1
|
Ramesh VG, Peters KJH, Rodriguez SRK. Arcsine Laws of Light. PHYSICAL REVIEW LETTERS 2024; 132:133801. [PMID: 38613295 DOI: 10.1103/physrevlett.132.133801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2022] [Accepted: 02/06/2024] [Indexed: 04/14/2024]
Abstract
We demonstrate that the time-integrated light intensity transmitted by a coherently driven resonator obeys Lévy's arcsine laws-a cornerstone of extreme value statistics. We show that convergence to the arcsine distribution is algebraic, universal, and independent of nonequilibrium behavior due to nonconservative forces or nonadiabatic driving. We furthermore verify, numerically, that the arcsine laws hold in the presence of frequency noise and in Kerr-nonlinear resonators supporting non-Gaussian states. The arcsine laws imply a weak ergodicity breaking which can be leveraged to enhance the precision of resonant optical sensors with zero energy cost, as shown in our companion manuscript [V. G. Ramesh et al., companion paper, Phys. Rev. Res. (2024).PPRHAI2643-1564]. Finally, we discuss perspectives for probing the possible breakdown of the arcsine laws in systems with memory.
Collapse
Affiliation(s)
- V G Ramesh
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
| | - K J H Peters
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
| | - S R K Rodriguez
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
| |
Collapse
|
2
|
Höll M, Nissan A, Berkowitz B, Barkai E. Controls that expedite first-passage times in disordered systems. Phys Rev E 2023; 108:034124. [PMID: 37849182 DOI: 10.1103/physreve.108.034124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2022] [Accepted: 08/28/2023] [Indexed: 10/19/2023]
Abstract
First-passage time statistics in disordered systems exhibiting scale invariance are studied widely. In particular, long trapping times in energy or entropic traps are fat-tailed distributed, which slow the overall transport process. We study the statistical properties of the first-passage time of biased processes in different models, and we employ the big-jump principle that shows the dominance of the maximum trapping time on the first-passage time. We demonstrate that the removal of this maximum significantly expedites transport. As the disorder increases, the system enters a phase where the removal shows a dramatic effect. Our results show how we may speed up transport in strongly disordered systems exploiting scale invariance. In contrast to the disordered systems studied here, the removal principle has essentially no effect in homogeneous systems; this indicates that improving the conductance of a poorly conducting system is, theoretically, relatively easy as compared to a homogeneous system.
Collapse
Affiliation(s)
- Marc Höll
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
| | - Alon Nissan
- Institute of Environmental Engineering, ETH Zurich, Zurich, Switzerland
| | - Brian Berkowitz
- Department of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot 7610001, Israel
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
| |
Collapse
|
3
|
Defaveri L, Dos Santos MAF, Kessler DA, Barkai E, Anteneodo C. Non-normalizable quasiequilibrium states under fractional dynamics. Phys Rev E 2023; 108:024133. [PMID: 37723721 DOI: 10.1103/physreve.108.024133] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2023] [Accepted: 07/31/2023] [Indexed: 09/20/2023]
Abstract
Particles anomalously diffusing in contact with a thermal bath are initially released from an asymptotically flat potential well. For temperatures that are sufficiently low compared to the potential depth, the dynamical and thermodynamical observables of the system remain almost constant for long times. We show how these stagnated states are characterized as non-normalizable quasiequilibrium (NNQE) states. We use the fractional-time Fokker-Planck equation (FTFPE) and continuous-time random walk approaches to calculate ensemble averages. We obtain analytical estimates of the durations of NNQE states, depending on the fractional order, from approximate theoretical solutions of the FTFPE. We study and compare two types of observables, the mean square displacement typically used to characterize diffusion, and the thermodynamic energy. We show that the typical timescales for transient stagnation depend exponentially on the value of the depth of the potential well, in units of temperature, multiplied by a function of the fractional exponent.
