Continuous Time Random Walk Processes Research Articles

Dynamics of Dual‐Mode Bedload Transport With Three‐Dimensional Alternate Bars Migration in Subcritical Flow: Experiments and Model Analysis

AbstractBedload transport often exhibits dual‐mode behavior due to interactions of spatiotemporal controlling factors with the migrating three‐dimensional bedforms (characterized by the fully developed patterns in the bed, such as alternate bars, pools, and clusters). This study explores dual‐mode bedload transport based on experimental measurements and develops Einstein's exponential‐based model to characterize large fluctuations of bedload sediment discharge. The particle waiting time, particle flux, and bed elevation are measured in a series of well‐controlled laboratory experiments. Flume experiments show that the waiting time distribution of sediments gives a bimodal characteristic, two distinct modes can be identified from the measured data. This study encapsulates this dual‐mode bedload transport behavior in a hyperexponential distribution of sediment resting times and introduces it into the continuous time random walk (CTRW) framework. Considering the scaling limit of the thin/heavy‐tailed CTRW processes, a single‐rate mass transfer (SRMT) and fractional‐derivative SRMT (F‐SRMT) models are obtained, and the model parameters are determined from the hyperexponential distribution. Further analyses reveal that the dual‐mode bedload transport behavior is controlled by mass exchange between the mobile and immobile zones, and a dimensionless index η can quantify the intensity of dual‐mode behavior. Applications show that the dual‐mode bedload transport models are much more accurate in characterizing bedload transport in a mixed‐size gravel bed than the traditional exponential‐based model, and the nonlocal movement of bedload sediments is significant in the mixed‐size gravel bed. Further investigations will focus on the applicability test of the dual‐mode models to other flow regimes and conditions.

A real-time data assimilative forecasting system for animal tracking.

Monitoring technologies now provide real‐time animal location information, which opens up the possibility of developing forecasting systems to fuse these data with movement models to predict future trajectories. State‐space modeling approaches are well established for retrospective location estimation and behavioral inference through state and parameter estimation. Here we use a state‐space model within a comprehensive data assimilative framework for probabilistic animal movement forecasting. Real‐time location information is combined with stochastic movement model predictions to provide forecasts of future animal locations and trajectories, as well as estimates of key behavioral parameters. Implementation uses ensemble‐based sequential Monte Carlo methods (a particle filter). We first apply the framework to an idealized case using a nondimensional animal movement model based on a continuous‐time random walk process. A set of numerical forecasting experiments demonstrates the workflow and key features, such as the online estimation of behavioral parameters using state augmentation, the use of potential functions for habitat preference, and the role of observation error and sampling frequency on forecast skill. For a realistic demonstration, we adapt the framework to short‐term forecasting of the endangered southern resident killer whale (SRKW) in the Salish Sea using visual sighting information wherein the potential function reflects historical habitat utilization of SRKW. We successfully estimate whale locations up to 2.5 h in advance with a moderate prediction error (<5 km), providing reasonable lead‐in time to mitigate vessel–whale interactions. It is argued that this forecasting framework can be used to synthesize diverse data types and improve animal movement models and behavioral understanding and has the potential to lead to important advances in movement ecology.

Infinite density and relaxation for Lévy walks in an external potential: Hermite polynomial approach.

Lévy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding qth-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed v(τ)=v_{0}/sqrt[τ]. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly.

Enhanced Facilitated Diffusion of Membrane-Associating Proteins under Symmetric Confinement.

The facilitated surface diffusion of transiently adsorbing molecules in a planar confined microenvironment (i.e., slit-like confinement) is highly relevant to biological phenomena, such as extracellular signaling, as well as numerous biotechnology systems. Here, we studied the surface diffusion of individual proteins confined between two symmetric lipid bilayer membranes, under a continuum of confinement heights, using single-molecule tracking and convex lens-induced confinement as well as hybrid, kinetic Monte Carlo simulations of a generalized continuous time random walk process. Surface diffusion was observed to vary non-monotonically with confinement height, exhibiting a maximum at a height of ∼750 nm, where diffusion was nearly 40% greater than that for a semi-infinite system. This demonstrated that planar confinement can, in fact, increase surface diffusion, qualitatively validating previous theoretical predictions. Simulations reproduced the experimental results and suggested that confinement enhancement of surface diffusion for symmetric systems is limited to cases where the adsorbate exhibits weak surface sticking.

