in responses of underlying microscopic variables, like dihedral angles as conformational change takes place. We perform all-atom simulations and modelling via Langevin equation to illustrate the changes in structural and dynamic responses of dihedral angles of calcium ion binding residues of different proteins in metal ion free (apo) and bound (holo) states. The equilibrium distributions of dihedral angles in apo- and holo-states represent structural response. Our studies show the presence of dihedrals with multiple peaks (isomeric states) separated by barrier heights is more frequent in apo- than in holo-state. The relaxation time-scale of dihedral fluctuations is found to increase linearly with decreasing barrier height due to more frequent barrier re-crossing events. The slow kinetic response of the dihedrals also contributes to slowing down of macro-scale fluctuations, which may be useful to understand kinetics of various bio-molecular processes.
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Affiliation(s)
- Samapan Sikdar
- Department of Chemical, Biological & Macro-Molecular Sciences, S. N. Bose National Centre for Basic Sciences, Sector III, Block JD, Salt Lake, Kolkata 700106, India.
| | - Mahua Ghosh
- Department of Chemical, Biological & Macro-Molecular Sciences, S. N. Bose National Centre for Basic Sciences, Sector III, Block JD, Salt Lake, Kolkata 700106, India
| | - Arunava Adak
- Department of Chemical, Biological & Macro-Molecular Sciences, S. N. Bose National Centre for Basic Sciences, Sector III, Block JD, Salt Lake, Kolkata 700106, India
| | - J Chakrabarti
- Department of Chemical, Biological & Macro-Molecular Sciences, S. N. Bose National Centre for Basic Sciences, Sector III, Block JD, Salt Lake, Kolkata 700106, India; The Thematic Unit of Excellence on Computational Materials Science, S. N. Bose National Centre for Basic Sciences, Sector III, Block JD, Salt Lake, Kolkata 700106, India.
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21
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Bian Y, Yan R, Li P, Zhao N. Unusual crowding-induced chain looping kinetics in hard-sphere fluids: a contrastive study with polymer solutions. SOFT MATTER 2019; 15:4976-4988. [PMID: 31173026 DOI: 10.1039/c9sm00400a] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
A theoretical framework is developed to investigate the looping kinetics of a chain in hard-sphere (HS) fluids, based on a generalized Smoluchowski diffusion-reaction equation. A contrastive study with polymer solutions is performed. The crowding-associated effective viscosity and collapse effects are properly taken into account, which obey different scaling relations in HS and polymer fluids. We examine the dependence of the looping time on both concentration and size of crowders, demonstrating unusual and distinct discrepancies in the two crowded media. Firstly, in the solution of large polymers, the looping rate grows monotonically with polymer concentration. On the other hand, in the solution of large HSs, a caging regime can be observed, where the looping time tends to the value in the absence of crowders. Secondly, polymers in moderate size generally impede chain looping due to the enhanced viscosity. However, in HS fluids, the looping time exhibits a rather complicated variation with increasing HS size. We show a possible mechanism where in the case of small crowders with a relatively strong compaction in the probed chain, the looping kinetics can be facilitated. As the crowder size increases, the collapse effect is reduced and looping is dominated by viscosity-induced inhibition. Simultaneously, our theory rationalizes another possibility of the mechanism observed by recent simulation work. We conclude that the looping kinetics in specific systems actually should be governed by the critical competition between the two crowding factors. By giving reasonable measurements of effective viscosity and collapse, our theoretical framework can provide a unified strategy to analyze crowding effects on the looping rate in a systematic manner.
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Affiliation(s)
- Yukun Bian
- College of Physical Science and Technology, Sichuan University, Chengdu 610064, China
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22
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Postnikov EB. Density fluctuations and random walks in an overdamped and supercooled simple liquid. Phys Rev E 2019; 99:062117. [PMID: 31330632 DOI: 10.1103/physreve.99.062117] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2019] [Indexed: 11/07/2022]
Abstract
In this work, the short-time dynamics of simple liquid is explored both analytically and numerically with the focus on the interplay between the density fluctuations in a volume surrounding a chosen particle and its random walk motion. The particles interact via the Lennard-Jones potential with parameters corresponding to liquid argon. For large times, analytical calculations based on the fluctuation theory provides an explicit expression reproducing isothermal change of the self-diffusion coefficient in liquid argon corresponding to the experimental data. These results lead to the conclusion that such behavior is based on the reduced mobility of particles reflected in their density fluctuations that can be equivalently achieved in the cases of either low temperatures and pressures (supercooling) or moderate temperatures and high pressures (overdamping).
