1
|
Gerry M, Segal D. Random walks on modular chains: Detecting structure through statistics. Phys Rev E 2023; 108:024135. [PMID: 37723810 DOI: 10.1103/physreve.108.024135] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2023] [Accepted: 08/04/2023] [Indexed: 09/20/2023]
Abstract
We study kinetic transport through one-dimensional modular networks consisting of alternating domains using both analytical and numerical methods. We demonstrate that the mean velocity is insensitive to the local structure of the network, and it depends only on global, structural-averaged properties. However, by examining high-order cumulants characterizing the kinetics, we reveal information on the degree of inhomogeneity of blocks and the size of repeating units in the network. Specifically, in unbiased diffusion, the kurtosis is the first transport coefficient that exposes structural information, whereas in biased chains, the diffusion coefficient already reveals structural motifs. Nevertheless, this latter dependence is weak, and it disappears at both low and high biasing. Our study demonstrates that high-order moments of the population distribution over sites provide information about the network structure that is not captured by the first moment (mean velocity) alone. These results are useful towards deciphering mechanisms and determining architectures underlying long-range charge transport in biomolecules and biological and chemical reaction networks.
Collapse
Affiliation(s)
- Matthew Gerry
- Department of Physics, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada
| | - Dvira Segal
- Department of Physics, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada
- Chemical Physics Theory Group, Department of Chemistry and Centre for Quantum Information and Quantum Control, University of Toronto, 80 Saint George St., Toronto, Ontario M5S 3H6, Canada
| |
Collapse
|
2
|
Meerson B. Negative autocorrelations of disorder strongly suppress thermally activated particle motion in short-correlated quenched Gaussian disorder potentials. Phys Rev E 2022; 105:034106. [PMID: 35428118 DOI: 10.1103/physreve.105.034106] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2021] [Accepted: 02/17/2022] [Indexed: 04/20/2023]
Abstract
We evaluate the mean escape time of overdamped particles over potential barriers in short-correlated quenched Gaussian disorder potentials in one dimension at low temperature. The thermally activated escape is very sensitive to the form of the tail of the potential barrier probability distribution. We evaluate this tail by using the optimal fluctuation method. For monotone decreasing autocovariances, we reproduce the tail obtained by Lopatin and Vinokur (2001). However, for nonmonotonic autocovariances of the disorder potential which exhibit negative autocorrelations, we show that the tail is higher. This leads to an exponential increase of the mean escape time. The transition between the two regimes has the character of a first-order transition.
Collapse
Affiliation(s)
- Baruch Meerson
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| |
Collapse
|
3
|
Goychuk I, Pöschel T. Fingerprints of viscoelastic subdiffusion in random environments: Revisiting some experimental data and their interpretations. Phys Rev E 2021; 104:034125. [PMID: 34654105 DOI: 10.1103/physreve.104.034125] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2021] [Accepted: 09/01/2021] [Indexed: 11/07/2022]
Abstract
Many experimental studies revealed subdiffusion of various nanoparticles in diverse polymer and colloidal solutions, cytosol and plasma membrane of biological cells, which are viscoelastic and, at the same time, highly inhomogeneous randomly fluctuating environments. The observed subdiffusion often combines features of ergodic fractional Brownian motion (reflecting viscoelasticity) and nonergodic jumplike non-Markovian diffusional processes (reflecting disorder). Accordingly, several theories were proposed to explain puzzling experimental findings. Below we show that some of the significant and profound published experimental results are better rationalized within the viscoelastic subdiffusion approach in random environments, which is based on generalized Langevin dynamics in random potentials, than some earlier proposed theories.
Collapse
Affiliation(s)
- Igor Goychuk
- Institute for Multiscale Simulation, Department of Chemical and Biological Engineering, Friedrich-Alexander University of Erlangen-Nürnberg, Cauerstr. 3, 91058 Erlangen, Germany
| | - Thorsten Pöschel
- Institute for Multiscale Simulation, Department of Chemical and Biological Engineering, Friedrich-Alexander University of Erlangen-Nürnberg, Cauerstr. 3, 91058 Erlangen, Germany
| |
Collapse
|
4
|
Kosztołowicz T. Boundary conditions at a thin membrane that generate non-Markovian normal diffusion. Phys Rev E 2020; 102:022123. [PMID: 32942412 DOI: 10.1103/physreve.102.022123] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/08/2019] [Accepted: 07/22/2020] [Indexed: 06/11/2023]
Abstract
We show that some boundary conditions assumed at a thin membrane may result in normal diffusion not being the stochastic Markov process. We consider boundary conditions defined in terms of the Laplace transform in which there is a linear combination of probabilities and probability fluxes defined on both membrane surfaces. The coefficients of the combination may depend on the Laplace transform parameter. Such boundary conditions are most commonly used when considering diffusion in a membrane system unless collective or nonlocal processes in particles diffusion occur. We find Bachelier-Smoluchowski-Chapmann-Kolmogorov (BSCK) equation in terms of the Laplace transform and we derive the criterion to check whether the boundary conditions lead to fundamental solutions of diffusion equation satisfying this equation. If the BSCK equation is not met, then the Markov property is broken. When a probability flux is continuous at the membrane, the general forms of the boundary conditions for which the fundamental solutions meet the BSCK equation are derived. A measure of broken of semi-group property is also proposed. The relation of this measure to the non-Markovian property measure is discussed.