Collapse
Affiliation(s)
| | - Maike A F Dos Santos
- Department of Physics, PUC-Rio, Rua Marquês de São Vicente 225, 22451-900 Gávea, Rio de Janeiro, Brazil
| | - David A Kessler
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Celia Anteneodo
- Department of Physics, PUC-Rio, Rua Marquês de São Vicente 225, 22451-900 Gávea, Rio de Janeiro, Brazil
- National Institute of Science and Technology for Complex Systems, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil
| |
Collapse
|
4
|
Dey R, Kundu A, Das B, Banerjee A. Experimental verification of arcsine laws in mesoscopic nonequilibrium systems. Phys Rev E 2022; 106:054113. [PMID: 36559344 DOI: 10.1103/physreve.106.054113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2022] [Accepted: 10/04/2022] [Indexed: 06/17/2023]
Abstract
A large number of processes in the mesoscopic world occur out of equilibrium, where the time evolution of a system becomes immensely important since it is driven principally by dissipative effects. Nonequilibrium steady states (NESS) represent a crucial category in such systems, where relaxation timescales are comparable to the operational timescales. In this study, we employ a model NESS stochastic system, which is comprised of a colloidal microparticle optically trapped in a viscous fluid, externally driven by a temporally correlated noise, and show that time-integrated observables such as the entropic current, the work done on the system or the work dissipated by it, follow the three Lévy arcsine laws [A. C. Barato et al., Phys. Rev. Lett. 121, 090601 (2018)0031-900710.1103/PhysRevLett.121.090601], in the large time limit. We discover that cumulative distributions converge faster to arcsine distributions when it is near equilibrium and the rate of entropy production is small, because in that case the entropic current has weaker temporal autocorrelation. We study this phenomenon by changing the strength of the added noise as well as by perturbing our system with a flow field produced by a microbubble at close proximity to the trapped particle. We confirm our experimental findings with theoretical simulations of the systems. Our work provides an interesting insight into the NESS statistics of the meso-regime, where stochastic fluctuations play a pivotal role.
Collapse
Affiliation(s)
- Raunak Dey
- Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Mohanpur, West Bengal 741246, India and School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
| | - Avijit Kundu
- Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Mohanpur, West Bengal 741246, India
| | - Biswajit Das
- Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Mohanpur, West Bengal 741246, India
| | - Ayan Banerjee
- Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Mohanpur, West Bengal 741246, India
| |
Collapse
|
5
|
Albers T, Radons G. Nonergodicity of d-dimensional generalized Lévy walks and their relation to other space-time coupled models. Phys Rev E 2022; 105:014113. [PMID: 35193310 DOI: 10.1103/physreve.105.014113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2021] [Accepted: 12/23/2021] [Indexed: 06/14/2023]
Abstract
We investigate the nonergodicity of the generalized Lévy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)PRLTAO0031-900710.1103/PhysRevLett.58.1100] with respect to the squared displacements. We present detailed analytical derivations of our previous findings outlined in a recent letter [Phys. Rev. Lett. 120, 104501 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.104501], give detailed interpretations, and in particular emphasize three surprising results. First, we find that the mean-squared displacements can diverge for a certain range of parameter values. Second, we show that an ensemble of trajectories can spread subdiffusively, whereas individual time-averaged squared displacements show superdiffusion. Third, we recognize that the fluctuations of the time-averaged squared displacements can become so large that the ergodicity breaking parameter diverges, what we call infinitely strong ergodicity breaking. This phenomenon can also occur for paramter values where the lag-time dependence of the mean-squared displacements is linear indicating normal diffusion. In order to numerically determine the full distribution of time-averaged squared displacements, we use importance sampling. For an embedding of our findings into existing results in the literature, we define a more general model which we call variable speed generalized Lévy walk and which includes well-known models from the literature as special cases such as the space-time coupled Lévy flight or the anomalous Drude model. We discuss and interpret our findings regarding the generalized Lévy walk in detail and compare them with the nonergodicity of the other space-time coupled models following from the more general model.
Collapse
Affiliation(s)
- Tony Albers
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
| | - Günter Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany and Institute of Mechatronics, 09126 Chemnitz, Germany
| |
Collapse
|
6
|
Scher Y, Reuveni S. Unified Approach to Gated Reactions on Networks. PHYSICAL REVIEW LETTERS 2021; 127:018301. [PMID: 34270310 DOI: 10.1103/physrevlett.127.018301] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2021] [Revised: 03/20/2021] [Accepted: 06/02/2021] [Indexed: 06/13/2023]
Abstract
For two molecules to react they first have to meet. Yet, reaction times are rarely on par with the first-passage times that govern such molecular encounters. A prime reason for this discrepancy is stochastic transitions between reactive and nonreactive molecular states, which results in effective gating of product formation and altered reaction kinetics. To better understand this phenomenon we develop a unifying approach to gated reactions on networks. We first show that the mean and distribution of the gated reaction time can always be expressed in terms of ungated first-passage and return times. This relation between gated and ungated kinetics is then explored to reveal universal features of gated reactions. The latter are exemplified using a diverse set of case studies which are also used to expose the exotic kinetics that arises due to molecular gating.
Collapse
Affiliation(s)
- Yuval Scher
- School of Chemistry, Center for the Physics & Chemistry of Living Systems, Ratner Institute for Single Molecule Chemistry, and the Sackler Center for Computational Molecular & Materials Science, Tel Aviv University, 6997801 Tel Aviv, Israel
| | - Shlomi Reuveni
- School of Chemistry, Center for the Physics & Chemistry of Living Systems, Ratner Institute for Single Molecule Chemistry, and the Sackler Center for Computational Molecular & Materials Science, Tel Aviv University, 6997801 Tel Aviv, Israel
| |
Collapse
|
7
|
Abstract
One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided.