From continuous-time random walks to the fractional Jeffreys equation: Solution and properties

Jeffreys equation provides an increasingly popular extension of the diffusive laws of Fourier and Fick for heat and particle transport. Similar to generalisations of the diffusion equation, we here investigate the connection between a time-fractional generalisation of the Jeffreys equation and a continuous-time random walk process based on a generalised waiting time density with diverging mean. We demonstrate that the mean squared displacement exhibits a variety of anomalous behaviors, such as retarding and accelerating subdiffusion, as well as a crossover from superdiffusion to subdiffusion. Moreover, we provide two alternative approaches, namely, a fractional Taylor series and distributed-order derivatives, that transform Fourier’s or Fick’s law into the time-fractional Jeffreys equation. Our discussion provides physics-based support for the fractional Jeffreys equation and shows its versatility for practical applications.

Continuous-time random walks and Lévy walks with stochastic resetting

Intermittent stochastic processes appear in a wide field, such as chemistry, biology, ecology, and computer science. This paper builds up the theory of intermittent continuous time random walk (CTRW) and L\'{e}vy walk, in which the particles are stochastically reset to a given position with a resetting rate $r$. The mean squared displacements of the CTRW and L\'{e}vy walks with stochastic resetting are calculated, uncovering that the stochastic resetting always makes the CTRW process localized and L\'{e}vy walk diffuse slower. The asymptotic behaviors of the probability density function of L\'evy walk with stochastic resetting are carefully analyzed under different scales of $x$, and a striking influence of stochastic resetting is observed.

Anomalous intracellular transport phases depend on cytoskeletal network features.

Intracellular transport in eukaryotic cells consists of phases of passive, diffusion-based transport and active, motor-driven transport along filaments that make up the cell's cytoskeleton. The interplay between superdiffusive transport along cytoskeletal filaments and the anomalous nature of subdiffusion in the bulk can lead to novel effects in transport behavior at the cellular scale. Here we develop a computational model of the process with cargo being ballistically transported along explicitly modeled cytoskeletal filament networks and passively transported in the cytoplasm by a subdiffusive continuous-time random walk (CTRW). We show that, over a physiologically relevant range of filament lengths and numbers, the network introduces a filament-length sensitive superdiffusive phase at early times which crosses over to a phase where the CTRW is dominant and produces subdiffusion at late times. We apply our approach to the problem of insulin secretion from cells and show that the superdiffusive phase introduced by the filament network manifests as a peak in the secretion at early times followed by an extended sustained release phase that is dominated by the CTRW process at late times. Our results are consistent with in vivo observations of insulin transport in healthy cells and shed light on the potential for the cell to tune functionally important transport phases by altering its cytoskeletal network.

Ageing effects in ultraslow continuous time random walks

In ageing systems physical observables explicitly depend on the time span elapsing between the original initiation of the system and the actual start of the recording of the particle motion. We here study the signatures of ageing in the framework of ultraslow continuous time random walk processes with super-heavy tailed waiting time densities. We derive the density for the forward or recurrent waiting time of the motion as function of the ageing time, generalise the Montroll–Weiss equation for this process, and analyse the ageing behaviour of the ensemble and time averaged mean squared displacements.

Ageing first passage time density in continuous time random walks and quenched energy landscapes

We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function (). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time ta, the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs with describe systems such as charge carriers in amorphous semiconducters, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range , for sufficiently long ta. We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the ageing CTRW results. With small bias or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape.

Aging scaled Brownian motion.

Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.

Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes.

We study the stochastic behavior of heterogeneous diffusion processes with the power-law dependence D(x) ∼ |x|(α) of the generalized diffusion coefficient encompassing sub- and superdiffusive anomalous diffusion. Based on statistical measures such as the amplitude scatter of the time-averaged mean-squared displacement of individual realizations, the ergodicity breaking and non-Gaussianity parameters, as well as the probability density function P(x,t), we analyze the weakly nonergodic character of the heterogeneous diffusion process and, particularly, the degree of irreproducibility of individual realizations. As we show, the fluctuations between individual realizations increase with growing modulus |α| of the scaling exponent. The fluctuations appear to diverge when the critical value α = 2 is approached, while for even larger α the fluctuations decrease, again. At criticality, the power-law behavior of the mean-squared displacement changes to an exponentially fast growth, and the fluctuations of the time-averaged mean-squared displacement do not converge for increasing number of realizations. From a systematic comparison we observe some striking similarities of the heterogeneous diffusion process with the familiar subdiffusive continuous time random walk process with power-law waiting time distribution and diverging characteristic waiting time.