Collapse
Affiliation(s)
- Tadeusz Kosztołowicz
- Institute of Physics, Jan Kochanowski University, Uniwersytecka 7, 25-406 Kielce, Poland
| |
Collapse
|
5
|
Bao JD. Generalization of the Kubo relation for confined motion and ergodicity breakdown. Phys Rev E 2020; 101:062131. [PMID: 32688488 DOI: 10.1103/physreve.101.062131] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2019] [Accepted: 05/07/2020] [Indexed: 06/11/2023]
Abstract
A time-dependent generalized Kubo relation is derived by introducing the notion of a diffusion function for a particle confined in a harmonic potential. The relation reduces to the standard Kubo relation as a special case but holds for anomalous diffusion, nonergodic processes, and bounded motion. We analyze in detail the behaviors of the diffusion and memory functions and report a generalized Stokes-Einstein relation concerning anomalous diffusion. Furthermore, we demonstrate that when a high finite-frequency cutoff is imposed on the noise spectral density, a breakdown in ergodicity accompanied by the appearance of nonstationarity in the velocity autocorrelation function occurs in forced systems. This breakdown is taken as explicit evidence for either decay-spring-memory or recovering-force effects leading to nonexponential relaxation kinematics.
Collapse
Affiliation(s)
- Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China
| |
Collapse
|
6
|
Goychuk I. Fractional Hydrodynamic Memory and Superdiffusion in Tilted Washboard Potentials. PHYSICAL REVIEW LETTERS 2019; 123:180603. [PMID: 31763886 DOI: 10.1103/physrevlett.123.180603] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2019] [Indexed: 06/10/2023]
Abstract
Diffusion in tilted washboard potentials can paradoxically exceed free normal diffusion. The effect becomes much stronger in the underdamped case due to inertial effects. What happens upon inclusion of usually neglected fractional hydrodynamics memory effects (Basset-Boussinesq frictional force), which result in a heavy algebraic tail of the velocity autocorrelation function of the potential-free diffusion making it transiently superdiffusive? Will a giant enhancement of diffusion become even stronger, and the transient superdiffusion last even longer? These are the questions that we answer in this Letter based on an accurate numerical investigation. We show that a resonancelike enhancement of normal diffusion becomes indeed much stronger and sharper. Moreover, a long-lasting transient regime of superdiffusion, including Richardson-like diffusion, ⟨δx^{2}(t)⟩∝t^{3} and ballistic supertransport, ⟨δx(t)⟩∝t^{2}, is revealed.
Collapse
Affiliation(s)
- Igor Goychuk
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
| |
Collapse
|
7
|
Tadavani SK, Yethiraj A. Anomalous dynamics in tracer-particle motions in an electrohydrodynamically driven oil-in-oil system. Phys Rev E 2018; 98:022602. [PMID: 30253573 DOI: 10.1103/physreve.98.022602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2018] [Indexed: 11/07/2022]
Abstract
We characterize the superdiffusive dynamics of tracer particles in an electrohydrodynamically driven emulsion of oil droplets in an immiscible oil medium, where the amplitude and frequency of an external electric field are the control parameters. In the weakly driven electrohydrodynamic regime, the droplets are trapped dielectrophoretically on a patterned electrode, and the driving is therefore spatially varying. We find excellent agreement with a 〈x^{2}〉∼t^{1.5} power law and find that this superdiffusive dynamics arises from an underlying displacement distribution that is distinctly non-Gaussian and exponential for small displacements and short times. While these results are comparable with a random-velocity field model, the tracer particle speeds are in fact spatially varying in two dimensions, arising from a spatially varying electrohydrodynamic driving force. This suggests that the important ingredient for the superdiffusive t^{1.5} behavior observed is a velocity field that is isotropic in the plane and spatially correlated. Finally, we can extract, from the superdiffusive dynamics, a experimental length scale that corresponds to the lateral range of the hydrodynamic flows. This experimental length scale is non zero only above a threshold ion mobility length.
Collapse
Affiliation(s)
- Somayeh Khajehpour Tadavani
- Physics and Physical Oceanography, Memorial University of Newfoundland, St. John's, Newfoundland, A1B 3X7 Canada
| | - Anand Yethiraj
- Physics and Physical Oceanography, Memorial University of Newfoundland, St. John's, Newfoundland, A1B 3X7 Canada
| |
Collapse
|
8
|
Goychuk I. Viscoelastic subdiffusion in a random Gaussian environment. Phys Chem Chem Phys 2018; 20:24140-24155. [PMID: 30206605 DOI: 10.1039/c8cp05238g] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
Viscoelastic subdiffusion governed by a fractional Langevin equation is studied numerically in a random Gaussian environment modeled by stationary Gaussian potentials with decaying spatial correlations. This anomalous diffusion is archetypal for living cells, where cytoplasm is known to be viscoelastic and a spatial disorder also naturally emerges. We obtain some first important insights into it within a model one-dimensional study. Two basic types of potential correlations are studied: short-range exponentially decaying and algebraically slow decaying with an infinite correlation length, both for a moderate (several kBT, in the units of thermal energy), and strong (5-10kBT) disorder. For a moderate disorder, it is shown that on the ensemble level viscoelastic subdiffusion can easily overcome the medium's disorder. Asymptotically, it is not distinguishable from the disorder-free subdiffusion. However, a strong scatter in single-trajectory averages is nevertheless seen even for a moderate disorder. It features a weak ergodicity breaking, which occurs on a very long yet transient time scale. Furthermore, for a strong disorder, a very long transient regime of logarithmic, Sinai-type diffusion emerges. It can last longer and be faster in the absolute terms for weakly decaying correlations as compared with the short-range correlations. Residence time distributions in a finite spatial domain are of a generalized log-normal type and are reminiscent also of a stretched exponential distribution. They can be easily confused for power-law distributions in view of the observed weak ergodicity breaking. This suggests a revision of some experimental data and their interpretation.
