1
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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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2
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Waigh TA, Korabel N. Heterogeneous anomalous transport in cellular and molecular biology. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2023; 86:126601. [PMID: 37863075 DOI: 10.1088/1361-6633/ad058f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2022] [Accepted: 10/20/2023] [Indexed: 10/22/2023]
Abstract
It is well established that a wide variety of phenomena in cellular and molecular biology involve anomalous transport e.g. the statistics for the motility of cells and molecules are fractional and do not conform to the archetypes of simple diffusion or ballistic transport. Recent research demonstrates that anomalous transport is in many cases heterogeneous in both time and space. Thus single anomalous exponents and single generalised diffusion coefficients are unable to satisfactorily describe many crucial phenomena in cellular and molecular biology. We consider advances in the field ofheterogeneous anomalous transport(HAT) highlighting: experimental techniques (single molecule methods, microscopy, image analysis, fluorescence correlation spectroscopy, inelastic neutron scattering, and nuclear magnetic resonance), theoretical tools for data analysis (robust statistical methods such as first passage probabilities, survival analysis, different varieties of mean square displacements, etc), analytic theory and generative theoretical models based on simulations. Special emphasis is made on high throughput analysis techniques based on machine learning and neural networks. Furthermore, we consider anomalous transport in the context of microrheology and the heterogeneous viscoelasticity of complex fluids. HAT in the wavefronts of reaction-diffusion systems is also considered since it plays an important role in morphogenesis and signalling. In addition, we present specific examples from cellular biology including embryonic cells, leucocytes, cancer cells, bacterial cells, bacterial biofilms, and eukaryotic microorganisms. Case studies from molecular biology include DNA, membranes, endosomal transport, endoplasmic reticula, mucins, globular proteins, and amyloids.
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Affiliation(s)
- Thomas Andrew Waigh
- Biological Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
| | - Nickolay Korabel
- Department of Mathematics, The University of Manchester, Manchester M13 9PL, United Kingdom
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3
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Liang Y, Wang W, Metzler R, Cherstvy AG. Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks. Phys Rev E 2023; 108:034113. [PMID: 37849140 DOI: 10.1103/physreve.108.034113] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2023] [Accepted: 08/07/2023] [Indexed: 10/19/2023]
Abstract
How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.
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Affiliation(s)
- Yingjie Liang
- College of Mechanics and Materials, Hohai University, 211100 Nanjing, China
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
| | - Wei Wang
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
| | - Ralf Metzler
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
- Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
| | - Andrey G Cherstvy
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
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4
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Sakamoto K, Akimoto T, Muramatsu M, Sansom MSP, Metzler R, Yamamoto E. Heterogeneous biological membranes regulate protein partitioning via fluctuating diffusivity. PNAS NEXUS 2023; 2:pgad258. [PMID: 37593200 PMCID: PMC10427746 DOI: 10.1093/pnasnexus/pgad258] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/06/2023] [Revised: 06/22/2023] [Accepted: 07/27/2023] [Indexed: 08/19/2023]
Abstract
Cell membranes phase separate into ordered L o and disordered L d domains depending on their compositions. This membrane compartmentalization is heterogeneous and regulates the localization of specific proteins related to cell signaling and trafficking. However, it is unclear how the heterogeneity of the membranes affects the diffusion and localization of proteins in L o and L d domains. Here, using Langevin dynamics simulations coupled with the phase-field (LDPF) method, we investigate several tens of milliseconds-scale diffusion and localization of proteins in heterogeneous biological membrane models showing phase separation into L o and L d domains. The diffusivity of proteins exhibits temporal fluctuations depending on the field composition. Increases in molecular concentrations and domain preference of the molecule induce subdiffusive behavior due to molecular collisions by crowding and confinement effects, respectively. Moreover, we quantitatively demonstrate that the protein partitioning into the L o domain is determined by the difference in molecular diffusivity between domains, molecular preference of domain, and molecular concentration. These results pave the way for understanding how biological reactions caused by molecular partitioning may be controlled in heterogeneous media. Moreover, the methodology proposed here is applicable not only to biological membrane systems but also to the study of diffusion and localization phenomena of molecules in various heterogeneous systems.