First-passage statistics for aging diffusion in systems with annealed and quenched disorder.

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.

Correlated continuous time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics

Standard continuous time random walk (CTRW) models are renewal processes in the sense that at each jump a new, independent pair of jump length and waiting time are chosen. Globally, anomalous diffusion emerges through scale-free forms of the jump length and/or waiting time distributions by virtue of the generalized central limit theorem. Here we present a modified version of recently proposed correlated CTRW processes, where we incorporate a power-law correlated noise on the level of both jump length and waiting time dynamics. We obtain a very general stochastic model, that encompasses key features of several paradigmatic models of anomalous diffusion: discontinuous, scale-free displacements as in Lévy flights, scale-free waiting times as in subdiffusive CTRWs, and the long-range temporal correlations of fractional Brownian motion (FBM). We derive the exact solutions for the single-time probability density functions and extract the scaling behaviours. Interestingly, we find that different combinations of the model parameters lead to indistinguishable shapes of the emerging probability density functions and identical scaling laws. Our model will be useful for describing recent experimental single particle tracking data that feature a combination of CTRW and FBM properties.

Asymptotic behaviour of random walks with correlated temporal structure.

We introduce a continuous-time random walk process with correlated temporal structure. The dependence between consecutive waiting times is generated by weighted sums of independent random variables combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. We derive the corresponding diffusion limit and prove its subdiffusive character. Analysing the set of corresponding coupled Langevin equations,we verify the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking.

Distributions of time averages for weakly chaotic systems: The role of infinite invariant density

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.

Anomalous beam diffusion near beam-beam synchrobetatron resonances

The diffusion process near low order synchro-betatron resonances driven by beam-beam interactions at a crossing angle is investigated. Macroscopic observables such as beam emittance, lifetime and beam profiles are calculated. These are followed with detailed studies of microscopic quantities such as the evolution of the variance at several initial transverse amplitudes and single particle probability distribution functions. We present evidence to show that the observed diffusion is anomalous and the dynamics follows a non-Markovian continuous time random walk process. We derive a modified master equation to replace the Chapman-Kolmogorov equation in action-angle space and a fractional diffusion equation to describe the density evolution for this class of processes.

Dephasing by a continuous-time random walk process

Stochastic treatments of magnetic resonance spectroscopy and optical spectroscopy require evaluations of functions such as (exp(i ∫(0)(t) Q(s)ds)), where t is time, Q(s) is the value of a stochastic process at time s, and the angular brackets denote ensemble averaging. This paper gives an exact evaluation of these functions for the case where Q is a continuous-time random walk process. The continuous-time random walk describes an environment that undergoes slow steplike changes in time. It also has a well-defined Gaussian limit and so allows for non-Gaussian and Gaussian stochastic dynamics to be studied within a single framework. We apply the results to extract qubit-lattice interaction parameters from dephasing data of P-doped Si semiconductors (data collected elsewhere) and to calculate the two-dimensional spectrum of a three-level harmonic oscillator undergoing random frequency modulations.

Exit times in non-Markovian drifting continuous-time random-walk processes

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.

Exit Times in Non-Markovian Drifting Continuous-Time Random Walk Processes

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.

Fluorescence Correlation Spectroscopy: The Case of Subdiffusion

The theory of fluorescence correlation spectroscopy is revisited here for the case of subdiffusing molecules. Subdiffusion is assumed to stem from a continuous-time random walk process with a fat-tailed distribution of waiting times and can therefore be formulated in terms of a fractional diffusion equation (FDE). The FDE plays the central role in developing the fluorescence correlation spectroscopy expressions, analogous to the role played by the simple diffusion equation for regular systems. Due to the nonstationary nature of the continuous-time random walk/FDE, some interesting properties emerge that are amenable to experimental verification and may help in discriminating among subdiffusion mechanisms. In particular, the current approach predicts 1), a strong dependence of correlation functions on the initial time (aging); 2), sensitivity of correlation functions to the averaging procedure, ensemble versus time averaging (ergodicity breaking); and 3), that the basic mean-squared displacement observable depends on how the mean is taken.