Collapse
Affiliation(s)
- Igor Goychuk
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany.
| |
Collapse
|
9
|
Liao JJ, Zhu WJ, Ai BQ. Transport and diffusion of paramagnetic ellipsoidal particles in a rotating magnetic field. Phys Rev E 2018; 97:062151. [PMID: 30011563 DOI: 10.1103/physreve.97.062151] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2018] [Indexed: 06/08/2023]
Abstract
Transport and diffusion of paramagnetic ellipsoidal particles under the action of a rotating magnetic field are numerically investigated in a two-dimensional channel. It is found that paramagnetic ellipsoidal particles in a rotating magnetic field can be rectified in the upper-lower asymmetric channel. The transport and the effective diffusion coefficient are much more different and complicated for active particles, while they have similar behaviors and change a little when applying rotating magnetic fields of different frequencies for passive particles. For active particles, the back-and-forth rotational motion facilitates the effective diffusion coefficient and reduces the rectification, whereas the rotational motion synchronous with the magnetic field suppresses the effective diffusion coefficient and enhances the rectification. There exist optimized values of the parameters (the anisotropic degree, the amplitude and frequency of magnetic field, the self-propelled velocity, and the rotational diffusion rate) at which the average velocity and diffusion take their maximal values. Particles with different shapes, self-propelled speeds, or rotational diffusion rates will move to the opposite directions and can be separated by applying rotating magnetic fields of suitable strength and frequency. Our results can be used to separate particles, orient the particles along any direction at will during motion, and control the particle diffusion.
Collapse
Affiliation(s)
- Jing-Jing Liao
- Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
- College of Applied Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
| | - Wei-Jing Zhu
- Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
| | - Bao-Quan Ai
- Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
| |
Collapse
|
10
|
Hu M, Bao JD. Diffusion crossing over a barrier in a random rough metastable potential. Phys Rev E 2018; 97:062143. [PMID: 30011451 DOI: 10.1103/physreve.97.062143] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2018] [Indexed: 06/08/2023]
Abstract
We carry out a detailed study of escape dynamics of a particle driven by a white noise over a metastable potential corrugated by spatial disorder in the form of zero-mean random correlated potential. The approach of double-averaging over test particles and statistic ensemble is proposed to calculate the escape rate in a finite-size random rough metastable potential, moreover, the interference mechanism of test particles multi-passing over the saddle point is considered. Through analyzing the dependence of the steady escape rate on various modelled potentials and parameters, we demonstrate that the obstruction induced by roughness leads to a decrease in the steady escape rate with the increase of rough intensity. We also add the random correlated potential into the vicinity of the saddle-point of metastable potentials of three kinds and show an enhancement phenomenon of escape rate similar to the previous study of a surmounting fluctuating sharp barrier.
Collapse
Affiliation(s)
- Meng Hu
- Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China
| | - Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China
| |
Collapse
|
11
|
Abstract
This work focuses on quantitative representation of transport in systems with quenched disorder. Explicit mapping of the quenched trap model to continuous time random walk is presented. Linear temporal transformation, t→t/Λ^{1/α}, for a transient process in the subdiffusive regime is sufficient for asymptotic mapping. An exact form of the constant Λ^{1/α} is established. A disorder averaged position probability density function for a quenched trap model is obtained, and analytic expressions for the diffusion coefficient and drift are provided.
Collapse
Affiliation(s)
- Stanislav Burov
- Physics Department, Bar-Ilan University, Ramat Gan 5290002, Israel
| |
Collapse
|
12
|
Goychuk I, Kharchenko VO, Metzler R. Persistent Sinai-type diffusion in Gaussian random potentials with decaying spatial correlations. Phys Rev E 2017; 96:052134. [PMID: 29347675 DOI: 10.1103/physreve.96.052134] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2017] [Indexed: 06/07/2023]
Abstract
Logarithmic or Sinai-type subdiffusion is usually associated with random force disorder and nonstationary potential fluctuations whose root-mean-squared amplitude grows with distance. We show here that extremely persistent, macroscopic logarithmic diffusion also universally emerges at sufficiently low temperatures in stationary Gaussian random potentials with spatially decaying correlations, known to exist in a broad range of physical systems. Combining results from extensive simulations with a scaling approach we elucidate the physical mechanism of this unusual subdiffusion. In particular, we explain why with growing temperature and/or time a first crossover occurs to standard, power-law subdiffusion, with a time-dependent power-law exponent, and then a second crossover occurs to normal diffusion with a disorder-renormalized diffusion coefficient. Interestingly, the initial, nominally ultraslow diffusion turns out to be much faster than the universal de Gennes-Bässler-Zwanzig limit of the renormalized normal diffusion, which realistically cannot be attained at sufficiently low temperatures and/or for strong disorder. The ultraslow diffusion is also shown to be nonergodic and it displays a local bias phenomenon. Our simple scaling theory not only explains our numerical findings but qualitatively also has a predictive character.
Collapse
Affiliation(s)
- Igor Goychuk
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Vasyl O Kharchenko
- Institute of Applied Physics, National Academy of Sciences of Ukraine, 40030 Sumy, Ukraine
| | - Ralf Metzler
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| |
Collapse
|
13
|
Figliozzi P, Sule N, Yan Z, Bao Y, Burov S, Gray SK, Rice SA, Vaikuntanathan S, Scherer NF. Driven optical matter: Dynamics of electrodynamically coupled nanoparticles in an optical ring vortex. Phys Rev E 2017; 95:022604. [PMID: 28298004 DOI: 10.1103/physreve.95.022604] [Citation(s) in RCA: 39] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2016] [Indexed: 11/07/2022]
Abstract
To date investigations of the dynamics of driven colloidal systems have focused on hydrodynamic interactions and often employ optical (laser) tweezers for manipulation. However, the optical fields that provide confinement and drive also result in electrodynamic interactions that are generally neglected. We address this issue with a detailed study of interparticle dynamics in an optical ring vortex trap using 150-nm diameter Ag nanoparticles. We term the resultant electrodynamically interacting nanoparticles a driven optical matter system. We also show that a superior trap is created by using a Au nanoplate mirror in a retroreflection geometry, which increases the electric field intensity, the optical drive force, and spatial confinement. Using nanoparticles versus micron sized colloids significantly reduces the surface hydrodynamic friction allowing us to access small values of optical topological charge and drive force. We quantify a further 50% reduction of hydrodynamic friction when the nanoparticles are driven over the Au nanoplate mirrors versus over a mildly electrostatically repulsive glass surface. Further, we demonstrate through experiments and electrodynamics-Langevin dynamics simulations that the optical drive force and the interparticle interactions are not constant around the ring for linearly polarized light, resulting in a strong position-dependent variation in the nanoparticle velocity. The nonuniformity in the optical drive force is also manifest as an increase in fluctuations of interparticle separation, or effective temperature, as the optical driving force is increased. Finally, we resolve an open issue in the literature on periodic modulation of interparticle separation with comparative measurements of driven 300-nm-diameter polystyrene beads that also clearly reveal the significance of electrodynamic forces and interactions in optically driven colloidal systems. Therefore, the modulations in the optical forces and electrodynamic interactions that we demonstrate should not be neglected for dielectric particles and might give rise to some structural and dynamic features that have previously been attributed exclusively to hydrodynamic interactions.
