Ergodic properties of fractional Brownian-Langevin motion (original) (raw)
Physical Review Letters, 2009
Fractional Brownian motion with Hurst index less then 1=2 and continuous-time random walk with heavy tailed waiting times (and the corresponding fractional Fokker-Planck equation) are two different processes that lead to a subdiffusive behavior widespread in complex systems. We propose a simple test, based on the analysis of the so-called p variations, which allows distinguishing between the two models on the basis of one realization of the unknown process. We apply the test to the data of Golding and Cox [Phys. Rev. Lett. 96, 098102 (2006)], describing the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells. It is found that the data does not follow heavy tailed continuous-time random walk. The test shows that it is likely that fractional Brownian motion is the underlying process.
Physical Review E, 2013
We analyze a class of estimators of the generalized diffusion coefficient for fractional Brownian motion B t of known Hurst index H , based on weighted functionals of the single-time square displacement. We show that for a certain choice of the weight function these functionals possess an ergodic property and thus provide the true ensemble-average generalized diffusion coefficient to any necessary precision from single-trajectory data, but at the expense of a progressively higher experimental resolution. Convergence is fastest around H 0.30, a value in the subdiffusive regime.
Fractional Brownian motion and generalized Langevin equation motion in confined geometries
2010
Motivated by subdiffusive motion of bio-molecules observed in living cells we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. We investigate by analytic calculations and simulations how time-averaged observables (e.g., the time averaged mean squared displacement and displacement correlation) are affected by spatial confinement and dimensionality. In particular we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain. The general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume, and with increasing dimensionality. We define the displacement correlation function and find that this quantity shows distinct features for fractional Brownian motion, fractional Langevin equation, and continuous time subdiffusion, such that it appears an efficient measure to distinguish these different processes based on single particle trajectory data.
Fractional kinetics emerging from ergodicity breaking in random media
Physical review. E, 2016
We present a modeling approach for diffusion in a complex medium characterized by a random length scale. The resulting stochastic process shows subdiffusion with a behavior in qualitative agreement with single-particle tracking experiments in living cells, such as ergodicity breaking, p variation, and aging. In particular, this approach recapitulates characteristic features previously described in part by the fractional Brownian motion and in part by the continuous-time random walk. Moreover, for a proper distribution of the length scale, a single parameter controls the ergodic-to-nonergodic transition and, remarkably, also drives the transition of the diffusion equation of the process from nonfractional to fractional, thus demonstrating that fractional kinetics emerges from ergodicity breaking.
Dynamics of Fractional Brownian Walks
The Journal of Physical Chemistry, 1995
We investigate the dynamics of polymers whose solution configurations are represented by fractional Brownian walks. The calculation of the two dynamical quantities considered here, the longest relaxation time t, and the intrinsic viscosity [VI, is formulated in terms of Langevin equations and is carried out within the continuum approach developed in an earlier paper. Our results for t, and [v] reproduce known scaling relations and provide reasonable numerical estimates of scaling amplitudes. The possible relevance of the work to the study of globular proteins and other compact polymeric phases is discussed.
Physical Review E
Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x) = D 0 |x| α. Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble-and time-averaged mean-squared displacements couple the scaling exponents α of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y ∼ |x| 1/(2/(2−α)) /t H coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), P HDP-FBM (y) = e −y 2 / √ π. Its universal shape agrees well with theoretical predictions for both uni-and bimodal PDF distributions.
Fractional Brownian motion as a nonstationary process: An alternative paradigm for DNA sequences
Physical Review E, 1998
The long-range correlations in DNA sequences are currently interpreted as an example of stationary fractional Brownian motion ͑FBM͒. First we show that the dynamics of a dichotomous stationary process with long-range correlations such as that used to model DNA sequences should correspond to Lévy statistics and not to FBM. To explain why, in spite of this, the statistical analysis of the data seems to be compatible with FBM, we notice that an initial Gaussian condition, generated by a process foreign to the mechanism establishing the long-range correlations and consequently implying a departure from the stationary condition, is maintained approximately unchanged for very long times. This is so because due to the nature itself of the long-range correlation process, it takes virtually an infinite time for the system to reach the genuine stationary state. Then we discuss a possible generator of initial Gaussian conditions, based on a folding mechanism of the nucleic acid in the cell nucleus. The model adopted is compatible with the known biological and physical constraints, namely, it is shown to be consistent with the information of current biological literature on folding as well as with the statistical analyses of DNA sequences.
The random walk's guide to anomalous diffusion: a fractional dynamics approach
Physics Reports, 2000
What can fractional equations do, what can they do better, and why should one care at all? 1 1.2. What is the scope of this report? 5 2. Introduction 5 2.1. Anomalous dynamics in complex systems 5 2.2. Historical remarks 7 2.3. Anomalous di!usion: experiments and models 9 3. From continuous time random walk to fractional di!usion equations 13 3.1. Revisiting the realm of Brownian motion 14 3.2. The continuous time random walk model 15 3.3. Back to Brownian motion 17 3.4. Long rests: a fractional di!usion equation describing subdi!usion 18 3.5. Long jumps: LeH vy #ights 25 3.6. The competition between long rests and long jumps 29 3.7. What's the course, helmsman? 30 4. Fractional di!usion}advection equations 31 4.1. The Galilei invariant fractional di!usion}advection equation 32 4.2. The Gallilei variant fractional di!usion}advection equation 4.3. Alternative approaches for LeH vy #ights 5. The fractional Fokker}Planck equation: anomalous di!usion in an external force "eld 5.1. The Fokker}Planck equation 5.2. The fractional Fokker}Planck equation 5.3. Separation of variables and the fractional Ornstein}Uhlenbeck process 5.4. The connection between the fractional solution and its Brownian counterpart 5.5. The fractional analogue of Kramers escape theory from a potential well 5.6. The derivation of the fractional Fokker}Planck equation 5.7. A fractional Fokker}Planck equation for LeH vy #ights 5.8. A generalised Kramers}Moyal expansion 6. From the Langevin equation to fractional di!usion: microscopic foundation of dispersive transport close to thermal equilibrium 6.1. Langevin dynamics and the three stages to subdi!usion
New Journal of Physics, 2019
Fractional Brownian motion is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically fractional Brownian motion confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected fractional Brownian motion at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement X 2 (t) t α with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The mean squared displacement in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of fractional Brownian motion for processes such as molecule or tracer diffusion in the confined of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.