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Affiliation(s)
- Ken Sakamoto
- Department of System Design Engineering, Keio University, Yokohama, Kanagawa 223-8522, Japan
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Mayu Muramatsu
- Department of Mechanical Engineering, Keio University, Yokohama, Kanagawa 223-8522, Japan
| | - Mark S P Sansom
- Department of Biochemistry, University of Oxford, South Parks Road, Oxford OX1 3QU, UK
| | - Ralf Metzler
- Institute of Physics & Astronomy, University of Potsdam, Potsdam-Golm 14476, Germany
- Asia Pacific Centre for Theoretical Physics, Pohang 37673, Republic of Korea
| | - Eiji Yamamoto
- Department of System Design Engineering, Keio University, Yokohama, Kanagawa 223-8522, Japan
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5
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Shi H, Du L, Huang F, Guo W. Weak ergodicity breaking and anomalous diffusion in collective motion of active particles under spatiotemporal disorder. Phys Rev E 2023; 107:024114. [PMID: 36932613 DOI: 10.1103/physreve.107.024114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2021] [Accepted: 01/12/2023] [Indexed: 06/18/2023]
Abstract
The effects of spatiotemporal disorder, i.e., both the noise and quenched disorder, on the dynamics of active particles in two dimensions are investigated. We demonstrate that within the tailored parameter regime, nonergodic superdiffusion and nonergodic subdiffusion occur in the system, identified by the observable quantities (the mean squared displacement and ergodicity-breaking parameter) averaged over both the noise and realizations of quenched disorder. Their origins are attributed to the competition effects between the neighbor alignment and spatiotemporal disorder on the collective motion of active particles. These results may be helpful for further understanding the nonequilibrium transport process of active particles, as well as for detection of the transport of self-propelled particles in complex and crowded environments.
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Affiliation(s)
- Hongda Shi
- Key Laboratory of Artificial Microstructures in Yunnan Higher Education Institutions, School of Physical Science and Technology, Kunming University, Kunming 650214, China
| | - Luchun Du
- Department of Physics, Yunnan University, Kunming 650091, China
| | - Feijie Huang
- Key Laboratory of Artificial Microstructures in Yunnan Higher Education Institutions, School of Physical Science and Technology, Kunming University, Kunming 650214, China
| | - Wei Guo
- Key Laboratory of Artificial Microstructures in Yunnan Higher Education Institutions, School of Physical Science and Technology, Kunming University, Kunming 650214, China
- Yunnan Key Laboratory of Metal-Organic Molecular Materials and Devices, Kunming University, Kunming 650214, China
- National Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, China
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6
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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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7
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Wang X, Chen Y. Random diffusivity processes in an external force field. Phys Rev E 2022; 106:024112. [PMID: 36109990 DOI: 10.1103/physreve.106.024112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/03/2022] [Accepted: 07/21/2022] [Indexed: 06/15/2023]
Abstract
Brownian yet non-Gaussian processes have recently been observed in numerous biological systems, and corresponding theories have been constructed based on random diffusivity models. Considering the particularity of random diffusivity, this paper studies the effect of an external force acting on two kinds of random diffusivity models whose difference is embodied in whether the fluctuation-dissipation theorem is valid. Based on the two random diffusivity models, we derive the Fokker-Planck equations with an arbitrary external force, and we analyze various observables in the case with a constant force, including the Einstein relation, the moments, the kurtosis, and the asymptotic behaviors of the probability density function of particle displacement at different timescales. Both the theoretical results and numerical simulations of these observables show a significant difference between the two kinds of random diffusivity models, which implies the important role of the fluctuation-dissipation theorem in random diffusivity systems.
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Affiliation(s)
- Xudong Wang
- School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, People's Republic of China
| | - Yao Chen
- College of Sciences, Nanjing Agricultural University, Nanjing 210095, People's Republic of China
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8
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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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9
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Wang X, Chen Y. Ergodic property of random diffusivity system with trapping events. Phys Rev E 2022; 105:014106. [PMID: 35193240 DOI: 10.1103/physreve.105.014106] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2021] [Accepted: 12/10/2021] [Indexed: 12/25/2022]
Abstract
A Brownian yet non-Gaussian phenomenon has recently been observed in many biological and active matter systems. The main idea of explaining this phenomenon is to introduce a random diffusivity for particles moving in inhomogeneous environment. This paper considers a Langevin system containing a random diffusivity and an α-stable subordinator with α<1. This model describes the particle's motion in complex media where both the long trapping events and random diffusivity exist. We derive the general expressions of ensemble- and time-averaged mean-squared displacements which only contain the values of the inverse subordinator and diffusivity. Further taking specific time-dependent diffusivity, we obtain the analytic expressions of ergodicity breaking parameter and probability density function of the time-averaged mean-squared displacement. The results imply the nonergodicity of the random diffusivity model with any kind of diffusivity, including the critical case where the model presents normal diffusion.