Collapse
Affiliation(s)
- Patrick Figliozzi
- Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, USA
| | - Nishant Sule
- James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
| | - Zijie Yan
- James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
| | - Ying Bao
- James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
| | - Stanislav Burov
- James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
| | - Stephen K Gray
- Center for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, USA
| | - Stuart A Rice
- Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, USA.,James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
| | - Suriyanarayanan Vaikuntanathan
- Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, USA.,James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
| | - Norbert F Scherer
- Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, USA.,James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
| |
Collapse
|
14
|
Goychuk I. Quantum ergodicity breaking in semi-classical electron transfer dynamics. Phys Chem Chem Phys 2017; 19:3056-3066. [DOI: 10.1039/c6cp07206b] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Strange statistics of single-electron transfer events in a classically adiabatic electron transfer with strictly exponential ensemble kinetics.
Collapse
Affiliation(s)
- Igor Goychuk
- Institute for Physics and Astronomy
- University of Potsdam
- 14476 Potsdam-Golm
- Germany
| |
Collapse
|
15
|
Spiechowicz J, Łuczka J, Hänggi P. Transient anomalous diffusion in periodic systems: ergodicity, symmetry breaking and velocity relaxation. Sci Rep 2016; 6:30948. [PMID: 27492219 PMCID: PMC4974640 DOI: 10.1038/srep30948] [Citation(s) in RCA: 52] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/16/2016] [Accepted: 07/11/2016] [Indexed: 11/18/2022] Open
Abstract
We study far from equilibrium transport of a periodically driven inertial Brownian particle moving in a periodic potential. As detected for a SQUID ratchet dynamics, the mean square deviation of the particle position from its average may involve three distinct intermediate, although extended diffusive regimes: initially as superdiffusion, followed by subdiffusion and finally, normal diffusion in the asymptotic long time limit. Even though these anomalies are transient effects, their lifetime can be many, many orders of magnitude longer than the characteristic time scale of the setup and turns out to be extraordinarily sensitive to the system parameters like temperature or the potential asymmetry. In the paper we reveal mechanisms of diffusion anomalies related to ergodicity of the system, symmetry breaking of the periodic potential and ultraslow relaxation of the particle velocity towards its steady state. Similar sequences of the diffusive behaviours could be detected in various systems including, among others, colloidal particles in random potentials, glass forming liquids and granular gases.
Collapse
Affiliation(s)
- Jakub Spiechowicz
- Institute of Physics, University of Silesia, 40-007 Katowice, Poland
- Silesian Center for Education and Interdisciplinary Research, University of Silesia, 41-500 Chorzów, Poland
| | - Jerzy Łuczka
- Institute of Physics, University of Silesia, 40-007 Katowice, Poland
- Silesian Center for Education and Interdisciplinary Research, University of Silesia, 41-500 Chorzów, Poland
| | - Peter Hänggi
- Institute of Physics, University of Augsburg, 86135 Augsburg, Germany
- Nanosystems Initiative Munich, Schellingstr, 4, D-80799 Műnchen, Germany
| |
Collapse
|
16
|
Budini AA. Weak ergodicity breaking induced by global memory effects. Phys Rev E 2016; 94:022108. [PMID: 27627247 DOI: 10.1103/physreve.94.022108] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2016] [Indexed: 06/06/2023]
Abstract
We study the phenomenon of weak ergodicity breaking for a class of globally correlated random walk dynamics defined over a finite set of states. The persistence in a given state or the transition to another one depends on the whole previous temporal history of the system. A set of waiting time distributions, associated to each state, sets the random times between consecutive steps. Their mean value is finite for all states. The probability density of time-averaged observables is obtained for different memory mechanisms. This statistical object explicitly shows departures between time and ensemble averages. While the residence time in each state may have a divergent mean value, we demonstrate that this condition is in general not necessary for breaking ergodicity. Hence, we conclude that global memory effects are an alternative mechanism able to induce ergodicity breaking without involving power-law statistics. Analytical and numerical calculations support these results.
Collapse
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
| |
Collapse
|
17
|
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.
Collapse
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
| |
Collapse
|
18
|
Mardoukhi Y, Jeon JH, Metzler R. Geometry controlled anomalous diffusion in random fractal geometries: looking beyond the infinite cluster. Phys Chem Chem Phys 2015; 17:30134-47. [PMID: 26503611 DOI: 10.1039/c5cp03548a] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022]
Abstract
We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law ∼T(-h) with h < 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided.