Collapse
|
8
|
Levin M, Bel G, Roichman Y. Measurements and characterization of the dynamics of tracer particles in an actin network. J Chem Phys 2021; 154:144901. [PMID: 33858166 DOI: 10.1063/5.0045278] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023] Open
Abstract
The underlying physics governing the diffusion of a tracer particle in a viscoelastic material is a topic of some dispute. The long-term memory in the mechanical response of such materials should induce diffusive motion with a memory kernel, such as fractional Brownian motion (fBM). This is the reason that microrheology is able to provide the shear modulus of polymer networks. Surprisingly, the diffusion of a tracer particle in a network of a purified protein, actin, was found to conform to the continuous time random walk type (CTRW). We set out to resolve this discrepancy by studying the tracer particle diffusion using two different tracer particle sizes, in actin networks of different mesh sizes. We find that the ratio of tracer particle size to the characteristic length scale of a bio-polymer network plays a crucial role in determining the type of diffusion it performs. We find that the diffusion of the tracer particles has features of fBm when the particle is large compared to the mesh size, of normal diffusion when the particle is much smaller than the mesh size, and of the CTRW in between these two limits. Based on our findings, we propose and verify numerically a new model for the motion of the tracer in all regimes. Our model suggests that diffusion in actin networks consists of fBm of the tracer particle coupled with caging events with power-law distributed escape times.
Collapse
Affiliation(s)
- Maayan Levin
- Raymond and Beverly Sackler School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Golan Bel
- Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus 8499000, Israel
| | - Yael Roichman
- Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel
| |
Collapse
|
9
|
A Continuous-Time Random Walk Extension of the Gillis Model. ENTROPY 2020; 22:e22121431. [PMID: 33353053 PMCID: PMC7766702 DOI: 10.3390/e22121431] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/24/2020] [Revised: 12/14/2020] [Accepted: 12/15/2020] [Indexed: 11/26/2022]
Abstract
We consider a continuous-time random walk which is the generalization, by means of the introduction of waiting periods on sites, of the one-dimensional non-homogeneous random walk with a position-dependent drift known in the mathematical literature as Gillis random walk. This modified stochastic process allows to significantly change local, non-local and transport properties in the presence of heavy-tailed waiting-time distributions lacking the first moment: we provide here exact results concerning hitting times, first-time events, survival probabilities, occupation times, the moments spectrum and the statistics of records. Specifically, normal diffusion gives way to subdiffusion and we are witnessing the breaking of ergodicity. Furthermore we also test our theoretical predictions with numerical simulations.
Collapse
|
10
|
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
|
11
|
Buonocore S, Sen M, Semperlotti F. Stochastic scattering model of anomalous diffusion in arrays of steady vortices. Proc Math Phys Eng Sci 2020; 476:20200183. [PMID: 32831596 DOI: 10.1098/rspa.2020.0183] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2020] [Accepted: 05/04/2020] [Indexed: 11/12/2022] Open
Abstract
We investigate the occurrence of anomalous transport phenomena associated with tracer particles propagating through arrays of steady vortices. The mechanism responsible for the occurrence of anomalous transport is identified in the particle dynamic, which is characterized by long collision-less trajectories (Lévy flights) interrupted by chaotic interactions with vortices. The process is studied via stochastic molecular models that are able to capture the underlying non-local nature of the transport mechanism. These models, however, are not well suited for problems where computational efficiency is an enabling factor. We show that fractional-order continuum models provide an excellent alternative that is able to capture the non-local nature of anomalous transport processes in turbulent environments. The equivalence between stochastic molecular and fractional continuum models is demonstrated both theoretically and numerically. In particular, the onset and the temporal evolution of heavy-tailed diffused fields are shown to be accurately captured, from a macroscopic perspective, by a fractional diffusion equation. The resulting anomalous transport mechanism, for the selected ranges of density of the vortices, shows a superdiffusive nature.
Collapse
Affiliation(s)
- Salvatore Buonocore
- Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
| | - Mihir Sen
- Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
| | - Fabio Semperlotti
- Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA.,Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA
| |
Collapse
|
12
|
Danieli C, Mithun T, Kati Y, Campbell DK, Flach S. Dynamical glass in weakly nonintegrable Klein-Gordon chains. Phys Rev E 2019; 100:032217. [PMID: 31639954 DOI: 10.1103/physreve.100.032217] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2018] [Indexed: 06/10/2023]
Abstract
Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a nonintegrable perturbation creates a coupling network in action space which can be short or long ranged. We analyze the dynamics of observables which become the conserved actions in the integrable limit. We compute distributions of their finite time averages and obtain the ergodization time scale T_{E} on which these distributions converge to δ distributions. We relate T_{E} to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations σ_{τ}^{+} dominating the means μ_{τ}^{+} and establish that T_{E}∼(σ_{τ}^{+})^{2}/μ_{τ}^{+}. The Lyapunov time T_{Λ} (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long- and short-range coupling networks by tuning its energy density. For long-range coupling networks T_{Λ}≈σ_{τ}^{+}, which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a dynamical glass, where T_{E} grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which satisfies T_{Λ}≲μ_{τ}^{+}. This effect arises from the formation of highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of nonchaotic regions. These structures persist up to the ergodization time T_{E}.
