Time-averaged Mean Square Displacement Research Articles

Scaled Brownian motion with random anomalous diffusion exponent

The scaled Brownian motion (SBM) is regarded as one of the paradigmatic random processes, characterized by anomalous diffusion through the diffusion exponent. It is a Gaussian, self-similar process with independent increments and has found applications across various fields, from turbulence and stochastic hydrology to biophysics. In our paper, inspired by recent single particle tracking biological experiments, we introduce a process called scaled Brownian motion with random exponent (SBMRE), which retains the key features of SBM at the level of individual trajectories, but with anomalous diffusion exponents that vary randomly across trajectories.We discuss the main probabilistic properties of SBMRE, including its probability density function (pdf) and the qth absolute moment. Additionally, we present the expected value of the time-averaged mean squared displacement (TAMSD) and the ergodicity breaking parameter. Furthermore, we analyze the pdf of the first hitting time in a semi-infinite domain, the martingale property of SBMRE, and its stochastic exponential. As special cases, we consider two distributions for the anomalous diffusion exponent, namely the mixture of two point distributions and beta distribution, and explore the asymptotics of the corresponding characteristics. Theoretical results for SBMRE are validated through numerical simulations and compared with the analogous characteristics of SBM.

Langevin Equation in Heterogeneous Landscapes: How to Choose the Interpretation.

The Langevin equation is a common tool to model diffusion at a single-particle level. In nonhomogeneous environments, such as aqueous two-phase systems or biological condensates with different diffusion coefficients in different phases, the solution to a Langevin equation is not unique unless the interpretation of stochastic integrals involved is selected. We analyze the diffusion of particles in such systems and evaluate the mean, the mean square displacement, and the distribution of particles, as well as the variance of the time-averaged mean-square displacements. Our analytical results provide a method to choose the interpretation parameter from single-particle tracking experiments.

Extreme heterogeneity in the microrheology of lamellar surfactant gels analyzed with neural networks.

The heterogeneity of the viscoelasticity of a lamellar gel network based on cetyl-trimethylammonium chloride and cetostearyl alcohol was studied using particle-tracking microrheology. A recurrent neural network (RNN) architecture was used for estimating the Hurst exponent, H, on small sectionsof tracks of probe spheres moving with fractional Brownian motion. Thus, dynamic segmentation of tracks via neural networks was used in microrheology and it is significantly more accurate than using mean square displacements (MSDs). An ensemble of 414 particles produces a MSD that is subdiffusive in time, t, with a power law of the form t^{0.74±0.02}, indicating power law viscoelasticity. RNN analysis of the probability distributions of H, combined with detailed analysis of the time-averaged MSDs of individual tracks, revealed diverse diffusion processes belied by the simple scaling of the ensemble MSD, such as caging phenomena, which give rise to the complex viscoelasticity of lamellar gels.

Aging and confinement in subordinated fractional Brownian motion.

We study the effects of aging properties of subordinated fractional Brownian motion (FBM) with drift and in harmonic confinement, when the measurement of the stochastic process starts a time t_{a}>0 after its original initiation at t=0. Specifically, we consider the aged versions of the ensemble mean-squared displacement (MSD) and the time-averaged MSD (TAMSD), along with the aging factor. Our results are favorably compared with simulations results. The aging subordinated FBM exhibits a disparity between MSD and TAMSD and is thus weakly nonergodic, while strong aging is shown to effect a convergence of the MSD and TAMSD. The information on the aging factor with respect to the lag time exhibits an identical form to the aging behavior of subdiffusive continuous-time random walks (CTRW). The statistical properties of the MSD and TAMSD for the confined subordinated FBM are also derived. At long times, the MSD in the harmonic potential has a stationary value, that depends on the Hurst index of the parental (nonequilibrium) FBM. The TAMSD of confined subordinated FBM does not relax to a stationary value but increases sublinearly with lag time, analogously to confined CTRW. Specifically, short aging times t_{a} in confined subordinated FBM do not affect the aged MSD, while for long aging times the aged MSD has a power-law increase and is identical to the aged TAMSD.

Aging and confinement in subordinated fractional Brownian motion

We study the effects of aging properties of subordinated fractional Brownian motion (FBM) with drift and in harmonic confinement, when the measurement of the stochastic process starts a time ta > 0 after its original initiation at t = 0. Specifically, we consider the aged versions of the ensemble mean squared displacement (MSD) and the time averaged MSD (TAMSD). The aging subordinated FBM exhibits a disparity between MSD and TAMSD and is thus weakly nonergodic, while strong aging is shown to effect a convergence of the MSD and TAMSD. At long times, the MSD in the harmonic potential has a stationary value, that depends on the Hurst index of the parental (non-equilibrium) FBM. The TAMSD of confined subordinated FBM does not relax to a stationary value but increases sublinearly with lag time, analogously to confined CTRW. Interestingly, for the confined subordinated FBM, small aging time ta has no effect on the aged MSD, while at large aging time the aged MSD has monotonic increase ultimately with a power-law and is identical to the aged TAMSD.