Collapse
Affiliation(s)
- Yousof Mardoukhi
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany.
| | | | | |
Collapse
|
19
|
Spiechowicz J, Łuczka J. Diffusion anomalies in ac-driven Brownian ratchets. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062104. [PMID: 26172658 DOI: 10.1103/physreve.91.062104] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/31/2015] [Indexed: 06/04/2023]
Abstract
We study diffusion in ratchet systems. As a particular experimental realization we consider an asymmetric SQUID subjected to an external ac current and a constant magnetic flux. We analyze mean-square displacement of the Josephson phase and find that within selected parameter regimes it evolves in three distinct stages: initially as superdiffusion, next as subdiffusion, and finally as normal diffusion in the asymptotic long-time limit. We show how crossover times that separates these stages can be controlled by temperature and an external magnetic flux. The first two stages can last many orders longer than characteristic time scales of the system, thus being comfortably detectable experimentally. The origin of abnormal behavior is noticeable related to the ratchet form of the potential revealing an entirely new mechanism of emergence of anomalous diffusion. Moreover, a normal diffusion coefficient exhibits nonmonotonic dependence on temperature leading to an intriguing phenomenon of thermal noise suppressed diffusion. The proposed setup for experimental verification of our findings provides a new and promising testing ground for investigating anomalies in diffusion phenomena.
Collapse
Affiliation(s)
- Jakub Spiechowicz
- Institute of Physics, University of Silesia, 40-007 Katowice, Poland
| | - Jerzy Łuczka
- Institute of Physics, University of Silesia, 40-007 Katowice, Poland
- Silesian Center for Education and Interdisciplinary Research, University of Silesia, 41-500 Chorzów, Poland
| |
Collapse
|
20
|
Godec A, Metzler R. Optimization and universality of Brownian search in a basic model of quenched heterogeneous media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052134. [PMID: 26066146 DOI: 10.1103/physreve.91.052134] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2015] [Indexed: 06/04/2023]
Abstract
The kinetics of a variety of transport-controlled processes can be reduced to the problem of determining the mean time needed to arrive at a given location for the first time, the so-called mean first-passage time (MFPT) problem. The occurrence of occasional large jumps or intermittent patterns combining various types of motion are known to outperform the standard random walk with respect to the MFPT, by reducing oversampling of space. Here we show that a regular but spatially heterogeneous random walk can significantly and universally enhance the search in any spatial dimension. In a generic minimal model we consider a spherically symmetric system comprising two concentric regions with piecewise constant diffusivity. The MFPT is analyzed under the constraint of conserved average dynamics, that is, the spatially averaged diffusivity is kept constant. Our analytical calculations and extensive numerical simulations demonstrate the existence of an optimal heterogeneity minimizing the MFPT to the target. We prove that the MFPT for a random walk is completely dominated by what we term direct trajectories towards the target and reveal a remarkable universality of the spatially heterogeneous search with respect to target size and system dimensionality. In contrast to intermittent strategies, which are most profitable in low spatial dimensions, the spatially inhomogeneous search performs best in higher dimensions. Discussing our results alongside recent experiments on single-particle tracking in living cells, we argue that the observed spatial heterogeneity may be beneficial for cellular signaling processes.
Collapse
Affiliation(s)
- Aljaž Godec
- Institute of Physics & Astronomy, University of Potsdam, 14776 Potsdam-Golm, Germany
- National Institute of Chemistry, 1000 Ljubljana, Slovenia
| | - Ralf Metzler
- Institute of Physics & Astronomy, University of Potsdam, 14776 Potsdam-Golm, Germany
- Department of Physics, Tampere University of Technology, 33101 Tampere, Finland
| |
Collapse
|
21
|
Wu JC, Chen Q, Wang R, Ai BQ. Diffusion and mobility of anisotropic particles in tilted periodic structures. CHAOS (WOODBURY, N.Y.) 2015; 25:023114. [PMID: 25725650 DOI: 10.1063/1.4913491] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We numerically investigated the transport of anisotropic particles in tilted periodic structures. The diffusion and mobility of the particles demonstrate distinct behaviors dependence on the shape of the particles. In two-dimensional (2D) periodic potentials, we find that the mobility is influenced a little by the anisotropy of the particle, while the diffusion increases monotonically with the increasing of the particle anisotropy for large enough biased force. However, due to the sensitivity of the channels for the particle anisotropy, the transport in smooth channels is obviously different from that in energy potentials. The mobility decreases monotonically with the increasing of the particle anisotropy, while the diffusion can be a non-monotonic function of the particle anisotropy with a peak under appropriate biased force.
Collapse
Affiliation(s)
- Jian-chun Wu
- Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, 510006 Guangzhou, China
| | - Qun Chen
- Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, 510006 Guangzhou, China
| | - Rang Wang
- Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, 510006 Guangzhou, China
| | - Bao-quan Ai
- Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, 510006 Guangzhou, China
| |
Collapse
|
22
|
Dean DS, Guérin T. Approach to asymptotically diffusive behavior for Brownian particles in media with periodic diffusivities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062114. [PMID: 25615051 DOI: 10.1103/physreve.90.062114] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2014] [Indexed: 06/04/2023]
Abstract
We analyze the mean squared displacement of a Brownian particle in a medium with a spatially varying local diffusivity, which is assumed to be periodic. When the system is asymptotically diffusive, the mean-squared displacement, characterizing the dispersion in the system, is, at late times, a linear function of time. A Kubo-type formula is given for the mean-squared displacement, which allows the recovery of some known results for the effective diffusion constant D(e) in a direct way, but also allows an understanding of the asymptotic approach to the diffusive limit. In particular, as well as as computing the slope of a linear fit to the late-time mean-squared displacement, we find a formula for the constant where the fit intersects the y axis.