Collapse
Affiliation(s)
- Carlo Danieli
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea
| | - Thudiyangal Mithun
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea
| | - Yagmur Kati
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea
- Basic Science Program, Korea University of Science and Technology (UST), Daejeon 34113, Republic of Korea
| | - David K Campbell
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - Sergej Flach
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea
- New Zealand Institute for Advanced Study, Massey University, Auckland 02215, New Zealand
| |
Collapse
|
13
|
Wang X, Deng W, Chen Y. Ergodic properties of heterogeneous diffusion processes in a potential well. J Chem Phys 2019; 150:164121. [DOI: 10.1063/1.5090594] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- 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
| | - 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
| |
Collapse
|
14
|
Mithun T, Kati Y, Danieli C, Flach S. Weakly Nonergodic Dynamics in the Gross-Pitaevskii Lattice. PHYSICAL REVIEW LETTERS 2018; 120:184101. [PMID: 29775355 DOI: 10.1103/physrevlett.120.184101] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/09/2017] [Indexed: 06/08/2023]
Abstract
The microcanonical Gross-Pitaevskii (also known as the semiclassical Bose-Hubbard) lattice model dynamics is characterized by a pair of energy and norm densities. The grand canonical Gibbs distribution fails to describe a part of the density space, due to the boundedness of its kinetic energy spectrum. We define Poincaré equilibrium manifolds and compute the statistics of microcanonical excursion times off them. The tails of the distribution functions quantify the proximity of the many-body dynamics to a weakly nonergodic phase, which occurs when the average excursion time is infinite. We find that a crossover to weakly nonergodic dynamics takes place inside the non-Gibbs phase, being unnoticed by the largest Lyapunov exponent. In the ergodic part of the non-Gibbs phase, the Gibbs distribution should be replaced by an unknown modified one. We relate our findings to the corresponding integrable limit, close to which the actions are interacting through a short range coupling network.
Collapse
Affiliation(s)
- Thudiyangal Mithun
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea
| | - Yagmur Kati
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea
- Basic Science Program, Korea University of Science and Technology (UST), Daejeon 34113, Republic of Korea
| | - Carlo Danieli
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea
| | - Sergej Flach
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea
| |
Collapse
|
15
|
Abstract
Anomalous diffusion is being discovered in a fast growing number of systems. The exact nature of this anomalous diffusion provides important information on the physical laws governing the studied system. One of the central properties analysed for finite particle motion time series is the intrinsic variability of the apparent diffusivity, typically quantified by the ergodicity breaking parameter EB. Here we demonstrate that frequently EB is insufficient to provide a meaningful measure for the observed variability of the data. Instead, important additional information is provided by the higher order moments entering by the skewness and kurtosis. We analyse these quantities for three popular anomalous diffusion models. In particular, we find that even for the Gaussian fractional Brownian motion a significant skewness in the results of physical measurements occurs and needs to be taken into account. Interestingly, the kurtosis and skewness may also provide sensitive estimates of the anomalous diffusion exponent underlying the data. We also derive a new result for the EB parameter of fractional Brownian motion valid for the whole range of the anomalous diffusion parameter. Our results are important for the analysis of anomalous diffusion but also provide new insights into the theory of anomalous stochastic processes.