Physical Chemistry Chemical Physics, 2011
Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristics even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytical expressions for the velocity correlation functions. The knowledge of the results presented here is expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.
Physical Chemistry Chemical Physics, 2014
Anomalous diffusion is frequently described by scaled Brownian motion (SBM), a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is x 2 (t) ≃ K (t)t with K (t) ≃ t α−1 for 0 < α < 2. SBM may provide a seemingly adequate description in the case of unbounded diffusion, for which its probability density function coincides with that of fractional Brownian motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a significant amplitude scatter of the time averaged mean squared displacement. More severely, we demonstrate that under confinement, the dynamics encoded by SBM is fundamentally different from both fractional Brownian motion and continuous time random walks. SBM is highly non-stationary and cannot provide a physical description for particles in a thermalised stationary system. Our findings have direct impact on the modelling of single particle tracking experiments, in particular, under confinement inside cellular compartments or when optical tweezers tracking methods are used.
Statistical analysis of superstatistical fractional Brownian motion and applications
Physical Review E
Recent advances in experimental techniques for complex systems and the corresponding theoretical findings show that in many cases random parametrization of the diffusion coefficients gives adequate descriptions of the observed fractional dynamics. In this paper we introduce two statistical methods which can be effectively applied to analyze and estimate parameters of superstatistical fractional Brownian motion with random scale parameter. The first method is based on the analysis of the increments of the process, the second one takes advantage of the variation of the trajectories of the process. We prove the effectiveness of the methods using simulated data. Also, we apply it to the experimental data describing random motion of individual molecules inside the cell of E.coli. We show that fractional Brownian motion with Weibull-distributed diffusion coefficient gives adequate description of this experimental data.
Physical review. E, 2016
In this paper, we study ergodic properties of α-stable autoregressive fractionally integrated moving average (ARFIMA) processes which form a large class of anomalous diffusions. A crucial practical question is how long trajectories one needs to observe in an experiment in order to claim that the analyzed data are ergodic or not. This will be solved by checking the asymptotic convergence to 0 of the empirical estimator F(n) for the dynamical functional D(n) defined as a Fourier transform of the n-lag increments of the ARFIMA process. Moreover, we introduce more flexible concept of the ε-ergodicity.
Brownian motion and anomalous diffusion revisited via a fractional Langevin equation
2010
In this paper we revisit the Brownian motion on the basis of {the fractional Langevin equation which turns out to be a particular case of the generalized Langevin equation introduced by Kubo in 1966. The importance of our approach is to model the Brownian motion more realistically than the usual one based on the classical Langevin equation, in that it
Single particle tracking has become a standard tool to investigate diffusive properties, especially in small systems such as biological cells. Usually the resulting time series are analyzed in terms of time averages over individual trajectories. Here we study confined normal as well as anomalous diffusion modeled by fractional Brownian motion and the fractional Langevin equation, and show that even for such ergodic systems time-averaged quantities behave differently from their ensemble averaged counterparts, irrespective of how long the measurement time becomes. Knowledge of the exact behavior of time averages is therefore fundamental for the proper physical interpretation of measured time series, in particular, for extraction of the relaxation time scale from data.
Fractional diffusion Processes: Probability Distributions and Continuous Time Random Walk
Lecture Notes in Physics, 2003
A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the spacetime fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 1] . The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process. We view it as a generalized diffusion process that we call fractional diffusion process, and present an integral representation of the fundamental solution. A more general approach to anomalous diffusion is however known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.
A Dynamical Approach to Fractional Brownian Motion
Fractals, 1994
Herein we develop a dynamical foundation for fractional Brownian motion. A clear relation is established between the asymptotic behavior of the correlation function and diffusion in a dynamical system. Then, assuming that scaling is applicable, we establish a connection between diffusion (either standard or anomalous) and the dynamical indicator known as the Hurst coefficient. We argue on the basis of numerical simulations that although we have been able to prove scaling only for…
Physical Review E, 2012
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 τc which can greatly exceed mean residence time in any trap, τc ≫ τ , and even not being related to it. Asymptotically, on a macroscale transport becomes normal for t ≫ τc. However, mesoscopic transport is anomalous. Differently from viscoelastic subdiffusion no long-range anti-correlations 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 of macroscopic viscosity of cytoplasm is finite.
Spurious ergodicity breaking in normal and fractional Ornstein–Uhlenbeck process
New Journal of Physics, 2020
The Ornstein-Uhlenbeck process is a stationary and ergodic Gaussian process, that is fully determined by its covariance function and mean. We show here that the generic definitions of the ensemble-and time-averaged mean squared displacements fail to capture these properties consistently, leading to a spurious ergodicity breaking. We propose to remedy this failure by redefining the mean squared displacements such that they reflect unambiguously the statistical properties of any stochastic process. In particular we study the effect of the initial condition in the Ornstein-Uhlenbeck process and its fractional extension. For the fractional Ornstein-Uhlenbeck process representing typical experimental situations in crowded environments such as living biological cells, we show that the stationarity of the process delicately depends on the initial condition.