Collapse
Affiliation(s)
- David S Dean
- Université de Bordeaux and CNRS, Laboratoire Ondes et Matière d'Aquitaine (LOMA), UMR 5798, F-33400 Talence, France
| | - Thomas Guérin
- Université de Bordeaux and CNRS, Laboratoire Ondes et Matière d'Aquitaine (LOMA), UMR 5798, F-33400 Talence, France
| |
Collapse
|
23
|
Nagar H, Roichman Y. Collective excitations of hydrodynamically coupled driven colloidal particles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042302. [PMID: 25375489 DOI: 10.1103/physreve.90.042302] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2013] [Indexed: 06/04/2023]
Abstract
Two colloidal particles, driven around an optical vortex trap, have been recently shown to pair due to an interplay between hydrodynamic interactions and the curved path they are forced to follow. We demonstrate here that this pairing interaction can be tuned experimentally, and we study its effect on the collective excitations of many particles driven around such an optical trap. We find that even though the system is overdamped, hydrodynamic interactions due to driving give rise to nondecaying excitations with characteristic dispersion relations. The collective excitations of the colloidal ring reflect fluctuations of particle pairs rather than those of single particles.
Collapse
Affiliation(s)
- Harel Nagar
- Raymond & Beverly Sackler School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Yael Roichman
- Raymond & Beverly Sackler School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
| |
Collapse
|
24
|
Goychuk I, Kharchenko VO. Anomalous features of diffusion in corrugated potentials with spatial correlations: faster than normal, and other surprises. PHYSICAL REVIEW LETTERS 2014; 113:100601. [PMID: 25238342 DOI: 10.1103/physrevlett.113.100601] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2014] [Indexed: 05/07/2023]
Abstract
Normal diffusion in corrugated potentials with spatially uncorrelated Gaussian energy disorder famously explains the origin of non-Arrhenius exp[-σ2/(kBT2)] temperature dependence in disordered systems. Here we show that unbiased diffusion remains asymptotically normal also in the presence of spatial correlations decaying to zero. However, because of a temporal lack of self-averaging, transient subdiffusion emerges on the mesoscale, and it can readily reach macroscale even for moderately strong disorder fluctuations of σ∼4-5kT. Because of its nonergodic origin, such subdiffusion exhibits a large scatter in single-trajectory averages. However, at odds with intuition, it occurs essentially faster than one expects from the normal diffusion in the absence of correlations. We apply these results to diffusion of regulatory proteins on DNA molecules and predict that such diffusion should be anomalous, but much faster than earlier expected on a typical length of genes for a realistic energy disorder of several room kBT, or merely 0.05-0.075 eV.
Collapse
Affiliation(s)
- Igor Goychuk
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
| | - Vasyl O Kharchenko
- Institute of Applied Physics, 58 Petropavlovskaya street, 40030 Sumy, Ukraine
| |
Collapse
|
25
|
Salgado-García R. Effective diffusion coefficient in tilted disordered potentials: optimal relative diffusivity at a finite temperature. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032105. [PMID: 25314393 DOI: 10.1103/physreve.90.032105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2014] [Indexed: 06/04/2023]
Abstract
In this work we study the transport properties of non-interacting overdamped particles, moving on tilted disordered potentials, subjected to Gaussian white noise. We give exact formulas for the drift and diffusion coefficients for the case of random potentials resulting from the interaction of a particle with a "random polymer". In our model the polymer is made up, by means of some stochastic process, of monomers that can be taken from a finite or countable infinite set of possible monomer types. For the case of uncorrelated random polymers we found that the diffusion coefficient exhibits a non-monotonous behavior as a function of the noise intensity. Particularly interesting is the fact that the relative diffusivity becomes optimal at a finite temperature, a behavior which is reminiscent of stochastic resonance. We explain this effect as an interplay between the deterministic and noisy dynamics of the system. We also show that this behavior of the diffusion coefficient at a finite temperature is more pronounced for the case of weakly disordered potentials. We test our findings by means of numerical simulations of the corresponding Langevin dynamics of an ensemble of noninteracting overdamped particles diffusing on uncorrelated random potentials.
Collapse
Affiliation(s)
- R Salgado-García
- Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Avenida Universidad 1001, Colonia Chamilpa, 62209, Cuernavaca Morelos, Mexico
| |
Collapse
|
26
|
Hurowitz D, Cohen D. Nonequilibrium version of the Einstein relation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032129. [PMID: 25314417 DOI: 10.1103/physreve.90.032129] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2014] [Indexed: 06/04/2023]
Abstract
The celebrated Einstein relation between the diffusion coefficient D and the drift velocity v is violated in nonequilibrium circumstances. We analyze how this violation emerges for the simplest example of a Brownian motion on a lattice, taking into account the interplay between the periodicity, the randomness, and the asymmetry of the transition rates. Based on the nonequilibrium fluctuation theorem the v/D ratio is found to be a nonlinear function of the affinity. Hence it depends in a nontrivial way on the microscopics of the sample.
Collapse
Affiliation(s)
- Daniel Hurowitz
- Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
| | - Doron Cohen
- Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
| |
Collapse
|
27
|
Dean DS, Oshanin G. Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials: extracting information from transients. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022112. [PMID: 25215694 DOI: 10.1103/physreve.90.022112] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2014] [Indexed: 06/03/2023]
Abstract
A Langevin process describing diffusion in a periodic potential landscape has a time-dependent diffusion constant, which means that its average mean-squared displacement (MSD) only becomes linear at late times. The long-time, or effective diffusion, constant can be estimated from the slope of a linear fit of the MSD at late times. Due to the crossover between a short time microscopic diffusion constant, which is independent of the potential, to the effective late-time diffusion constant, a linear fit of the MSD will not in general pass through the origin and will have a nonzero constant term. Here we address how to compute the constant term and provide explicit results for Brownian particles in one dimension in periodic potentials. We show that the constant is always positive and that at low temperatures it depends on the curvature of the minimum of the potential. For comparison we also consider the same question for the simpler problem of a symmetric continuous time random walk in discrete space. Here the constant can be positive or negative and can be used to determine the variance of the hopping time distribution.
