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Chen Y, Wang X. Different effects of external force fields on aging Lévy walk. CHAOS (WOODBURY, N.Y.) 2023; 33:013102. [PMID: 36725624 DOI: 10.1063/5.0124654] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2022] [Accepted: 12/06/2022] [Indexed: 06/18/2023]
Abstract
Aging phenomena have been observed in numerous physical systems. Many statistical quantities depend on the aging time ta for aging anomalous diffusion processes. This paper pays more attention to how an external force field affects the aging Lévy walk. Based on the Langevin picture of the Lévy walk and the generalized Green-Kubo formula, we investigate the quantities that include the ensemble- and time-averaged mean-squared displacements in both weak aging ta≪t and strong aging ta≫t cases and compare them to the ones in the absence of any force field. Two typical force fields, constant force F and time-dependent periodic force F(t)=f0sin(ωt), are considered for comparison. The generalized Einstein relation is also discussed in the case with the constant force. We find that the constant force is the key to causing the aging phenomena and enhancing the diffusion behavior of the aging Lévy walk, while the time-dependent periodic force is not. The different effects of the two kinds of forces on the aging Lévy walk are verified by both theoretical analyses and numerical simulations.
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Affiliation(s)
- Yao Chen
- College of Sciences, Nanjing Agricultural University, Nanjing 210095, 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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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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3
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Xu P, Metzler R, Wang W. Infinite density and relaxation for Lévy walks in an external potential: Hermite polynomial approach. Phys Rev E 2022; 105:044118. [PMID: 35590616 DOI: 10.1103/physreve.105.044118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2022] [Accepted: 03/22/2022] [Indexed: 06/15/2023]
Abstract
Lévy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding qth-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed v(τ)=v_{0}/sqrt[τ]. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly.
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Affiliation(s)
- Pengbo Xu
- School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
| | - Ralf Metzler
- Institute of Physics & Astronomy, University of Potsdam, 14476 Potsdam, Germany
| | - Wanli Wang
- Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China
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4
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Magdziarz M, Szczotka W. Lévy walks with rests: Long-time analysis. Phys Rev E 2022; 105:014114. [PMID: 35193294 DOI: 10.1103/physreve.105.014114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2021] [Accepted: 12/27/2021] [Indexed: 06/14/2023]
Abstract
In this paper we analyze the asymptotic behavior of Lévy walks with rests. Applying recent results in the field of functional convergence of continuous-time random walks we find the corresponding limiting processes. Depending on the parameters of the model, we show that in the limit we can obtain standard Lévy walk or the process describing competition between subdiffusion and Lévy flights. Some other more complicated limit forms are also possible to obtain. Finally we present some numerical results, which confirm our findings.
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Affiliation(s)
- Marcin Magdziarz
- Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland
| | - Wladyslaw Szczotka
- Institute of Mathematics, University of Wroclaw, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland
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5
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McClure JE, Berg S, Armstrong RT. Thermodynamics of fluctuations based on time-and-space averages. Phys Rev E 2021; 104:035106. [PMID: 34654200 DOI: 10.1103/physreve.104.035106] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2020] [Accepted: 06/23/2021] [Indexed: 11/07/2022]
Abstract
We develop nonequilibrium theory by using averages in time and space as a generalized way to upscale thermodynamics in nonergodic systems. The approach offers a classical perspective on the energy dynamics in fluctuating systems. The rate of entropy production is shown to be explicitly scale dependent when considered in this context. We show that while any stationary process can be represented as having zero entropy production, second law constraints due to the Clausius theorem are preserved due to the fact that heat and work are related based on conservation of energy. As a demonstration, we consider the energy dynamics for the Carnot cycle and for Maxwell's demon. We then consider nonstationary processes, applying time-and-space averages to characterize nonergodic effects in heterogeneous systems where energy barriers such as compositional gradients are present. We show that the derived theory can be used to understand the origins of anomalous diffusion phenomena in systems where Fick's law applies at small length scales, but not at large length scales. We further characterize fluctuations in capillary-dominated systems, which are nonstationary due to the irreversibility of cooperative events.