Collapse
|
16
|
Denys M, Gubiec T, Kutner R, Jagielski M, Stanley HE. Universality of market superstatistics. Phys Rev E 2016; 94:042305. [PMID: 27841535 DOI: 10.1103/physreve.94.042305] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2015] [Indexed: 11/07/2022]
Abstract
We use a key concept of the continuous-time random walk formalism, i.e., continuous and fluctuating interevent times in which mutual dependence is taken into account, to model market fluctuation data when traders experience excessive (or superthreshold) losses or excessive (or superthreshold) profits. We analytically derive a class of "superstatistics" that accurately model empirical market activity data supplied by Bogachev, Ludescher, Tsallis, and Bunde that exhibit transition thresholds. We measure the interevent times between excessive losses and excessive profits and use the mean interevent discrete (or step) time as a control variable to derive a universal description of empirical data collapse. Our dominant superstatistic value is a power-law corrected by the lower incomplete gamma function, which asymptotically tends toward robustness but initially gives an exponential. We find that the scaling shape exponent that drives our superstatistics subordinates itself and a "superscaling" configuration emerges. Thanks to the Weibull copula function, our approach reproduces the empirically proven dependence between successive interevent times. We also use the approach to calculate a dynamic risk function and hence the dynamic VaR, which is significant in financial risk analysis. Our results indicate that there is a functional (but not literal) balance between excessive profits and excessive losses that can be described using the same body of superstatistics but different calibration values and driving parameters. We also extend our original approach to cover empirical seismic activity data (e.g., given by Corral), the interevent times of which range from minutes to years. Superpositioned superstatistics is another class of superstatistics that protects power-law behavior both for short- and long-time behaviors. These behaviors describe well the collapse of seismic activity data and capture so-called volatility clustering phenomena.
Collapse
Affiliation(s)
- Mateusz Denys
- Faculty of Physics, University of Warsaw, Pasteur 5, PL-02093 Warsaw, Poland
| | - Tomasz Gubiec
- Faculty of Physics, University of Warsaw, Pasteur 5, PL-02093 Warsaw, Poland
| | - Ryszard Kutner
- Faculty of Physics, University of Warsaw, Pasteur 5, PL-02093 Warsaw, Poland
| | - Maciej Jagielski
- Department of Management, Technology and Economics, ETHZ, Scheuchzerstrasse 7, CH-8092 Zürich, Switzerland; Faculty of Physics, University of Warsaw, Pasteur 5, PL-02093 Warsaw, Poland; and Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - H Eugene Stanley
- Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| |
Collapse
|
17
|
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
|
18
|
Metzler R, Jeon JH, Cherstvy AG. Non-Brownian diffusion in lipid membranes: Experiments and simulations. BIOCHIMICA ET BIOPHYSICA ACTA-BIOMEMBRANES 2016; 1858:2451-2467. [PMID: 26826272 DOI: 10.1016/j.bbamem.2016.01.022] [Citation(s) in RCA: 126] [Impact Index Per Article: 15.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2015] [Revised: 01/21/2016] [Accepted: 01/23/2016] [Indexed: 12/14/2022]
Abstract
The dynamics of constituents and the surface response of cellular membranes-also in connection to the binding of various particles and macromolecules to the membrane-are still a matter of controversy in the membrane biophysics community, particularly with respect to crowded membranes of living biological cells. We here put into perspective recent single particle tracking experiments in the plasma membranes of living cells and supercomputing studies of lipid bilayer model membranes with and without protein crowding. Special emphasis is put on the observation of anomalous, non-Brownian diffusion of both lipid molecules and proteins embedded in the lipid bilayer. While single component, pure lipid bilayers in simulations exhibit only transient anomalous diffusion of lipid molecules on nanosecond time scales, the persistence of anomalous diffusion becomes significantly longer ranged on the addition of disorder-through the addition of cholesterol or proteins-and on passing of the membrane lipids to the gel phase. Concurrently, experiments demonstrate the anomalous diffusion of membrane embedded proteins up to macroscopic time scales in the minute time range. Particular emphasis will be put on the physical character of the anomalous diffusion, in particular, the occurrence of ageing observed in the experiments-the effective diffusivity of the measured particles is a decreasing function of time. Moreover, we present results for the time dependent local scaling exponent of the mean squared displacement of the monitored particles. Recent results finding deviations from the commonly assumed Gaussian diffusion patterns in protein crowded membranes are reported. The properties of the displacement autocorrelation function of the lipid molecules are discussed in the light of their appropriate physical anomalous diffusion models, both for non-crowded and crowded membranes. In the last part of this review we address the upcoming field of membrane distortion by elongated membrane-binding particles. We discuss how membrane compartmentalisation and the particle-membrane binding energy may impact the dynamics and response of lipid membranes. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.
Collapse
Affiliation(s)
- R Metzler
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany; Department of Physics, Tampere University of Technology, 33101 Tampere, Finland.
| | - J-H Jeon
- Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea
| | - A G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| |
Collapse
|
19
|
Schulz JHP, Barkai E. Fluctuations around equilibrium laws in ergodic continuous-time random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062129. [PMID: 26172683 DOI: 10.1103/physreve.91.062129] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2014] [Indexed: 06/04/2023]
Abstract
We study occupation time statistics in ergodic continuous-time random walks. Under thermal detailed balance conditions, the average occupation time is given by the Boltzmann-Gibbs canonical law. But close to the nonergodic phase, the finite-time fluctuations around this mean are large and nontrivial. They exhibit dual time scaling and distribution laws: the infinite density of large fluctuations complements the Lévy-stable density of bulk fluctuations. Neither of the two should be interpreted as a stand-alone limiting law, as each has its own deficiency: the infinite density has an infinite norm (despite particle conservation), while the stable distribution has an infinite variance (although occupation times are bounded). These unphysical divergences are remedied by consistent use and interpretation of both formulas. Interestingly, while the system's canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion, they also control the infinite and Lévy-stable densities of fluctuations. The duality of stable and infinite densities is in fact ubiquitous for these dynamics, as it concerns the time averages of general physical observables.