Collapse
Affiliation(s)
- David S Dean
- Université de Bordeaux and CNRS, Laboratoire Ondes et Matière d'Aquitaine (LOMA), UMR 5798, F-33400 Talence, France
| | - Gleb Oshanin
- Sorbonne Universités, UPMC Univ Paris 06, UMR 7600, LPTMC, F-75005, Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005, Paris, France
| |
Collapse
|
28
|
Massignan P, Manzo C, Torreno-Pina JA, García-Parajo MF, Lewenstein M, Lapeyre GJ. Nonergodic subdiffusion from Brownian motion in an inhomogeneous medium. PHYSICAL REVIEW LETTERS 2014; 112:150603. [PMID: 24785018 DOI: 10.1103/physrevlett.112.150603] [Citation(s) in RCA: 86] [Impact Index Per Article: 8.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2014] [Indexed: 06/03/2023]
Abstract
Nonergodicity observed in single-particle tracking experiments is usually modeled by transient trapping rather than spatial disorder. We introduce models of a particle diffusing in a medium consisting of regions with random sizes and random diffusivities. The particle is never trapped but rather performs continuous Brownian motion with the local diffusion constant. Under simple assumptions on the distribution of the sizes and diffusivities, we find that the mean squared displacement displays subdiffusion due to nonergodicity for both annealed and quenched disorder. The model is formulated as a walk continuous in both time and space, similar to the Lévy walk.
Collapse
Affiliation(s)
- P Massignan
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain
| | - C Manzo
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain
| | - J A Torreno-Pina
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain
| | - M F García-Parajo
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain and ICREA-Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain
| | - M Lewenstein
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain and ICREA-Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain
| | - G J Lapeyre
- ICFO-Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels, Spain
| |
Collapse
|
29
|
Krüsemann H, Godec A, Metzler R. First-passage statistics for aging diffusion in systems with annealed and quenched disorder. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:040101. [PMID: 24827169 DOI: 10.1103/physreve.89.040101] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2014] [Indexed: 06/03/2023]
Abstract
Aging, the dependence of the dynamics of a physical process on the time ta since its original preparation, is observed in systems ranging from the motion of charge carriers in amorphous semiconductors over the blinking dynamics of quantum dots to the tracer dispersion in living biological cells. Here we study the effects of aging on one of the most fundamental properties of a stochastic process, the first-passage dynamics. We find that for an aging continuous time random walk process, the scaling exponent of the density of first-passage times changes twice as the aging progresses and reveals an intermediate scaling regime. The first-passage dynamics depends on ta differently for intermediate and strong aging. Similar crossovers are obtained for the first-passage dynamics for a confined and driven particle. Comparison to the motion of an aged particle in the quenched trap model with a bias shows excellent agreement with our analytical findings. Our results demonstrate how first-passage measurements can be used to unravel the age ta of a physical system.
Collapse
Affiliation(s)
- Henning Krüsemann
- Institute of Physics & Astronomy, University of Potsdam, 14776 Potsdam-Golm, Germany
| | - Aljaž Godec
- Institute of Physics & Astronomy, University of Potsdam, 14776 Potsdam-Golm, Germany and National Institute of Chemistry, 1000 Ljubljana, Slovenia
| | - Ralf Metzler
- Institute of Physics & Astronomy, University of Potsdam, 14776 Potsdam-Golm, Germany and Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland
| |
Collapse
|
30
|
Heuer A, Lühning L. Physical mechanisms of nonlinear conductivity: A model analysis. J Chem Phys 2014; 140:094508. [DOI: 10.1063/1.4867058] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
31
|
Cieśla M, Gudowska-Nowak E, Sagués F, Sokolov IM. Tracer diffusion inside fibrinogen layers. J Chem Phys 2014; 140:044706. [DOI: 10.1063/1.4862170] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
32
|
Metzler R, Jeon JH, Cherstvy AG, Barkai E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys Chem Chem Phys 2014; 16:24128-64. [DOI: 10.1039/c4cp03465a] [Citation(s) in RCA: 1046] [Impact Index Per Article: 104.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Abstract
This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.
Collapse
Affiliation(s)
- Ralf Metzler
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
- Physics Department
- Tampere University of Technology
| | - Jae-Hyung Jeon
- Physics Department
- Tampere University of Technology
- Tampere, Finland
- Korean Institute for Advanced Study (KIAS)
- Seoul, Republic of Korea
| | - Andrey G. Cherstvy
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
| | - Eli Barkai
- Physics Department and Institute of Nanotechnology and Advanced Materials
- Bar-Ilan University
- Ramat Gan, Israel
| |
Collapse
|
33
|
Jeon JH, Chechkin AV, Metzler R. Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion. Phys Chem Chem Phys 2014; 16:15811-7. [DOI: 10.1039/c4cp02019g] [Citation(s) in RCA: 133] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
|
34
|
Simon MS, Sancho JM, Lindenberg K. Transport and diffusion of overdamped Brownian particles in random potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062105. [PMID: 24483384 DOI: 10.1103/physreve.88.062105] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2013] [Indexed: 06/03/2023]
Abstract
We present a numerical study of the anomalies in transport and diffusion of overdamped Brownian particles in totally disordered potential landscapes in one and in two dimensions. We characterize and analyze the effects of three different disordered potentials. The anomalous regimes are characterized by the time exponents that exhibit the statistical moments of the ensemble of particle trajectories. The anomaly in the transport is always of the subtransport type, but diffusion presents a greater variety of anomalies: Both subdiffusion and superdiffusion are possible. In two dimensions we present a mixed anomaly: subdiffusion in the direction perpendicular to the force and superdiffusion in the parallel direction.