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Affiliation(s)
- James E McClure
- Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA
| | - Steffen Berg
- Shell Global Solutions International B.V., Grasweg 31, 1031HW Amsterdam, The Netherlands
| | - Ryan T Armstrong
- School of Minerals and Energy Resources Engineering, University of New South Wales, Sydney 2052, Australia
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6
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Chen Y, Deng W. Lévy-walk-like Langevin dynamics affected by a time-dependent force. Phys Rev E 2021; 103:012136. [PMID: 33601647 DOI: 10.1103/physreve.103.012136] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2020] [Accepted: 01/13/2021] [Indexed: 01/17/2023]
Abstract
The Lévy walk is a popular and more 'physical' model to describe the phenomena of superdiffusion, because of its finite velocity. The movements of particles are under the influence of external potentials at almost any time and anywhere. In this paper, we establish a Langevin system coupled with a subordinator to describe the Lévy walk in a time-dependent periodic force field. The effects of external force are detected and carefully analyzed, including the nonzero first moment (even though the force is periodic), adding an additional dispersion on the particle position, a consistent influence on the ensemble- and time-averaged mean-squared displacement, etc. Besides, the generalized Klein-Kramers equation is obtained, not only for the time-dependent force but also for the space-dependent one.
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Affiliation(s)
- Yao Chen
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People's Republic of China
| | - Weihua Deng
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People's Republic of China
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7
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Chen Y, Wang X, Deng W. Langevin picture of Lévy walk in a constant force field. Phys Rev E 2020; 100:062141. [PMID: 31962521 DOI: 10.1103/physreve.100.062141] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2019] [Indexed: 11/06/2022]
Abstract
Lévy walk is a practical model and has wide applications in various fields. Here we focus on the effect of an external constant force on the Lévy walk with the exponent of the power-law-distributed flight time α∈(0,2). We add the term Fη(s) [η(s) is the Lévy noise] on a subordinated Langevin system to characterize such a constant force, as it is effective on the velocity process for all physical time after the subordination. We clearly show the effect of the constant force F on this Langevin system and find this system is like the continuous limit of the collision model. The first moments of velocity processes for these two models are consistent. In particular, based on the velocity correlation function derived from our subordinated Langevin equation, we investigate more interesting statistical quantities, such as the ensemble- and time-averaged mean-squared displacements. Under the influence of constant force, the diffusion of particles becomes faster. Finally, the superballistic diffusion and the nonergodic behavior are verified by the simulations with different α.
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Affiliation(s)
- Yao Chen
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China
| | - Xudong Wang
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China
| | - Weihua Deng
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China
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8
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Abstract
The phenomena of subdiffusion are widely observed in physical and biological systems. To investigate the effects of external potentials, say, harmonic potential, linear potential, and time-dependent force, we study the subdiffusion described by the subordinated Langevin equation with white Gaussian noise or, equivalently, by the single Langevin equation with compound noise. If the force acts on the subordinated process, it keeps working all the time; otherwise, the force just exerts an influence on the system at the moments of jump. Some common statistical quantities, such as the ensemble- and time-averaged mean squared displacements, position autocorrelation function, correlation coefficient, and generalized Einstein relation, are discussed to distinguish the effects of various forces and different patterns of acting. The corresponding Fokker-Planck equations are also presented. All the stochastic processes discussed here are nonstationary, nonergodic, and aging.
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Affiliation(s)
- Yao Chen
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People's Republic of China
| | - Xudong Wang
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People's Republic of China
| | - Weihua Deng
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People's Republic of China
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9
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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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10
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Abstract
A growing body of literature examines the effects of superdiffusive subballistic movement premeasurement (aging or time lag) on observations arising from single-particle tracking. A neglected aspect is the finite lifetime of these Lévy walkers, be they proteins, cells, or larger structures. We examine the effects of aging on the motility of mortal walkers, and discuss the means by which permanent stopping of walkers may be categorized as arising from "natural" death or experimental artifacts such as low photostability or radiation damage. This is done by comparison of the walkers' mean squared displacement (MSD) with the front velocity of propagation of a group of walkers, which is found to be invariant under time lags. For any running time distribution of a mortal random walker, the MSD is tempered by the stopping rate θ. This provides a physical interpretation for truncated heavy-tailed diffusion processes and serves as a tool by which to better classify the underlying running time distributions of random walkers. Tempering of aged MSDs raises the issue of misinterpreting superdiffusive motion which appears Brownian or subdiffusive over certain time scales.