Collapse
Affiliation(s)
- Johannes H P Schulz
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
| |
Collapse
|
20
|
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
|
21
|
Korabel N, Barkai E. Distributions of time averages for weakly chaotic systems: the role of infinite invariant density. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032114. [PMID: 24125221 DOI: 10.1103/physreve.88.032114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2013] [Indexed: 06/02/2023]
Abstract
Distributions of time averaged observables are investigated using deterministic maps with N indifferent fixed points and N-state continuous time random walk processes associated with them. In a weakly chaotic phase, namely when separation of trajectories is subexponential, maps are characterized by an infinite invariant density. We find that the infinite density can be used to calculate the distribution of time averages of integrable observables with a formula recently obtained by Rebenshtok and Barkai. As an example we calculate distributions of the average position of the particle and average occupation fractions. Our work provides the distributional limit theorem for time averages for a wide class of nonintegrable observables with respect to the infinite invariant density, in other words it deals with the situation where the Darling-Kac-Aaronson theorem does not hold.
Collapse
Affiliation(s)
- Nickolay Korabel
- School of Natural Sciences, University of California, Merced, California 95343, USA
| | | |
Collapse
|
22
|
Carmi S, Barkai E. Fractional Feynman-Kac equation for weak ergodicity breaking. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:061104. [PMID: 22304037 DOI: 10.1103/physreve.84.061104] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2011] [Indexed: 05/31/2023]
Abstract
The continuous-time random walk (CTRW) is a model of anomalous subdiffusion in which particles are immobilized for random times between successive jumps. A power-law distribution of the waiting times, ψ(τ) ~ τ(-(1+α)), leads to subdiffusion (x(2) ~ t(α)) for 0 < α < 1. In closed systems, the long stagnation periods cause time averages to divert from the corresponding ensemble averages, which is a manifestation of weak ergodicity breaking. The time average of a general observable U(t) = 1/t ∫(0)(t) U[x(τ)]dτ is a functional of the path and is described by the well-known Feynman-Kac equation if the motion is Brownian. Here, we derive forward and backward fractional Feynman-Kac equations for functionals of CTRW in a binding potential. We use our equations to study two specific time averages: the fraction of time spent by a particle in half-box, and the time average of the particle's position in a harmonic field. In both cases, we obtain the probability density function of the time averages for t → ∞ and the first two moments. Our results show that both the occupation fraction and the time-averaged position are random variables even for long times, except for α = 1, when they are identical to their ensemble averages. Using our fractional Feynman-Kac equation, we also study the dynamics leading to weak ergodicity breaking, namely the convergence of the fluctuations to their asymptotic values.
Collapse
Affiliation(s)
- Shai Carmi
- Department of Physics & Advanced Materials and Nanotechnology Institute, Bar-Ilan University, Ramat Gan 52900, Israel
| | | |
Collapse
|
23
|
Yang H. Change-Point Localization and Wavelet Spectral Analysis of Single-Molecule Time Series. SINGLE-MOLECULE BIOPHYSICS 2011. [DOI: 10.1002/9781118131374.ch9] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/24/2023]
|
24
|
Korabel N, Barkai E. Boundary conditions of normal and anomalous diffusion from thermal equilibrium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:051113. [PMID: 21728496 DOI: 10.1103/physreve.83.051113] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2010] [Revised: 03/03/2011] [Indexed: 05/31/2023]
Abstract
Infiltration of diffusing particles from one material to another, where the diffusion mechanism is either normal or anomalous, is a widely observed phenomenon. Starting with an underlying continuous-time random-walk model, we derive the boundary conditions for the diffusion equations describing this problem. We discuss a simple method showing how the boundary conditions can be determined from equilibrium experiments. When the diffusion processes are close to thermal equilibrium, the boundary conditions are determined by a thermal Boltzmann factor, which in turn controls the solution of the problem.