Collapse
Affiliation(s)
- Marc Suñé Simon
- Departament d'Estructura i Constituents de la Matèria, Facultat de Física, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain
| | - J M Sancho
- Departament d'Estructura i Constituents de la Matèria, Facultat de Física, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain
| | - Katja Lindenberg
- Department of Chemistry and Biochemistry and BioCircuits Institute, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093-0340, USA
| |
Collapse
|
35
|
Bao JD, Liu J. Group superballistic diffusion: bimodal velocity inducing coexistence of two states in a corrugated plane. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022153. [PMID: 24032820 DOI: 10.1103/physreve.88.022153] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/12/2013] [Revised: 07/10/2013] [Indexed: 06/02/2023]
Abstract
We consider anomalous diffusion of a particle moving in a tilted periodic potential in the presence of Lévy noise and nonlinear friction. Using Monte Carlo simulations, we have found some interesting characteristics of diffusion in such a nonlinear system: when the noise intensity is weak and the external force is close to the critical value at which local minima of the potential just vanish, the nonmonotonic behavior of the effective diffusion index and the superballistic diffusion are observed. This is due to the bimodal nature of the velocity distribution, and thus the test particles exist in either a running state or a long-tailed behind state in the spatial coordinate; the latter is disintegrated into small pieces of the probability peaks. We provide a relation between the group diffusion coefficient and the phase diffusion coefficient. It is shown that the distance between the above two-state centers increasing with time plays the definitive role in the superballistic group diffusion.
Collapse
Affiliation(s)
- Jing-Dong Bao
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | | |
Collapse
|
36
|
Balankin AS, Mena B, Martínez-González CL, Matamoros DM. Random walk in chemical space of Cantor dust as a paradigm of superdiffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:052101. [PMID: 23214828 DOI: 10.1103/physreve.86.052101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2012] [Revised: 07/31/2012] [Indexed: 06/01/2023]
Abstract
We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.
Collapse
Affiliation(s)
- Alexander S Balankin
- Grupo Mecánica Fractal, Instituto Politécnico Nacional, México Distrito Federal 07738, Mexico
| | | | | | | |
Collapse
|
37
|
Burov S, Barkai E. Weak subordination breaking for the quenched trap model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:041137. [PMID: 23214559 DOI: 10.1103/physreve.86.041137] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2012] [Indexed: 06/01/2023]
Abstract
We map the problem of diffusion in the quenched trap model onto a different stochastic process: Brownian motion that is terminated at the coverage time S(α)=∑(x=-∞)(∞)(n(x))(α), with n(x) being the number of visits to site x. Here 0<α=T/T(g)<1 is a measure of the disorder in the original model. This mapping allows us to treat the intricate correlations in the underlying random walk in the random environment. The operational time S(α) is changed to laboratory time t with a Lévy time transformation. Investigation of Brownian motion stopped at time S(α) yields the diffusion front of the quenched trap model, which is favorably compared with numerical simulations. In the zero-temperature limit of α→0 we recover the renormalization group solution obtained by Monthus [Phys. Rev. E 68, 036114 (2003)]. Our theory surmounts the critical slowing down that is found when α→1. Above the critical dimension 2, mapping the problem to a continuous time random walk becomes feasible, though still not trivial.
Collapse
Affiliation(s)
- S Burov
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel
| | | |
Collapse
|
38
|
Goychuk I. Fractional-time random walk subdiffusion and anomalous transport with finite mean residence times: faster, not slower. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:021113. [PMID: 23005728 DOI: 10.1103/physreve.86.021113] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2012] [Revised: 05/23/2012] [Indexed: 05/07/2023]
Abstract
Continuous time random walk (CTRW) subdiffusion along with the associated fractional Fokker-Planck equation (FFPE) is traditionally based on the premise of random clock with divergent mean period. This work considers an alternative CTRW and FFPE description which is featured by finite mean residence times (MRTs) in any spatial domain of finite size. Transient subdiffusive transport can occur on a very large time scale τ(c) which can greatly exceed mean residence time in any trap, τ(c) >>(τ), and even not being related to it. Asymptotically, on a macroscale transport becomes normal for t >> τ(c). However, mesoscopic transport is anomalous. Differently from viscoelastic subdiffusion no long-range anticorrelations among position increments are required. Moreover, our study makes it obvious that the transient subdiffusion and transport are faster than one expects from their normal asymptotic limit on a macroscale. This observation has profound implications for anomalous mesoscopic transport processes in biological cells because the macroscopic viscosity of cytoplasm is finite.
Collapse
Affiliation(s)
- Igor Goychuk
- Institut für Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany
| |
Collapse
|
39
|
Carretero-Campos C, Bernaola-Galván P, Ch. Ivanov P, Carpena P. Phase transitions in the first-passage time of scale-invariant correlated processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:011139. [PMID: 22400544 PMCID: PMC3518899 DOI: 10.1103/physreve.85.011139] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2011] [Revised: 11/30/2011] [Indexed: 05/31/2023]
Abstract
A key quantity describing the dynamics of complex systems is the first-passage time (FPT). The statistical properties of FPT depend on the specifics of the underlying system dynamics. We present a unified approach to account for the diversity of statistical behaviors of FPT observed in real-world systems. We find three distinct regimes, separated by two transition points, with fundamentally different behavior for FPT as a function of increasing strength of the correlations in the system dynamics: stretched exponential, power-law, and saturation regimes. In the saturation regime, the average length of FPT diverges proportionally to the system size, with important implications for understanding electronic delocalization in one-dimensional correlated-disordered systems.
Collapse
Affiliation(s)
| | | | - Plamen Ch. Ivanov
- Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02212, USA
- Harvard Medical School and Division of Sleep Medicine, Brigham and Women’s Hospital, Boston, Massachusetts 02115, USA
- Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
| | - Pedro Carpena
- Departamento de Física Aplicada II, Universidad de Málaga, E-29071 Málaga, Spain
| |
Collapse
|