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Affiliation(s)
- E B Postnikov
- Department of Theoretical Physics, Kursk State University, Radishcheva st., 33, 305000 Kursk, Russia
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23
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Affiliation(s)
- Razvan C. Stan
- Immunology DepartmentUniversity of São Paulo Brazil
- Cantacuzino Military-Medical Research-Development National Institute, Bucharest Romania
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24
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Abstract
In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle- τ Cantor set. The fractal mean square displacement of different random walks on the middle- τ Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details.
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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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Šiler M, Ornigotti L, Brzobohatý O, Jákl P, Ryabov A, Holubec V, Zemánek P, Filip R. Diffusing up the Hill: Dynamics and Equipartition in Highly Unstable Systems. PHYSICAL REVIEW LETTERS 2018; 121:230601. [PMID: 30576167 DOI: 10.1103/physrevlett.121.230601] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2018] [Revised: 08/20/2018] [Indexed: 06/09/2023]
Abstract
Stochastic motion of particles in a highly unstable potential generates a number of diverging trajectories leading to undefined statistical moments of the particle position. This makes experiments challenging and breaks down a standard statistical analysis of unstable mechanical processes and their applications. A newly proposed approach takes advantage of the local characteristics of the most probable particle motion instead of the divergent averages. We experimentally verify its theoretical predictions for a Brownian particle moving near an inflection in a highly unstable cubic optical potential. The most likely position of the particle atypically shifts against the force, despite the trajectories diverging in the opposite direction. The local uncertainty around the most likely position saturates even for strong diffusion and enables well-resolved position detection. Remarkably, the measured particle distribution quickly converges to a quasistationary one with the same atypical shift for different initial particle positions. The demonstrated experimental confirmation of the theoretical predictions approves the utility of local characteristics for highly unstable systems which can be exploited in thermodynamic processes to uncover energetics of unstable systems.
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Affiliation(s)
- Martin Šiler
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - Luca Ornigotti
- Department of Optics, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic
| | - Oto Brzobohatý
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - Petr Jákl
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - Artem Ryabov
- Charles University, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, 180 00 Praha 8, Czech Republic
| | - Viktor Holubec
- Charles University, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, 180 00 Praha 8, Czech Republic
- Universität Leipzig, Institut für Theoretische Physik, Postfach 100 920, D-04009 Leipzig, Germany
| | - Pavel Zemánek
- Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
| | - Radim Filip
- Department of Optics, Palacký University, 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic
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27
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Bian Y, Cao X, Li P, Zhao N. Understanding chain looping kinetics in polymer solutions: crowding effects of microviscosity and collapse. SOFT MATTER 2018; 14:8060-8072. [PMID: 30255917 DOI: 10.1039/c8sm01499j] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
A theoretical framework based on a generalized Langevin equation with fractional Gaussian noise is presented to describe the looping kinetics of chains in polymer solutions. Particular attention is paid to quantitatively revealing crowding effects on the loop formation rate in terms of microviscosity and collapse. By the aid of empirical relations for these two crowding associated physical quantities, we explicitly investigate the relationship between the looping rate and polymer concentration, the degree of polymerization, and system parameters. According to our analysis, the dependence of the looping rate on the crowder volume fraction exhibits three typical regimes: monotonic decreasing, a non-monotonic trend and monotonic increasing. We reveal that these non-trivial behaviors can be attributed to the competition between the two opposing factors of viscosity-associated inhibition and collapse-induced facilitation of loop formation. We apply our theory to analyze the kinetics of single-stranded DNA hairpin base pairing in polyethylene glycol solutions. The theoretical results can reproduce the experimental data on the closing rate of hairpins quantitatively to a certain degree with reasonable fitting parameters. The unexpected increase of the closing rate upon the addition of increasing amounts of polymer is well rationalised. Such good agreements clearly demonstrate the validity of our theory, appropriately addressing the very role of crowding effects in the relevant kinetics.