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Affiliation(s)
- Xudong Wang
- School of Science, Nanjing University of Science and Technology, Nanjing, 210094, P.R. China
| | - Yao Chen
- College of Sciences, Nanjing Agricultural University, Nanjing, 210094, P.R. China
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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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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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12
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Cherstvy AG, Vinod D, Aghion E, Sokolov IM, Metzler R. Scaled geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements. Phys Rev E 2021; 103:062127. [PMID: 34271619 DOI: 10.1103/physreve.103.062127] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2021] [Accepted: 05/24/2021] [Indexed: 01/21/2023]
Abstract
Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic stochastic process of geometric Brownian motion (GBM). The innate property of such models and of real stock-market prices is the roughly exponential growth of prices with time [on average, in crisis-free times]. We here explore the ensemble- and time averages of a multiplicative-noise stochastic process with power-law-like time-dependent volatility, σ(t)∼t^{α}, named scaled GBM (SGBM). For SGBM, the mean-squared displacement (MSD) computed for an ensemble of statistically equivalent trajectories can grow faster than exponentially in time, while the time-averaged MSD (TAMSD)-based on a sliding-window averaging along a single trajectory-is always linear at short lag times Δ. The proportionality factor between these the two averages of the time series is Δ/T at short lag times, where T is the trajectory length, similarly to GBM. This discrepancy of the scaling relations and pronounced nonequivalence of the MSD and TAMSD at Δ/T≪1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with σ(t)-modulation, the main focus of our analysis. The analytical predictions for the MSD and mean TAMSD for SGBM are in quantitative agreement with the results of stochastic computer simulations.
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Affiliation(s)
- Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany.,Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
| | - Deepak Vinod
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Erez Aghion
- Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
| | - Igor M Sokolov
- Institut für Physik and IRIS Adlershof, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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13
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Yamamoto E, Akimoto T, Mitsutake A, Metzler R. Universal Relation between Instantaneous Diffusivity and Radius of Gyration of Proteins in Aqueous Solution. PHYSICAL REVIEW LETTERS 2021; 126:128101. [PMID: 33834804 DOI: 10.1103/physrevlett.126.128101] [Citation(s) in RCA: 46] [Impact Index Per Article: 15.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/05/2020] [Accepted: 02/09/2021] [Indexed: 06/12/2023]
Abstract
Protein conformational fluctuations are highly complex and exhibit long-term correlations. Here, molecular dynamics simulations of small proteins demonstrate that these conformational fluctuations directly affect the protein's instantaneous diffusivity D_{I}. We find that the radius of gyration R_{g} of the proteins exhibits 1/f fluctuations that are synchronous with the fluctuations of D_{I}. Our analysis demonstrates the validity of the local Stokes-Einstein-type relation D_{I}∝1/(R_{g}+R_{0}), where R_{0}∼0.3 nm is assumed to be a hydration layer around the protein. From the analysis of different protein types with both strong and weak conformational fluctuations, the validity of the Stokes-Einstein-type relation appears to be a general property.
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Affiliation(s)
- Eiji Yamamoto
- Department of System Design Engineering, Keio University, Yokohama, Kanagawa 223-8522, Japan
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Ayori Mitsutake
- Department of Physics, Meiji University, Kawasaki, Kanagawa 214-8571, Japan
| | - Ralf Metzler
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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14
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Sabri A, Xu X, Krapf D, Weiss M. Elucidating the Origin of Heterogeneous Anomalous Diffusion in the Cytoplasm of Mammalian Cells. PHYSICAL REVIEW LETTERS 2020; 125:058101. [PMID: 32794890 DOI: 10.1103/physrevlett.125.058101] [Citation(s) in RCA: 48] [Impact Index Per Article: 12.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2019] [Accepted: 06/22/2020] [Indexed: 06/11/2023]
Abstract
Diffusion of tracer particles in the cytoplasm of mammalian cells is often anomalous with a marked heterogeneity even within individual particle trajectories. Despite considerable efforts, the mechanisms behind these observations have remained largely elusive. To tackle this problem, we performed extensive single-particle tracking experiments on quantum dots in the cytoplasm of living mammalian cells at varying conditions. Analyses of the trajectories reveal a strong, microtubule-dependent subdiffusion with antipersistent increments and a substantial heterogeneity. Furthermore, particles stochastically switch between different mobility states, most likely due to transient associations with the cytoskeleton-shaken endoplasmic reticulum network. Comparison to simulations highlight that all experimental observations can be fully described by an intermittent fractional Brownian motion, alternating between two states of different mobility.