Landscapes of random diffusivity processes in harmonic potential

A huge number of experimental evidences support that Brownian yet non-Gaussian diffusion prevalently exists in biological and soft matter systems. To interpret the physical mechanism of the Brownian yet non-Gaussian diffusion, the superstatistical approach with random diffusivity and further general random diffusivity processes attract extensive attention. This paper focuses on the anomalous diffusion and nonergodic property of the random diffusivity processes confined in a harmonic potential. Based on the theoretical analyses in the framework of Langevin equation, we find the inherent disagreement between ensemble-averaged mean-squared displacement and time-averaged mean-squared displacement for any form of the random diffusivity D(t), which implies the ergodicity breaking of the confined dynamics. Further, the asymptotic distribution of the time-averaged mean-squared displacement is strictly derived and shows the explicit dependence on the time average of the diffusivity D(t), which means the randomness of the time-averaged mean-squared displacement is completely contributed by the random diffusivity D(t). These results hold true for a general diffusivity D(t), which has been verified through the simulations of three specific kinds of diffusivities D(t).

Viscoelastic active diffusion governed by nonequilibrium fractional Langevin equations: Underdamped dynamics and ergodicity breaking

In this work, we investigate the active dynamics and ergodicity breaking of a nonequilibrium fractional Langevin equation (FLE) with a power-law memory kernel of the form K(t)∼t−(2−2H), where 1/2<H<1 represents the Hurst exponent. The system is subjected to two distinct noises: a thermal noise satisfying the fluctuation–dissipation theorem and an active noise characterized by an active Ornstein–Uhlenbeck process with a propulsion memory time τA. We provide analytic solutions for the underdamped active fractional Langevin equation, performing both analytical and computational investigations of dynamic observables such as velocity autocorrelation, the two-time position correlation, ensemble- and time-averaged mean-squared displacements (MSDs), and ergodicity-breaking parameters. Our results reveal that the interplay between the active noise and long-time viscoelastic memory effect leads to unusual and complex nonequilibrium dynamics in the active FLE systems. Furthermore, the active FLE displays a new type of discrepancy between ensemble- and time-averaged observables. The active component of the system exhibits ultraweak ergodicity breaking where both ensemble- and time-averaged MSDs have the same functional form with unequal amplitudes. However, the combined dynamics of the active and thermal components of the active FLE system are eventually ergodic in the infinite-time limit. Intriguingly, the system has a long-standing ergodicity-breaking state before recovering the ergodicity. This apparent ergodicity-breaking state becomes exceptionally long-lived as H→1, making it difficult to observe ergodicity within practical measurement times. Our findings provide insight into related problems, such as the transport dynamics for self-propelled particles in crowded or polymeric media.

Nonergodicity of confined superdiffusive fractional Brownian motion.

Using stochastic simulations supported by analytics we determine the degree of nonergodicity of box-confined fractional Brownian motion for both sub- and superdiffusive Hurst exponents H. At H>1/2 the nonequivalence of the ensemble- and time-averaged mean-squared displacements (TAMSDs) is found to be most pronounced (with a giant spread of individual TAMSDs at H→1), with two distinct short-lag-time TAMSD exponents.

Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks.

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.

Anomalous diffusion, non-Gaussianity, and nonergodicity for subordinated fractional Brownian motion with a drift.

The stochastic motion of a particle with long-range correlated increments (the moving phase) which is intermittently interrupted by immobilizations (the trapping phase) in a disordered medium is considered in the presence of an external drift. In particular, we consider trapping events whose times follow a scale-free distribution with diverging mean trapping time. We construct this process in terms of fractional Brownian motion with constant forcing in which the trapping effect is introduced by the subordination technique, connecting "operational time" with observable "real time." We derive the statistical properties of this process such as non-Gaussianity and nonergodicity, for both ensemble and single-trajectory (time) averages. We demonstrate nice agreement with extensive simulations for the probability density function, skewness, kurtosis, as well as ensemble and time-averaged mean-squared displacements. We place a specific emphasis on the comparisons between the cases with and without drift.

Time-fractional Caputo derivative versus other integrodifferential operators in generalized Fokker-Planck and generalized Langevin equations.