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Affiliation(s)
- Yukun Bian
- College of Physical Science and Technology, Sichuan University, Chengdu 610064, China
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28
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Zhokh A, Strizhak P. Non-Fickian Transport in Porous Media: Always Temporally Anomalous? Transp Porous Media 2018. [DOI: 10.1007/s11242-018-1066-6] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023]
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29
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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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30
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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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31
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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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32
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Abstract
Non-Gaussian diffusion is commonly considered as a result of fluctuating diffusivity, which is correlated in time or in space or both. In this work, we investigate the non-Gaussian diffusion in static disordered media via a quenched trap model, where the diffusivity is spatially correlated. Several unique effects due to quenched disorder are reported. We analytically estimate the diffusion coefficient D_{dis} and its fluctuation over samples of finite size. We show a mechanism of population splitting in the non-Gaussian diffusion. It results in a sharp peak in the distribution of displacement P(x,t) around x=0, that has frequently been observed in experiments. We examine the fidelity of the coarse-grained diffusion map, which is reconstructed from particle trajectories. Finally, we propose a procedure to estimate the correlation length in static disordered environments, where the information stored in the sample-to-sample fluctuation has been utilized.
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Affiliation(s)
- Liang Luo
- Department of Physics, Huazhong Agricultural University, Wuhan 430070, China
- Institute of Applied Physics, Huazhong Agricultural University, Wuhan 430070, China
| | - Ming Yi
- Department of Physics, Huazhong Agricultural University, Wuhan 430070, China
- Institute of Applied Physics, Huazhong Agricultural University, Wuhan 430070, China
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33
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Singh C, Mazza MG. Early-stage aggregation in three-dimensional charged granular gas. Phys Rev E 2018; 97:022904. [PMID: 29548210 DOI: 10.1103/physreve.97.022904] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2017] [Indexed: 06/08/2023]
Abstract
Neutral grains made of the same dielectric material can attain considerable charges due to collisions and generate long-range interactions. We perform molecular dynamic simulations in three dimensions for a dilute, freely cooling granular gas of viscoelastic particles that exchange charges during collisions. As compared to the case of clustering of viscoelastic particles solely due to dissipation, we find that the electrostatic interactions due to collisional charging alter the characteristic size, morphology, and growth rate of the clusters. The average cluster size grows with time as a power law, whose exponent is relatively larger in the charged gas than the neutral case. The growth of the average cluster size is found to be independent of the ratio of characteristic Coulomb to kinetic energy, or equivalently, of the typical Bjerrum length. However, this ratio alters the crossover time of the growth. Both simulations and mean-field calculations based on Smoluchowski's equation suggest that a suppression of particle diffusion due to the electrostatic interactions helps in the aggregation process.
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Affiliation(s)
- Chamkor Singh
- Max Planck Institute for Dynamics and Self-Organization (MPIDS), Am Faßberg 17, 37077, Göttingen, Germany
- Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
| | - Marco G Mazza
- Max Planck Institute for Dynamics and Self-Organization (MPIDS), Am Faßberg 17, 37077, Göttingen, Germany
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34
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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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35
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Meyer P, Barkai E, Kantz H. Scale-invariant Green-Kubo relation for time-averaged diffusivity. Phys Rev E 2017; 96:062122. [PMID: 29347404 DOI: 10.1103/physreve.96.062122] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2017] [Indexed: 06/07/2023]
Abstract
In recent years it was shown both theoretically and experimentally that in certain systems exhibiting anomalous diffusion the time- and ensemble-averaged mean-squared displacement are remarkably different. The ensemble-averaged diffusivity is obtained from a scaling Green-Kubo relation, which connects the scale-invariant nonstationary velocity correlation function with the transport coefficient. Here we obtain the relation between time-averaged diffusivity, usually recorded in single-particle tracking experiments, and the underlying scale-invariant velocity correlation function. The time-averaged mean-squared displacement is given by 〈δ^{2}[over ¯]〉∼2D_{ν}t^{β}Δ^{ν-β}, where t is the total measurement time and Δ is the lag time. Here ν is the anomalous diffusion exponent obtained from ensemble-averaged measurements 〈x^{2}〉∼t^{ν}, while β≥-1 marks the growth or decline of the kinetic energy 〈v^{2}〉∼t^{β}. Thus, we establish a connection between exponents that can be read off the asymptotic properties of the velocity correlation function and similarly for the transport constant D_{ν}. We demonstrate our results with nonstationary scale-invariant stochastic and deterministic models, thereby highlighting that systems with equivalent behavior in the ensemble average can differ strongly in their time average. If the averaged kinetic energy is finite, β=0, the time scaling of 〈δ^{2}[over ¯]〉 and 〈x^{2}〉 are identical; however, the time-averaged transport coefficient D_{ν} is not identical to the corresponding ensemble-averaged diffusion constant.