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Affiliation(s)
- Adal Sabri
- Experimental Physics I, University of Bayreuth, D-95440 Bayreuth, Germany
| | - Xinran Xu
- Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, Colorado 80523, USA
| | - Diego Krapf
- Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, Colorado 80523, USA
- School of Biomedical Engineering, Colorado State University, Fort Collins, Colorado 80523, USA
| | - Matthias Weiss
- Experimental Physics I, University of Bayreuth, D-95440 Bayreuth, Germany
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15
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Luo Y, Zeng C. Negative friction and mobilities induced by friction fluctuation. CHAOS (WOODBURY, N.Y.) 2020; 30:053115. [PMID: 32491875 DOI: 10.1063/1.5144556] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2020] [Accepted: 04/20/2020] [Indexed: 06/11/2023]
Abstract
We study the transport phenomena of an inertial Brownian particle in a symmetric potential with periodicity, which is driven by an external time-periodic force and an external constant bias for both cases of the deterministic dynamics and the existence of friction coefficient fluctuations. For the deterministic case, it is shown that for suitable parameters, the existence of certain appropriate friction coefficients can enhance the transport of the particle, which may be interpreted as the negative friction coefficient; additionally, there coexist absolute, differential negative, and giant positive mobilities with increasing friction coefficients in the system. We analyze physical mechanisms hinted behind these findings via basins of attraction. For the existence of friction coefficient fluctuations, it is shown that the fluctuation can enhance or weaken, even eliminate these phenomena. We present the probability distribution of the particle's velocity to interpret these mobilities and the suitable parameters' regimes of these phenomena. In order to further understand the physical mechanism, we also study diffusions corresponding to these mobilities and find that for the small fluctuation, the negative friction appears, and there coexists absolute negative mobility, superdiffusion, and ballistic diffusion, whereas all of them vanish for the large fluctuation. Our findings may extensively exist in materials, including different defects, strains, the number of interfacial hydrogen bonds, the arrangements of ions, or graphite concentrations, which hints at the existence of different friction coefficients.
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Affiliation(s)
- Yuhui Luo
- Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming 650500, China
| | - Chunhua Zeng
- Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming 650500, China
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16
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Luo L, Yi M. Quenched trap model on the extreme landscape: The rise of subdiffusion and non-Gaussian diffusion. Phys Rev E 2019; 100:042136. [PMID: 31770896 DOI: 10.1103/physreve.100.042136] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2019] [Indexed: 11/07/2022]
Abstract
Non-Gaussian diffusion has been intensively studied in recent years, which reflects the dynamic heterogeneity in the disordered media. The recent study on the non-Gaussian diffusion in a static disordered landscape suggests novel phenomena due to the quenched disorder. In this paper, we further investigate the random walk on this landscape under various effective temperatures μ, which continuously modulate the dynamic heterogeneity. We show in the long-time limit, the trap dynamics on the landscape is equivalent to the quenched trap model in which subdiffusion appears for μ<1. The non-Gaussian distribution of displacement has been analytically estimated for short t of which the stretched exponential tail is expected for μ≠1. Due to the localization in the ensemble of trajectory segments, an additional peak arises in P(x,t) around x=0 even for μ>1. Evolving in different timescales, the peak and the tail of P(x,t) are well split for a wide range of t. This theoretical paper reveals the connections among the subdiffusion, non-Gaussian diffusion, and the dynamic heterogeneity in the static disordered medium. It also offers an insight on how the cell would benefit from the quasistatic disordered structures.
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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
- School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
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17
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Miyaguchi T, Uneyama T, Akimoto T. Brownian motion with alternately fluctuating diffusivity: Stretched-exponential and power-law relaxation. Phys Rev E 2019; 100:012116. [PMID: 31499895 DOI: 10.1103/physreve.100.012116] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2019] [Indexed: 11/07/2022]
Abstract
We investigate Brownian motion with diffusivity alternately fluctuating between fast and slow states. We assume that sojourn-time distributions of these two states are given by exponential or power-law distributions. We develop a theory of alternating renewal processes to study a relaxation function which is expressed with an integral of the diffusivity over time. This relaxation function can be related to a position correlation function if the particle is in a harmonic potential and to the self-intermediate scattering function if the potential force is absent. It is theoretically shown that, at short times, the exponential relaxation or the stretched-exponential relaxation are observed depending on the power-law index of the sojourn-time distributions. In contrast, at long times, a power-law decay with an exponential cutoff is observed. The dependencies on the initial ensembles (i.e., equilibrium or nonequilibrium initial ensembles) are also elucidated. These theoretical results are consistent with numerical simulations.
