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Chen Y, Wang X, Wang W. Langevin picture of subdiffusion in nonuniformly expanding medium. CHAOS (WOODBURY, N.Y.) 2023; 33:113133. [PMID: 38029759 DOI: 10.1063/5.0166613] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2023] [Accepted: 10/30/2023] [Indexed: 12/01/2023]
Abstract
Anomalous diffusion phenomena have been observed in many complex physical and biological systems. One significant advance recently is the physical extension of particle's motion in a static medium to a uniformly and even nonuniformly expanding medium. The dynamic mechanism of the anomalous diffusion in the nonuniformly expanding medium has only been investigated by the approach of continuous-time random walk. To study more physical observables and to supplement the physical models of the anomalous diffusion in the expanding mediums, we characterize the nonuniformly expanding medium with a spatiotemporal dependent scale factor a(x,t) and build the Langevin picture describing the particle's motion in the nonuniformly expanding medium. Besides the existing comoving and physical coordinates, by introducing a new coordinate and assuming that a(x,t) is separable at a long-time limit, we build the relation between the nonuniformly expanding medium and the uniformly expanding one and further obtain the moments of the comoving and physical coordinates. Different forms of the scale factor a(x,t) are considered to uncover the combined effects of the particle's intrinsic diffusion and the nonuniform expansion of medium. The theoretical analyses and simulations provide the foundation for studying more anomalous diffusion phenomena in the expanding mediums.
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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
| | - Wanli Wang
- Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, People's Republic of China
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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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3
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Wang X, Chen Y. Langevin picture of anomalous diffusion processes in expanding medium. Phys Rev E 2023; 107:024105. [PMID: 36932587 DOI: 10.1103/physreve.107.024105] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2022] [Accepted: 01/11/2023] [Indexed: 06/18/2023]
Abstract
The expanding medium is very common in many different fields, such as biology and cosmology. It brings a nonnegligible influence on particle's diffusion, which is quite different from the effect of an external force field. The dynamic mechanism of a particle's motion in an expanding medium has only been investigated in the framework of a continuous-time random walk. To focus on more diffusion processes and physical observables, we build the Langevin picture of anomalous diffusion in an expanding medium, and conduct detailed analyses in the framework of the Langevin equation. With the help of a subordinator, both subdiffusion process and superdiffusion process in the expanding medium are discussed. We find that the expanding medium with different changing rate (exponential form and power-law form) leads to quite different diffusion phenomena. The particle's intrinsic diffusion behavior also plays an important role. Our detailed theoretical analyses and simulations present a panoramic view of investigating anomalous diffusion in an expanding medium under the framework of the Langevin equation.
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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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Zhou T, Xu P, Deng W. Lévy Walk Dynamics in an External Constant Force Field in Non-Static Media. JOURNAL OF STATISTICAL PHYSICS 2022; 187:9. [PMID: 35250092 PMCID: PMC8883250 DOI: 10.1007/s10955-022-02904-8] [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: 11/17/2021] [Accepted: 02/16/2022] [Indexed: 06/14/2023]
Abstract
Based on the recognition of the huge change of the transport properties for diffusion particles in non-static media, we consider a Lévy walk model subjected to an external constant force in non-static media. Since the physical and comoving coordinates of non-static media are related by scale factor, we equivalently transfer the process from physical coordinate into comoving coordinate and derive the master equation governing the probability density function of the position of the particles in comoving coordinate. Utilizing the Hermite orthogonal polynomial expansions, some statistical properties are obtained, including the asymptotic behaviors of the first two moments in both coordinates and kurtosis. For some representative types of non-static media and Lévy walks, the striking and interesting phenomena originating from the interplay between non-static media, external force, and intrinsic stochastic motion are observed. The stationary distribution are also analyzed for some cases through numerical simulations.
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Affiliation(s)
- Tian Zhou
- Gansu Key Laboratory of Applied Mathematics and Complex Systems, School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000 People’s Republic of China
| | - Pengbo Xu
- School of Mathematical Sciences, Peking University, Beijing, 100871 People’s Republic of China
| | - Weihua Deng
- Gansu Key Laboratory of Applied Mathematics and Complex Systems, School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000 People’s Republic of China
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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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Abad E, Angstmann CN, Henry BI, McGann AV, Le Vot F, Yuste SB. Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains. Phys Rev E 2020; 102:032111. [PMID: 33075977 DOI: 10.1103/physreve.102.032111] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/14/2020] [Accepted: 08/19/2020] [Indexed: 01/22/2023]
Abstract
Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. Inhomogeneously growing domains are frequently encountered in biological phenomena involving stochastic transport, such as tumor growth and morphogen gradient formation. A method for constructing analytic expressions for short-time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favorably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short-time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface. From a more general perspective, our findings help to mitigate the scarcity of analytic results for reaction-diffusion problems in geometries displaying nonuniform growth. They are also expected to pave the way for further results, including the treatment of first-passage problems associated with encounter-controlled reactions in such domains.