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Affiliation(s)
- Philipp Meyer
- Max Planck Institute for the Physics of Complex Systems Noethnitzer Strasse 38 D 01187 Dresden, Germany
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 52900, Israel
| | - Holger Kantz
- Max Planck Institute for the Physics of Complex Systems Noethnitzer Strasse 38 D 01187 Dresden, Germany
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36
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Ma YD, Luo KF. Anomalous and Normal Diffusion of Tracers in Crowded Environments: Effect of Size Disparity between Tracer and Crowders. CHINESE J CHEM PHYS 2017. [DOI: 10.1063/1674-0068/30/cjcp1609184] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
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37
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Fu X, Chen B, Tang J, Hassan MT, Zewail AH. Imaging rotational dynamics of nanoparticles in liquid by 4D electron microscopy. Science 2017; 355:494-498. [DOI: 10.1126/science.aah3582] [Citation(s) in RCA: 59] [Impact Index Per Article: 8.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2016] [Revised: 09/19/2016] [Accepted: 12/15/2016] [Indexed: 11/02/2022]
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38
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Shin J, Cherstvy AG, Kim WK, Zaburdaev V. Elasticity-based polymer sorting in active fluids: a Brownian dynamics study. Phys Chem Chem Phys 2017; 19:18338-18347. [DOI: 10.1039/c7cp02947k] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
Abstract
While the dynamics of polymer chains in equilibrium media is well understood by now, the polymer dynamics in active non-equilibrium environments can be very different.
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Affiliation(s)
- Jaeoh Shin
- Max Planck Institute for the Physics of Complex Systems
- 01187 Dresden
- Germany
| | - Andrey G. Cherstvy
- Institute for Physics & Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
| | - Won Kyu Kim
- Institut für Weiche Materie and Funktionale Materialen
- Helmholtz-Zentrum Berlin
- 14109 Berlin
- Germany
| | - Vasily Zaburdaev
- Max Planck Institute for the Physics of Complex Systems
- 01187 Dresden
- Germany
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39
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Safdari H, Cherstvy AG, Chechkin AV, Bodrova A, Metzler R. Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation. Phys Rev E 2017; 95:012120. [PMID: 28208482 DOI: 10.1103/physreve.95.012120] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2016] [Indexed: 06/06/2023]
Abstract
We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble- and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.
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Affiliation(s)
- Hadiseh Safdari
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
- Department of Physics, Shahid Beheshti University, 19839 Tehran, Iran
| | - Andrey G Cherstvy
- 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
- Department of Physics & Astronomy, University of Padova, "Galileo Galilei" - DFA, 35131 Padova, Italy
| | - Anna Bodrova
- Institute of Physics, Humboldt University Berlin, 12489 Berlin, Germany
- Faculty of Physics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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40
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Cherstvy AG, Metzler R. Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes. Phys Chem Chem Phys 2016; 18:23840-52. [DOI: 10.1039/c6cp03101c] [Citation(s) in RCA: 60] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/15/2023]
Abstract
We investigate the diffusive and ergodic properties of massive and confined particles in a model disordered medium, in which the local diffusivity fluctuates in time while its mean has a power law dependence on the diffusion time.
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Affiliation(s)
- Andrey G. Cherstvy
- 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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