Fractional diffusion and Fokker-Planck equationsare widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous-time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integrodifferential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We study here the dynamics of the associated generalized Fokker-Planck equationsfrom the perspective of the moments, the time-averaged mean-squared displacements, and the autocovariance functions. We also study generalized Langevin equationsbased on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integrodifferential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integrodifferential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equationsfor the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equationsare unveiled to share similar properties with scaled and fractional Brownian motion, respectively.

Single-particle tracking and machine-learning classification reveals heterogeneous Piezo1 diffusion.

The mechanically-activated ion channel Piezo1 is involved in numerous physiological processes. Piezo1 is activated by diverse mechanical cues and is gated by membrane tension. The channel has been found to be mobile in the plasma membrane. We employed single particle tracking (SPT) of endogenously-expressed, tdTomato-tagged Piezo1 using Total Internal Reflection Fluorescence Microscopy in two cell types, mouse embryonic fibroblasts and liver sinusoidal endothelial cells. Application of SPT unveiled a surprising heterogeneity of Piezo1 mobility in the plasma membrane. Leveraging a machine learning technique, Piezo1 trajectories were sorted into distinct classes (“mobile,” “intermediate,” and “trapped”) by partitioning features that describe the geometric properties of a trajectory. To evaluate the effects of the plasma membrane properties on Piezo1 diffusion, we manipulated membrane composition by depleting or supplementing cholesterol or by adding margaric acid to stiffen the membrane. To address effects of channel activation on Piezo1 mobility, we treated cells with Yoda1, a Piezo1 agonist, and GsMTx-4, a channel inhibitor. We collected thousands of trajectories for each condition, and found that “mobile” Piezo1 in cells supplemented with cholesterol or margaric acid exhibited decreased mobility, whereas Piezo1 in cholesterol-depleted membranes demonstrated increased mobility, compared to their respective controls. Additionally, activation by Yoda1 increased Piezo1 mobility and inhibition by GsMTx-4 decreased Piezo1 mobility compared to their respective controls. The “mobile” trajectories were analyzed further by fitting the time-averaged mean-squared displacement as a function of lag time to a power-law model, revealing Piezo1 consistently exhibits anomalous subdiffusion. This suggests Piezo1 is not freely mobile, but that its mobility may be hindered by subcellular interactions. These studies illuminate the fundamental properties governing Piezo1 diffusion in the plasma membrane and set the stage to determine how specific cellular interactions may influence channel activity and mobility.

Different effects of external force fields on aging Lévy walk.

Aging phenomena have been observed in numerous physical systems. Many statistical quantities depend on the aging time ta for aging anomalous diffusion processes. This paper pays more attention to how an external force field affects the aging Lévy walk. Based on the Langevin picture of the Lévy walk and the generalized Green-Kubo formula, we investigate the quantities that include the ensemble- and time-averaged mean-squared displacements in both weak aging ta≪t and strong aging ta≫t cases and compare them to the ones in the absence of any force field. Two typical force fields, constant force F and time-dependent periodic force F(t)=f0sin⁡(ωt), are considered for comparison. The generalized Einstein relation is also discussed in the case with the constant force. We find that the constant force is the key to causing the aging phenomena and enhancing the diffusion behavior of the aging Lévy walk, while the time-dependent periodic force is not. The different effects of the two kinds of forces on the aging Lévy walk are verified by both theoretical analyses and numerical simulations.

Dynamics of self-propelled tracer particles inside a polymer network.

The transport of tracer particles through mesh-like environments such as biological hydrogels and polymer matrices is ubiquitous in nature. These tracers can be passive, such as colloids, or active (self-propelled), for example, synthetic nanomotors or bacteria. Computer simulations in principle could be extremely useful in exploring the mechanism of the active transport of tracer particles through mesh-like environments. Therefore, we construct a polymer network on a diamond lattice and use computer simulations to investigate the dynamics of spherical self-propelled particles inside the network. Our main objective is to elucidate the effect of the self-propulsion on the tracer particle dynamics as a function of the tracer size and the stiffness of the polymer network. We compute the time-averaged mean-squared displacement (MSD) and the van-Hove correlations of the tracer. On the one hand, in the case of a bigger sticky particle, the caging caused by the network particles wins over the escape assisted by the self-propulsion. This results an intermediate-time subdiffusion. On the other hand, smaller tracers or tracers with high self-propulsion velocities can easily escape from the cages and show intermediate-time superdiffusion. The stiffer the network, the slower the dynamics of the tracer, and bigger tracers exhibit longer lived intermediate time superdiffusion, since the persistence time scales as ∼σ3, where σ is the diameter of the tracer. At the intermediate time, non-Gaussianity is more pronounced for active tracers. At the long time, the dynamics of the tracer, if passive or weakly active, becomes Gaussian and diffusive, but remains flat for tracers with high self-propulsion, accounting for their seemingly unrestricted motion inside the network.