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Affiliation(s)
- Tomoshige Miyaguchi
- Department of Mathematics, Naruto University of Education, Naruto, Tokushima 772-8502, Japan
| | - Takashi Uneyama
- Center for Computational Science, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa, Nagoya 464-8603, Japan
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
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18
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Granik N, Weiss LE, Nehme E, Levin M, Chein M, Perlson E, Roichman Y, Shechtman Y. Single-Particle Diffusion Characterization by Deep Learning. Biophys J 2019; 117:185-192. [PMID: 31280841 PMCID: PMC6701009 DOI: 10.1016/j.bpj.2019.06.015] [Citation(s) in RCA: 69] [Impact Index Per Article: 13.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2019] [Revised: 05/06/2019] [Accepted: 06/13/2019] [Indexed: 12/15/2022] Open
Abstract
Diffusion plays a crucial role in many biological processes including signaling, cellular organization, transport mechanisms, and more. Direct observation of molecular movement by single-particle-tracking experiments has contributed to a growing body of evidence that many cellular systems do not exhibit classical Brownian motion but rather anomalous diffusion. Despite this evidence, characterization of the physical process underlying anomalous diffusion remains a challenging problem for several reasons. First, different physical processes can exist simultaneously in a system. Second, commonly used tools for distinguishing between these processes are based on asymptotic behavior, which is experimentally inaccessible in most cases. Finally, an accurate analysis of the diffusion model requires the calculation of many observables because different transport modes can result in the same diffusion power-law α, which is typically obtained from the mean-square displacements (MSDs). The outstanding challenge in the field is to develop a method to extract an accurate assessment of the diffusion process using many short trajectories with a simple scheme that is applicable at the nonexpert level. Here, we use deep learning to infer the underlying process resulting in anomalous diffusion. We implement a neural network to classify single-particle trajectories by diffusion type: Brownian motion, fractional Brownian motion and continuous time random walk. Further, we demonstrate the applicability of our network architecture for estimating the Hurst exponent for fractional Brownian motion and the diffusion coefficient for Brownian motion on both simulated and experimental data. These networks achieve greater accuracy than time-averaged MSD analysis on simulated trajectories while only requiring as few as 25 steps. When tested on experimental data, both net and ensemble MSD analysis converge to similar values; however, the net needs only half the number of trajectories required for ensemble MSD to achieve the same confidence interval. Finally, we extract diffusion parameters from multiple extremely short trajectories (10 steps) using our approach.
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Affiliation(s)
- Naor Granik
- Department of Biomedical Engineering; Lorry I. Lokey Interdisciplinary Center for Life Sciences and Engineering
| | - Lucien E Weiss
- Department of Biomedical Engineering; Lorry I. Lokey Interdisciplinary Center for Life Sciences and Engineering
| | - Elias Nehme
- Department of Biomedical Engineering; Lorry I. Lokey Interdisciplinary Center for Life Sciences and Engineering; Department of Electrical Engineering, Technion - Israel Institute of Technology, Haifa, Israel
| | | | - Michael Chein
- Department of Physiology and Pharmacology, Sackler Faculty of Medicine; Sagol School of Neuroscience
| | - Eran Perlson
- Department of Physiology and Pharmacology, Sackler Faculty of Medicine; Sagol School of Neuroscience
| | - Yael Roichman
- Raymond & Beverly Sackler School of Chemistry; Raymond & Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel.
| | - Yoav Shechtman
- Department of Biomedical Engineering; Lorry I. Lokey Interdisciplinary Center for Life Sciences and Engineering.
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19
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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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20
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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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21
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Uneyama T, Miyaguchi T, Akimoto T. Relaxation functions of the Ornstein-Uhlenbeck process with fluctuating diffusivity. Phys Rev E 2019; 99:032127. [PMID: 30999488 DOI: 10.1103/physreve.99.032127] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2019] [Indexed: 06/09/2023]
Abstract
We study the relaxation behavior of the Ornstein-Uhlenbeck (OU) process with time-dependent and fluctuating diffusivity. In this process, the dynamics of the position vector is modeled by the Langevin equation with a linear restoring force and a fluctuating diffusivity (FD). This process can be interpreted as a simple model of relaxational dynamics with internal degrees of freedom or in a heterogeneous environment. By utilizing the functional integral expression and the transfer matrix method, we show that the relaxation function can be expressed in terms of the eigenvalues and eigenfunctions of the transfer matrix for general FD processes. We apply our general theory to two simple FD processes where the FD is described by the Markovian two-state model or an OU-type process. We show analytic expressions of the relaxation functions in these models and their asymptotic forms. We also show that the relaxation behavior of the OU process with an FD is qualitatively different from those obtained from conventional models such as the generalized Langevin equation.