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Affiliation(s)
- Helena Stage
- School of Mathematics, The University of Manchester, Manchester M13 9PL, United Kingdom
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11
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Budini AA. Memory-induced diffusive-superdiffusive transition: Ensemble and time-averaged observables. Phys Rev E 2017; 95:052110. [PMID: 28618554 DOI: 10.1103/physreve.95.052110] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2017] [Indexed: 06/07/2023]
Abstract
The ensemble properties and time-averaged observables of a memory-induced diffusive-superdiffusive transition are studied. The model consists in a random walker whose transitions in a given direction depend on a weighted linear combination of the number of both right and left previous transitions. The diffusion process is nonstationary, and its probability develops the phenomenon of aging. Depending on the characteristic memory parameters, the ensemble behavior may be normal, superdiffusive, or ballistic. In contrast, the time-averaged mean squared displacement is equal to that of a normal undriven random walk, which renders the process nonergodic. In addition, and similarly to Lévy walks [Godec and Metzler, Phys. Rev. Lett. 110, 020603 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.020603], for trajectories of finite duration the time-averaged displacement apparently become random with properties that depend on the measurement time and also on the memory properties. These features are related to the nonstationary power-law decay of the transition probabilities to their stationary values. Time-averaged response to a bias is also calculated. In contrast with Lévy walks [Froemberg and Barkai, Phys. Rev. E 87, 030104(R) (2013)PLEEE81539-375510.1103/PhysRevE.87.030104], the response always vanishes asymptotically.
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Affiliation(s)
- Adrián A Budini
- Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Centro Atómico Bariloche, Avenida E. Bustillo Km 9.5, (8400) Bariloche, Argentina and Universidad Tecnológica Nacional (UTN-FRBA), Fanny Newbery 111, (8400) Bariloche, Argentina
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12
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Abstract
Aging can be observed for numerous physical systems. In such systems statistical properties [like probability distribution, mean square displacement (MSD), first-passage time] depend on a time span t_{a} between the initialization and the beginning of observations. In this paper we study aging properties of ballistic Lévy walks and two closely related jump models: wait-first and jump-first. We calculate explicitly their probability distributions and MSDs. It turns out that despite similarities these models react very differently to the delay t_{a}. Aging weakly affects the shape of probability density function and MSD of standard Lévy walks. For the jump models the shape of the probability density function is changed drastically. Moreover for the wait-first jump model we observe a different behavior of MSD when t_{a}≪t and t_{a}≫t.
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Affiliation(s)
- Marcin Magdziarz
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland
| | - Tomasz Zorawik
- 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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13
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Budini AA. Inhomogeneous diffusion and ergodicity breaking induced by global memory effects. Phys Rev E 2016; 94:052142. [PMID: 27967169 DOI: 10.1103/physreve.94.052142] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2016] [Indexed: 06/06/2023]
Abstract
We introduce a class of discrete random-walk model driven by global memory effects. At any time, the right-left transitions depend on the whole previous history of the walker, being defined by an urnlike memory mechanism. The characteristic function is calculated in an exact way, which allows us to demonstrate that the ensemble of realizations is ballistic. Asymptotically, each realization is equivalent to that of a biased Markovian diffusion process with transition rates that strongly differs from one trajectory to another. Using this "inhomogeneous diffusion" feature, the ergodic properties of the dynamics are analytically studied through the time-averaged moments. Even in the long-time regime, they remain random objects. While their average over realizations recovers the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second time-averaged moment, an ergodic limit and the limit of infinite lag times do not commutate. All these effects are induced by the memory effects. A generalized Einstein fluctuation-dissipation relation is also obtained for the time-averaged moments.
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Affiliation(s)
- Adrián A Budini
- Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Centro Atómico Bariloche, Avenida E. Bustillo Km 9.5, (8400) Bariloche, Argentina and Universidad Tecnológica Nacional (UTN-FRBA), Fanny Newbery 111, (8400) Bariloche, Argentina
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14
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Cherstvy AG, Metzler R. Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes. J Chem Phys 2016; 142:144105. [PMID: 25877560 DOI: 10.1063/1.4917077] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We study noisy heterogeneous diffusion processes with a position dependent diffusivity of the form D(x) ∼ D0|x|(α0) in the presence of annealed and quenched disorder of the environment, corresponding to an effective variation of the exponent α in time and space. In the case of annealed disorder, for which effectively α0 = α0(t), we show how the long time scaling of the ensemble mean squared displacement (MSD) and the amplitude variation of individual realizations of the time averaged MSD are affected by the disorder strength. For the case of quenched disorder, the long time behavior becomes effectively Brownian after a number of jumps between the domains of a stratified medium. In the latter situation, the averages are taken over both an ensemble of particles and different realizations of the disorder. As physical observables, we analyze in detail the ensemble and time averaged MSDs, the ergodicity breaking parameter, and higher order moments of the time averages.
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Affiliation(s)
- 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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15
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Kärger J, Ruthven DM. Diffusion in nanoporous materials: fundamental principles, insights and challenges. NEW J CHEM 2016. [DOI: 10.1039/c5nj02836a] [Citation(s) in RCA: 124] [Impact Index Per Article: 15.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022]
Abstract
The increasing complexity of nanoporous catalysts and adsorbents presents a challenge to both the experimental measurement and theoretical modeling of transport behavior.