Collapse
Affiliation(s)
- Nickolay Korabel
- Physics Department, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, Israel
| | | |
Collapse
|
25
|
Saa A, Venegeroles R. Ergodic transitions in continuous-time random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:031110. [PMID: 21230028 DOI: 10.1103/physreve.82.031110] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2010] [Indexed: 05/30/2023]
Abstract
We consider continuous-time random walk models described by arbitrary sojourn time probability density functions. We find a general expression for the distribution of time-averaged observables for such systems, generalizing some recent results presented in the literature. For the case where sojourn times are identically distributed independent random variables, our results shed some light on the recently proposed transitions between ergodic and weakly nonergodic regimes. On the other hand, for the case of nonidentical trapping time densities over the lattice points, the distribution of time-averaged observables reveals that such systems are typically nonergodic, in agreement with some recent experimental evidences on the statistics of blinking quantum dots. Some explicit examples are considered in detail. Our results are independent of the lattice topology and dimensionality.
Collapse
Affiliation(s)
- Alberto Saa
- Departamento de Matemática Aplicada, UNICAMP, 13083-859 Campinas, SP, Brazil.
| | | |
Collapse
|
26
|
Bel G, Munsky B, Nemenman I. The simplicity of completion time distributions for common complex biochemical processes. Phys Biol 2009; 7:016003. [PMID: 20026876 DOI: 10.1088/1478-3975/7/1/016003] [Citation(s) in RCA: 51] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
Biochemical processes typically involve huge numbers of individual reversible steps, each with its own dynamical rate constants. For example, kinetic proofreading processes rely upon numerous sequential reactions in order to guarantee the precise construction of specific macromolecules. In this work, we study the transient properties of such systems and fully characterize their first passage (completion) time distributions. In particular, we provide explicit expressions for the mean and the variance of the completion time for a kinetic proofreading process and computational analyses for more complicated biochemical systems. We find that, for a wide range of parameters, as the system size grows, the completion time behavior simplifies: it becomes either deterministic or exponentially distributed, with a very narrow transition between the two regimes. In both regimes, the dynamical complexity of the full system is trivial compared to its apparent structural complexity. Similar simplicity is likely to arise in the dynamics of many complex multistep biochemical processes. In particular, these findings suggest not only that one may not be able to understand individual elementary reactions from macroscopic observations, but also that such an understanding may be unnecessary.
Collapse
Affiliation(s)
- Golan Bel
- Center for Nonlinear Studies and the Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
| | | | | |
Collapse
|
27
|
Munsky B, Nemenman I, Bel G. Specificity and completion time distributions of biochemical processes. J Chem Phys 2009; 131:235103. [DOI: 10.1063/1.3274803] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
|
28
|
Esposito M, Lindenberg K. Continuous-time random walk for open systems: fluctuation theorems and counting statistics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:051119. [PMID: 18643038 DOI: 10.1103/physreve.77.051119] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/05/2008] [Indexed: 05/26/2023]
Abstract
We consider continuous-time random walks (CTRW) for open systems that exchange energy and matter with multiple reservoirs. Each waiting time distribution (WTD) for times between steps is characterized by a positive parameter alpha , which is set to alpha=1 if it decays at least as fast as t{-2} at long times and therefore has a finite first moment. A WTD with alpha<1 decays as t{-alpha-1} . A fluctuation theorem for the trajectory quantity R , defined as the logarithm of the ratio of the probability of a trajectory and the probability of the time reversed trajectory, holds for any CTRW. However, R can be identified as a trajectory entropy change only if the WTDs have alpha=1 and satisfy separability (also called "direction time independence"). For nonseparable WTDs with alpha=1 , R can only be identified as a trajectory entropy change at long times, and a fluctuation theorem for the entropy change then only holds at long times. For WTDs with 0<alpha<1 no meaningful fluctuation theorem can be derived. We also show that the (experimentally accessible) nth moments of the energy and matter transfers between the system and a given reservoir grow as t{nalpha} at long times.
Collapse
Affiliation(s)
- Massimiliano Esposito
- Department of Chemistry and Biochemistry and Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0340, USA
| | | |
Collapse
|
29
|
Shushin AI. Non-Markovian stochastic Liouville equation and its Markovian representation: Extensions of the continuous-time random-walk approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:031130. [PMID: 18517352 DOI: 10.1103/physreve.77.031130] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2007] [Revised: 09/10/2007] [Indexed: 05/26/2023]
Abstract
Some specific features and extensions of the continuous-time random-walk (CTRW) approach are analyzed in detail within the Markovian representation (MR) and CTRW-based non-Markovian stochastic Liouville equation (SLE). In the MR, CTRW processes are represented by multidimensional Markovian ones. In this representation the probability density function (PDF) W(t) of fluctuation renewals is associated with that of reoccurrences in a certain jump state of some Markovian controlling process. Within the MR the non-Markovian SLE, which describes the effect of CTRW-like noise on the relaxation of dynamic and stochastic systems, is generalized to take into account the influence of relaxing systems on the statistical properties of noise. Some applications of the generalized non-Markovian SLE are discussed. In particular, it is applied to study two modifications of the CTRW approach. One of them considers cascaded CTRWs in which the controlling process is actually a CTRW-like one controlled by another CTRW process, controlled in turn by a third one, etc. Within the MR a simple expression for the PDF W(t) of the total controlling process is obtained in terms of Markovian variants of controlling PDFs in the cascade. The expression is shown to be especially simple and instructive in the case of anomalous processes determined by the long-time tailed W(t) . The cascaded CTRWs can model the effect of the complexity of a system on the relaxation kinetics (in glasses, fractals, branching media, ultrametric structures, etc.). Another CTRW modification describes the kinetics of processes governed by fluctuating W(t) . Within the MR the problem is analyzed in a general form without restrictive assumptions on the correlations of PDFs of consecutive renewals. The analysis shows that fluctuations of W(t) can strongly affect the kinetics of the process. Possible manifestations of this effect are discussed.