Goodness-of-fit test for stochastic processes using even empirical moments statistic.

In this paper, we introduce a novel framework that allows efficient stochastic process discrimination. The underlying test statistic is based on even empirical moments and generalizes the time-averaged mean-squared displacement framework; the test is designed to allow goodness-of-fit statistical testing of processes with stationary increments and a finite-moment distribution. In particular, while our test statistic is based on a simple and intuitive idea, it enables efficient discrimination between finite- and infinite-moment processes even if the underlying laws are relatively close to each other. This claim is illustrated via an extensive simulation study, e.g., where we confront α-stable processes with stability index close to 2 with their standard Gaussian equivalents. For completeness, we also show how to embed our methodology into the real data analysis by studying the real metal price data.

Time-averaging and nonergodicity of reset geometric Brownian motion with drift.

How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Δ)≠TAMSD(Δ) and Variance(Δ)≠TAMSD(Δ) at short lag times Δ and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Δ/T≪1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.

Universal fluctuations and ergodicity of generalized diffusivity on critical percolation clusters

Despite a long history and a clear overall understanding of properties of random walks on an incipient infinite cluster in percolation, some important information on it seems to be missing in the literature. In the present work, we revisit the problem by performing massive numerical simulations for (sub)diffusion of particles on such clusters. Thus, we discuss the shape of the probability density function of particles’ displacements, and the way it converges to its long-time limiting scaling form. Moreover, we discuss the properties of the mean squared displacement (MSD) of a particle diffusing on the infinite cluster at criticality. This one is known not to be self-averaging. We show that the fluctuations of the MSD in different realizations of the cluster are universal, and discuss the properties of the distribution of these fluctuations. These strong fluctuations coexist with the ergodicity of subdiffusive behavior in the time domain. The dependence of the relative strength of fluctuations in time-averaged MSD on the total trajectory length (total simulation time) is characteristic for diffusion in a percolation system and can be used as an additional test to distinguish this process with disorder-induced memory from processes with otherwise similar behavior, like fractional Brownian motion with the same value of the Hurst exponent.

Attaining ergodicity in a Langevin equation with heterogeneous media

Generalizations of the mean-squared displacement and time-averaged mean-squared displacement are proposed to attaining the ergodicity of a Langevin equation with a broad class of the space-dependent diffusion coefficient.

Restoring ergodicity of stochastically reset anomalous-diffusion processes

How do different reset protocols affect ergodicity of a diffusion process in single-particle-tracking experiments? We here address the problem of resetting of an arbitrary stochastic anomalous-diffusion process (ADP) from the general mathematical points of view and assess ergodicity of such reset ADPs for an arbitrary resetting protocol. The process of stochastic resetting describes the events of the instantaneous restart of a particle's motion via randomly distributed returns to a preset initial position (or a set of those). The waiting times of such resetting events obey the Poissonian, Gamma, or more generic distributions with specified conditions regarding the existence of moments. Within these general approaches, we derive general analytical results and support them by computer simulations for the behavior of the reset mean-squared displacement (MSD), the new reset increment-MSD (iMSD), and the mean reset time-averaged MSD (TAMSD). For parental nonreset ADPs with the MSD($t$)$\ensuremath{\propto}{t}^{\ensuremath{\mu}}$ we find a generic behavior and a switch of the short-time growth of the reset iMSD and mean reset TAMSDs from $\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{\mu}}$ for subdiffusive to $\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{1}$ for superdiffusive reset ADPs. The critical condition for a reset ADP that recovers its ergodicity is found to be more general than that for the nonequilibrium stationary state, where obviously the iMSD and the mean TAMSD are equal. The consideration of the new statistical quantifier, the iMSD---as compared to the standard MSD---restores the ergodicity of an arbitrary reset ADP in all situations when the $\ensuremath{\mu}\mathrm{th}$ moment of the waiting-time distribution of resetting events is finite. Potential applications of these new resetting results are, inter alia, in the area of biophysical and soft-matter systems.

Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution for a single trajectory undergoing resetting. Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model. In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) time-averaged-mean-squared-displacement (TAMSD) which is of acute interest in physics. We find that both diffusion and drift–diffusion processes with resetting are ergodic at the mean level unlike their reset-free counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations. Our results can be verified in experimental set-ups that can track single particle trajectories and thus have strong implications in understanding the physics of resetting.