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Affiliation(s)
- Jörg Kärger
- Faculty of Physics and Earth Sciences
- University of Leipzig
- 04103 Leipzig
- Germany
| | - Douglas M. Ruthven
- Department of Chemical and Biological Engineering
- University of Maine
- Orono
- USA
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16
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Cho S, Choi YJ, Zheng S, Han J, Ko SY, Park JO, Park S. Modeling of chemotactic steering of bacteria-based microrobot using a population-scale approach. BIOMICROFLUIDICS 2015; 9:054116. [PMID: 26487902 PMCID: PMC4592439 DOI: 10.1063/1.4932304] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2015] [Accepted: 09/22/2015] [Indexed: 05/04/2023]
Abstract
The bacteria-based microrobot (Bacteriobot) is one of the most effective vehicles for drug delivery systems. The bacteriobot consists of a microbead containing therapeutic drugs and bacteria as a sensor and an actuator that can target and guide the bacteriobot to its destination. Many researchers are developing bacteria-based microrobots and establishing the model. In spite of these efforts, a motility model for bacteriobots steered by chemotaxis remains elusive. Because bacterial movement is random and should be described using a stochastic model, bacterial response to the chemo-attractant is difficult to anticipate. In this research, we used a population-scale approach to overcome the main obstacle to the stochastic motion of single bacterium. Also known as Keller-Segel's equation in chemotaxis research, the population-scale approach is not new. It is a well-designed model derived from transport theory and adaptable to any chemotaxis experiment. In addition, we have considered the self-propelled Brownian motion of the bacteriobot in order to represent its stochastic properties. From this perspective, we have proposed a new numerical modelling method combining chemotaxis and Brownian motion to create a bacteriobot model steered by chemotaxis. To obtain modeling parameters, we executed motility analyses of microbeads and bacteriobots without chemotactic steering as well as chemotactic steering analysis of the bacteriobots. The resulting proposed model shows sound agreement with experimental data with a confidence level <0.01.
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Affiliation(s)
- Sunghoon Cho
- School of Mechanical Engineering, Chonnam National University , 300, Yongbong-dong, Buk-gu, Gwangju, South Korea
| | - Young Jin Choi
- School of Mechanical Engineering, Chonnam National University , 300, Yongbong-dong, Buk-gu, Gwangju, South Korea
| | - Shaohui Zheng
- School of Mechanical Engineering, Chonnam National University , 300, Yongbong-dong, Buk-gu, Gwangju, South Korea
| | - Jiwon Han
- School of Mechanical Engineering, Chonnam National University , 300, Yongbong-dong, Buk-gu, Gwangju, South Korea
| | - Seong Young Ko
- School of Mechanical Engineering, Chonnam National University , 300, Yongbong-dong, Buk-gu, Gwangju, South Korea
| | - Jong-Oh Park
- School of Mechanical Engineering, Chonnam National University , 300, Yongbong-dong, Buk-gu, Gwangju, South Korea
| | - Sukho Park
- School of Mechanical Engineering, Chonnam National University , 300, Yongbong-dong, Buk-gu, Gwangju, South Korea
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17
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Cherstvy AG, Metzler R. Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012134. [PMID: 25122278 DOI: 10.1103/physreve.90.012134] [Citation(s) in RCA: 56] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2014] [Indexed: 06/03/2023]
Abstract
We study the stochastic behavior of heterogeneous diffusion processes with the power-law dependence D(x) ∼ |x|(α) of the generalized diffusion coefficient encompassing sub- and superdiffusive anomalous diffusion. Based on statistical measures such as the amplitude scatter of the time-averaged mean-squared displacement of individual realizations, the ergodicity breaking and non-Gaussianity parameters, as well as the probability density function P(x,t), we analyze the weakly nonergodic character of the heterogeneous diffusion process and, particularly, the degree of irreproducibility of individual realizations. As we show, the fluctuations between individual realizations increase with growing modulus |α| of the scaling exponent. The fluctuations appear to diverge when the critical value α = 2 is approached, while for even larger α the fluctuations decrease, again. At criticality, the power-law behavior of the mean-squared displacement changes to an exponentially fast growth, and the fluctuations of the time-averaged mean-squared displacement do not converge for increasing number of realizations. From a systematic comparison we observe some striking similarities of the heterogeneous diffusion process with the familiar subdiffusive continuous time random walk process with power-law waiting time distribution and diverging characteristic waiting time.
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Affiliation(s)
- A G Cherstvy
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - R Metzler
- Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany and Department of Physics, Tampere University of Technology, 33101 Tampere, Finland
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