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Affiliation(s)
- Takashi Uneyama
- Center for Computational Science, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa, Nagoya 464-8603, Japan
| | - Tomoshige Miyaguchi
- Department of Mathematics, Naruto University of Education, Naruto, Tokushima 772-8502, Japan
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
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22
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Maćkała A, Magdziarz M. Statistical analysis of superstatistical fractional Brownian motion and applications. Phys Rev E 2019; 99:012143. [PMID: 30780232 DOI: 10.1103/physreve.99.012143] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2018] [Indexed: 06/09/2023]
Abstract
Recent advances in experimental techniques for complex systems and the corresponding theoretical findings show that in many cases random parametrization of the diffusion coefficients gives adequate descriptions of the observed fractional dynamics. In this paper we introduce two statistical methods which can be effectively applied to analyze and estimate parameters of superstatistical fractional Brownian motion with random scale parameter. The first method is based on the analysis of the increments of the process, the second one takes advantage of the variation of the trajectories of the process. We prove the effectiveness of the methods using simulated data. Also, we apply it to the experimental data describing random motion of individual molecules inside the cell of E.coli. We show that fractional Brownian motion with Weibull-distributed diffusion coefficient gives adequate description of this experimental data.
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Affiliation(s)
- Arleta Maćkała
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland
| | - Marcin Magdziarz
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland
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23
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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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24
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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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25
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Vitali S, Sposini V, Sliusarenko O, Paradisi P, Castellani G, Pagnini G. Langevin equation in complex media and anomalous diffusion. J R Soc Interface 2018; 15:20180282. [PMID: 30158182 PMCID: PMC6127165 DOI: 10.1098/rsif.2018.0282] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2018] [Accepted: 07/30/2018] [Indexed: 11/30/2022] Open
Abstract
The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modelling approaches for the description of anomalous diffusion in biological systems, such as the very complex and heterogeneous cell environment. Nevertheless, many questions are still open, such as the joint manifestation of statistical features in agreement with different models that can also be somewhat alternative to each other, e.g. continuous time random walk and fractional Brownian motion. To overcome these limitations, we propose a stochastic diffusion model with additive noise and linear friction force (linear Langevin equation), thus involving the explicit modelling of velocity dynamics. The complexity of the medium is parametrized via a population of intensity parameters (relaxation time and diffusivity of velocity), thus introducing an additional randomness, in addition to white noise, in the particle's dynamics. We prove that, for proper distributions of these parameters, we can get both Gaussian anomalous diffusion, fractional diffusion and its generalizations.
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Affiliation(s)
- Silvia Vitali
- Department of Physics and Astronomy, Bologna University, Viale Berti Pichat 6/2, 40126 Bologna, Italy
| | - Vittoria Sposini
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
| | - Oleksii Sliusarenko
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
| | - Paolo Paradisi
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
- ISTI-CNR, Institute of Information Science and Technologies 'A. Faedo' (Consiglio Nazionale delle Ricerche), Via Moruzzi 1, 56124 Pisa, Italy
| | - Gastone Castellani
- Department of Physics and Astronomy, Bologna University, Viale Berti Pichat 6/2, 40126 Bologna, Italy
| | - Gianni Pagnini
- BCAM-Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spain
- Ikerbasque-Basque Foundation for Science, Calle de María Díaz de Haro 3, 48013 Bilbao, Basque Country, Spain
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26
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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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27
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De Vos O, Venable RM, Van Hecke T, Hummer G, Pastor RW, Ghysels A. Membrane Permeability: Characteristic Times and Lengths for Oxygen and a Simulation-Based Test of the Inhomogeneous Solubility-Diffusion Model. J Chem Theory Comput 2018; 14:3811-3824. [PMID: 29894626 DOI: 10.1021/acs.jctc.8b00115] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/07/2023]
Abstract
The balance of normal and radial (lateral) diffusion of oxygen in phospholipid membranes is critical for biological function. Based on the Smoluchowski equation for the inhomogeneous solubility-diffusion model, Bayesian analysis (BA) can be applied to molecular dynamics trajectories of oxygen to extract the free energy and the normal and radial diffusion profiles. This paper derives a theoretical formalism to convert these profiles into characteristic times and lengths associated with entering, escaping, or completely crossing the membrane. The formalism computes mean first passage times and holds for any process described by rate equations between discrete states. BA of simulations of eight model membranes with varying lipid composition and temperature indicate that oxygen travels 3 to 5 times further in the radial than in the normal direction when crossing the membrane in a time of 15 to 32 ns, thereby confirming the anisotropy of passive oxygen transport in membranes. Moreover, the preceding times and distances estimated from the BA are compared to the aggregate of 280 membrane exits explicitly observed in the trajectories. BA predictions for the distances of oxygen radial diffusion within the membrane are statistically indistinguishable from the corresponding simulation values, yet BA oxygen exit times from the membrane interior are approximately 20% shorter than the simulation values, averaged over seven systems. The comparison supports the BA approach and, therefore, the applicability of the Smoluchowski equation to membrane diffusion. Given the shorter trajectories required for the BA, these results validate the BA as a computationally attractive alternative to direct observation of exits when estimating characteristic times and radial distances. The effect of collective membrane undulations on the BA is also discussed.