Collapse
Affiliation(s)
- A I Shushin
- Institute of Chemical Physics, Russian Academy of Sciences, 117977, GSP-1, Kosygin Street 4, Moscow, Russia
| |
Collapse
|
30
|
Rebenshtok A, Barkai E. Distribution of time-averaged observables for weak ergodicity breaking. PHYSICAL REVIEW LETTERS 2007; 99:210601. [PMID: 18233203 DOI: 10.1103/physrevlett.99.210601] [Citation(s) in RCA: 79] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2007] [Indexed: 05/25/2023]
Abstract
We find a general formula for the distribution of time-averaged observables for systems modeled according to the subdiffusive continuous time random walk. For Gaussian random walks coupled to a thermal bath we recover ergodicity and Boltzmann's statistics, while for the anomalous subdiffusive case a weakly nonergodic statistical mechanical framework is constructed, which is based on Lévy's generalized central limit theorem. As an example we calculate the distribution of X, the time average of the position of the particle, for unbiased and uniformly biased particles, and show that X exhibits large fluctuations compared with the ensemble average <X>.
Collapse
Affiliation(s)
- A Rebenshtok
- Department of Physics, Bar Ilan University, Ramat-Gan 52900 Israel
| | | |
Collapse
|
31
|
Lomholt MA, Zaid IM, Metzler R. Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary. PHYSICAL REVIEW LETTERS 2007; 98:200603. [PMID: 17677681 DOI: 10.1103/physrevlett.98.200603] [Citation(s) in RCA: 44] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2007] [Indexed: 05/16/2023]
Abstract
We derive the boundary condition for a subdiffusive particle interacting with a reactive boundary with a finite reaction rate. Molecular crowding conditions, that are found to cause subdiffusion of larger molecules in biological cells, are shown to effect long-tailed distributions with an identical exponent for both the unbinding times from the boundary to the bulk and the rebinding times from the bulk. This causes a weak ergodicity breaking: typically, an individual particle either stays bound or remains in the bulk for very long times. We discuss why this may be beneficial for in vivo gene regulation by DNA-binding proteins, whose typical concentrations are nanomolar.
Collapse
Affiliation(s)
- Michael A Lomholt
- Physics Department, University of Ottawa, Pavillon MacDonald, Ottawa, Ontario K1N 6N5, Canada
| | | | | |
Collapse
|
32
|
Margolin G, Protasenko V, Kuno M, Barkai E. Photon Counting Statistics for Blinking CdSe−ZnS Quantum Dots: A Lévy Walk Process. J Phys Chem B 2006; 110:19053-60. [PMID: 16986903 DOI: 10.1021/jp061487m] [Citation(s) in RCA: 61] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
We analyze photon statistics of blinking CdSe-ZnS nanocrystals interacting with a continuous wave laser field, showing that the process is described by a ballistic Lévy walk. In particular, we show that Mandel's Q parameter, describing the fluctuations of the photon counts, is increasing with time even in the limit of long time. This behavior is in agreement with the theory of Silbey and co-workers (Jung et al. Chem. Phys. 2002, 284, 181), and in contrast to all existing examples where Q approaches a constant, independent of time in the long time limit. We then analyze the distribution of the time averaged intensities, showing that they exhibit a nonergodic behavior, namely, the time averages remain random even in the limit of a long measurement time. In particular, the distribution of occupation times in the on-state compares favorably to a theory of weak ergodicity breaking of blinking nanocrystals. We show how our data analysis yields information on the amplitudes of power-law decaying on and off time distributions, information not available using standard data analysis of on and off time histograms. Photon statistics reveals fluctuations in the intensity of the bright state indicating that it is composed of several states. Photon statistics exhibits a Lévy walk behavior also when an ensemble of 100 dots is investigated, indicating that the strange kinetics can be observed already at the level of small ensembles.
Collapse
Affiliation(s)
- G Margolin
- Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46556, USA
| | | | | | | |
Collapse
|