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Affiliation(s)
- E Abad
- Departamento de Física Aplicada and Instituto de Computación Científica Avanzada, Centro Universitario de Mérida, Universidad de Extremadura, 06800 Mérida, Spain
| | - C N Angstmann
- School of Mathematics and Statistics, UNSW, Sydney New South Wales, 2052, Australia
| | | | | | - F Le Vot
- Departamento de Física and Instituto de Computación Científica Avanzada, Universidad de Extremadura, 06071 Badajoz, Spain
| | - S B Yuste
- Departamento de Física and Instituto de Computación Científica Avanzada, Universidad de Extremadura, 06071 Badajoz, Spain
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Le Vot F, Yuste SB, Abad E. Standard and fractional Ornstein-Uhlenbeck process on a growing domain. Phys Rev E 2019; 100:012142. [PMID: 31499768 DOI: 10.1103/physreve.100.012142] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/30/2019] [Indexed: 11/07/2022]
Abstract
We study normal diffusive and subdiffusive processes in a harmonic potential (Ornstein-Uhlenbeck process) on a uniformly growing or contracting domain. Our starting point is a recently derived fractional Fokker-Planck equation, which covers both the case of Brownian diffusion and the case of a subdiffusive continuous-time random walk (CTRW). We find a high sensitivity of the random walk properties to the details of the domain growth rate, which gives rise to a variety of regimes with extremely different behaviors. At the origin of this rich phenomenology is the fact that the walkers still move while they wait to jump, since they are dragged by the deterministic drift arising from the domain growth. Thus, the increasingly long waiting times associated with the aging of the subdiffusive CTRW imply that, in the time interval between two consecutive jumps, the walkers might travel over much longer distances than in the normal diffusive case. This gives rise to seemingly counterintuitive effects. For example, on a static domain, both Brownian diffusion and subdiffusive CTRWs yield a stationary particle distribution with finite width when a harmonic potential is at play, thus indicating a confinement of the diffusing particle. However, for a sufficiently fast growing or contracting domain, this qualitative behavior breaks down, and differences between the Brownian case and the subdiffusive case are found. In the case of Brownian particles, a sufficiently fast exponential domain growth is needed to break the confinement induced by the harmonic force; in contrast, for subdiffusive particles such a breakdown may already take place for a sufficiently fast power-law domain growth. Our analytic and numerical results for both types of diffusion are fully confirmed by random walk simulations.
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Affiliation(s)
- F Le Vot
- Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx) Universidad de Extremadura, E-06071 Badajoz, Spain
| | - S B Yuste
- Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx) Universidad de Extremadura, E-06071 Badajoz, Spain
| | - E Abad
- Departamento de Física Aplicada and Instituto de Computación Científica Avanzada (ICCAEx) Centro Universitario de Mérida Universidad de Extremadura, E-06800 Mérida, Spain
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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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Angstmann CN, Henry BI, McGann AV. Generalized fractional diffusion equations for subdiffusion in arbitrarily growing domains. Phys Rev E 2018; 96:042153. [PMID: 29347596 DOI: 10.1103/physreve.96.042153] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2017] [Indexed: 11/07/2022]
Abstract
The ubiquity of subdiffusive transport in physical and biological systems has led to intensive efforts to provide robust theoretical models for this phenomena. These models often involve fractional derivatives. The important physical extension of this work to processes occurring in growing materials has proven highly nontrivial. Here we derive evolution equations for modeling subdiffusive transport in a growing medium. The derivation is based on a continuous-time random walk. The concise formulation of these evolution equations requires the introduction of a new, comoving, fractional derivative. The implementation of the evolution equation is illustrated with a simple model of subdiffusing proteins in a growing membrane.
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Affiliation(s)
- C N Angstmann
- School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia
| | - B I Henry
- School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia
| | - A V McGann
- School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia
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Le Vot F, Abad E, Yuste SB. Continuous-time random-walk model for anomalous diffusion in expanding media. Phys Rev E 2017; 96:032117. [PMID: 29347028 DOI: 10.1103/physreve.96.032117] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2017] [Indexed: 06/07/2023]
Abstract
Expanding media are typical in many different fields, e.g., in biology and cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties such as the set of positional moments and the Green's function. Here, we focus on the characterization of such effects when the diffusion process is described by the continuous-time random-walk (CTRW) model. As is well known, when the medium is static this model yields anomalous diffusion for a proper choice of the probability density function (pdf) for the jump length and the waiting time, but the behavior may change drastically if a medium expansion is superimposed on the intrinsic random motion of the diffusing particle. For the case where the jump length and the waiting time pdfs are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Lévy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Green's function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. In the specific case of a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This "big crunch" effect, totally absent in the case of normal diffusion, stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model in an expanding medium. From this hierarchy, the full time evolution of the second-order moment is obtained for some specific types of expansion. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained, whence the long-time behavior of moments of arbitrary order is subsequently inferred. Our analytical and numerical results for both Lévy flights and subdiffusive CTRWs confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Lévy flights, we quantify this effect by means of the so-called "Lévy horizon."
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Affiliation(s)
- F Le Vot
- Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEX), Universidad de Extremadura, E-06071 Badajoz, Spain
| | - E Abad
- Departamento de Física Aplicada and Instituto de Computación Científica Avanzada (ICCAEX), Centro Universitario de Mérida and Universidad de Extremadura, E-06800 Mérida, Spain
| | - S B Yuste
- Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEX), Universidad de Extremadura, E-06071 Badajoz, Spain
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