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Affiliation(s)
- Oriana De Vos
- Center for Molecular Modeling , Ghent University , Technologiepark 903 , 9052 Gent , Belgium
| | - Richard M Venable
- Laboratory of Computational Biology , National Heart Lung Blood Institute, National Institutes of Health , Bethesda , Maryland 20892 , United States
| | - Tanja Van Hecke
- Department of Information Technology , Ghent University , 9000 Gent , Belgium
| | - Gerhard Hummer
- Department of Theoretical Biophysics , Max Planck Institute of Biophysics , 60438 Frankfurt am Main , Germany.,Institute for Biophysics , Goethe University Frankfurt , 60438 Frankfurt am Main , Germany
| | - Richard W Pastor
- Laboratory of Computational Biology , National Heart Lung Blood Institute, National Institutes of Health , Bethesda , Maryland 20892 , United States
| | - An Ghysels
- Center for Molecular Modeling , Ghent University , Technologiepark 903 , 9052 Gent , Belgium
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28
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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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29
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Sikora G, Kepten E, Weron A, Balcerek M, Burnecki K. An efficient algorithm for extracting the magnitude of the measurement error for fractional dynamics. Phys Chem Chem Phys 2018; 19:26566-26581. [PMID: 28920611 DOI: 10.1039/c7cp04464j] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022]
Abstract
Modern live-imaging fluorescent microscopy techniques following the stochastic motion of labeled tracer particles, i.e. single particle tracking (SPT) experiments, have uncovered significant deviations from the laws of Brownian motion in a variety of biological systems. Accurately characterizing the anomalous diffusion for SPT experiments has become a central issue in biophysics. However, measurement errors raise difficulty in the analysis of single trajectories. In this paper, we introduce a novel surface calibration method based on a fractionally integrated moving average (FIMA) process as an effective tool for extracting both the magnitude of the measurement error and the anomalous exponent for autocorrelated processes of various origins. This method is developed using a toy model - fractional Brownian motion disturbed by independent Gaussian white noise - and is illustrated on both simulated and experimental biological data. We also compare this new method with the mean-squared displacement (MSD) technique, extended to capture the measurement noise in the toy model, which shows inferior results. The introduced procedure is expected to allow for more accurate analysis of fractional anomalous diffusion trajectories with measurement errors across different experimental fields and without the need for any calibration measurements.
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Affiliation(s)
- G Sikora
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland.
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30
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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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31
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Miyaguchi T. Elucidating fluctuating diffusivity in center-of-mass motion of polymer models with time-averaged mean-square-displacement tensor. Phys Rev E 2017; 96:042501. [PMID: 29347492 DOI: 10.1103/physreve.96.042501] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2017] [Indexed: 06/07/2023]
Abstract
There have been increasing reports that the diffusion coefficient of macromolecules depends on time and fluctuates randomly. Here a method is developed to elucidate this fluctuating diffusivity from trajectory data. Time-averaged mean-square displacement (MSD), a common tool in single-particle-tracking (SPT) experiments, is generalized to a second-order tensor with which both magnitude and orientation fluctuations of the diffusivity can be clearly detected. This method is used to analyze the center-of-mass motion of four fundamental polymer models: the Rouse model, the Zimm model, a reptation model, and a rigid rodlike polymer. It is found that these models exhibit distinctly different types of magnitude and orientation fluctuations of diffusivity. This is an advantage of the present method over previous ones, such as the ergodicity-breaking parameter and a non-Gaussian parameter, because with either of these parameters it is difficult to distinguish the dynamics of the four polymer models. Also, the present method of a time-averaged MSD tensor could be used to analyze trajectory data obtained in SPT experiments.
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Affiliation(s)
- Tomoshige Miyaguchi
- Department of Mathematics, Naruto University of Education, Tokushima 772-8502, Japan
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32
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Matse M, Chubynsky MV, Bechhoefer J. Test of the diffusing-diffusivity mechanism using near-wall colloidal dynamics. Phys Rev E 2017; 96:042604. [PMID: 29347613 DOI: 10.1103/physreve.96.042604] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2016] [Indexed: 05/14/2023]
Abstract
The mechanism of diffusing diffusivity predicts that, in environments where the diffusivity changes gradually, the displacement distribution becomes non-Gaussian, even though the mean-square displacement grows linearly with time. Here, we report single-particle tracking measurements of the diffusion of colloidal spheres near a planar substrate. Because the local effective diffusivity is known, we have been able to carry out a direct test of this mechanism for diffusion in inhomogeneous media.
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Affiliation(s)
- Mpumelelo Matse
- Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
| | - Mykyta V Chubynsky
- Department of Physics, University of Ottawa, 150 Louis-Pasteur, Ottawa, Ontario, Canada K1N 6N5
| | - John Bechhoefer
- Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
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33
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34
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Jain R, Sebastian KL. Lévy flight with absorption: A model for diffusing diffusivity with long tails. Phys Rev E 2017; 95:032135. [PMID: 28415215 DOI: 10.1103/physreve.95.032135] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2016] [Indexed: 06/07/2023]
Abstract
We consider diffusion of a particle in rearranging environment, so that the diffusivity of the particle is a stochastic function of time. In our previous model of "diffusing diffusivity" [Jain and Sebastian, J. Phys. Chem. B 120, 3988 (2016)JPCBFK1520-610610.1021/acs.jpcb.6b01527], it was shown that the mean square displacement of particle remains Fickian, i.e., 〈x^{2}(T)〉∝T at all times, but the probability distribution of particle displacement is not Gaussian at all times. It is exponential at short times and crosses over to become Gaussian only in a large time limit in the case where the distribution of D in that model has a steady state limit which is exponential, i.e., π_{e}(D)∼e^{-D/D_{0}}. In the present study, we model the diffusivity of a particle as a Lévy flight process so that D has a power-law tailed distribution, viz., π_{e}(D)∼D^{-1-α} with 0<α<1. We find that in the short time limit, the width of displacement distribution is proportional to sqrt[T], implying that the diffusion is Fickian. But for long times, the width is proportional to T^{1/2α} which is a characteristic of anomalous diffusion. The distribution function for the displacement of the particle is found to be a symmetric stable distribution with a stability index 2α which preserves its shape at all times.
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Affiliation(s)
- Rohit Jain
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
| | - K L Sebastian
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
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35
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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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Bhattacharya S, Sharma DK, De S, Mahato J, Chowdhury A. Heterogeneity during Plasticization of Poly(vinylpyrrolidone): Insights from Reorientational Mobility of Single Fluorescent Probes. J Phys Chem B 2016; 120:12404-12415. [DOI: 10.1021/acs.jpcb.6b08584] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Sukanya Bhattacharya
- Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
| | | | - Suman De
- Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
| | - Jaladhar Mahato
- Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
| | - Arindam Chowdhury
- Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
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37
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Samanta N, Chakrabarti R. Tracer diffusion in a sea of polymers with binding zones: mobile vs. frozen traps. SOFT MATTER 2016; 12:8554-8563. [PMID: 27714359 DOI: 10.1039/c6sm01943a] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We use molecular dynamics simulations to investigate the tracer diffusion in a sea of polymers with specific binding zones for the tracer. These binding zones act as traps. Our simulations show that the tracer can undergo normal yet non-Gaussian diffusion under certain circumstances, e.g., when the polymers with traps are frozen in space and the volume fraction and the binding strength of the traps are moderate. In this case, as the tracer moves, it experiences a heterogeneous environment and exhibits confined continuous time random walk (CTRW) like motion resulting in a non-Gaussian behavior. Also the long time dynamics becomes subdiffusive as the number or the binding strength of the traps increases. However, if the polymers are mobile then the tracer dynamics is Gaussian but could be normal or subdiffusive depending on the number and the binding strength of the traps. In addition, with increasing binding strength and number of polymer traps, the probability of the tracer being trapped increases. On the other hand, removing the binding zones does not result in trapping, even at comparatively high crowding. Our simulations also show that the trapping probability increases with the increasing size of the tracer and for a bigger tracer with the frozen polymer background the dynamics is only weakly non-Gaussian but highly subdiffusive. Our observations are in the same spirit as found in many recent experiments on tracer diffusion in polymeric materials and question the validity of using Gaussian theory to describe diffusion in a crowded environment in general.
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Affiliation(s)
- Nairhita Samanta
- Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India.
| | - Rajarshi Chakrabarti
- Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India.
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