Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes

Universal superdiffusion of random walks in media with embedded fractal networks of low diffusivity

Diffusion in composite media with high contrasts between diffusion coefficients in fractal sets of inclusions and in their embedding matrices is modeled by lattice random walks (RWs) with probabilities p<1 of hops from fractal sites and 1 from matrix sites. Superdiffusion is predicted in time intervals that depend on p and with diffusion exponents that depend on the dimensions of matrix (E) and fractal (D_{F}) as ν=1/(2+D_{F}-E). This contrasts with the nonuniversal subdiffusion of RWs confined to fractal media. Simulations with four fractals show the anomaly at several time decades for p≲10^{-3} and the crossover to the asymptotic normal diffusion. These results show that superdiffusion can be observed in isotropic RWs with finite moments of hop length distributions and allow the estimation of the dimension of the inclusion set from the diffusion exponent. However, displacements within single trajectories have normal scaling, which shows transient ergodicity breaking.

Multifractal perturbations to multiplicative cascades promote multifractal nonlinearity with asymmetric spectra

Biological and psychological processes have been conceptualized as emerging from intricate multiplicative interactions among component processes across various spatial and temporal scales. Among the statistical models employed to approximate these intricate nonlinear interactions across scales, one prominent framework is that of cascades. Despite decades of empirical work using multifractal formalisms, several fundamental questions persist concerning the proper interpretations of multifractal evidence of nonlinear cross-scale interactivity. Does multifractal spectrum width depend on multiplicative interactions, constituent noise processes participating in those interactions, or both? We conducted numerical simulations of cascade time series featuring component noise processes characterizing a range of nonlinear temporal correlations: nonlinearly multifractal, linearly multifractal (obtained via the iterative amplitude adjusted wavelet transform of nonlinearly multifractal), phase-randomized linearity (obtained via the iterative amplitude adjustment Fourier transform of nonlinearly multifractal), and phase and amplitude randomized (obtained via shuffling of nonlinearly multifractal). Our findings show that the multiplicative interactions coordinate with the nonlinear temporal correlations of noise components to dictate emergent multifractal properties. Multiplicative cascades with stronger nonlinear temporal correlations make multifractal spectra more asymmetric with wider left sides. However, when considering multifractal spectral differences between the original and surrogate time series, even multiplicative cascades produce multifractality greater than in surrogate time series, even with linearized multifractal noise components. In contrast, additivity among component processes leads to a linear outcome. These findings provide a robust framework for generating multifractal expectations for biological and psychological models in which cascade dynamics flow from one part of an organism to another.

Beyond microtubules: The cellular environment at the endoplasmic reticulum attracts proteins to the nucleus, enabling nuclear transport

All proteins are translated in the cytoplasm, yet many, including transcription factors, play vital roles in the nucleus. While previous research has concentrated on molecular motors for the transport of these proteins to the nucleus, recent observations reveal perinuclear accumulation even in the absence of an energy source, hinting at alternative mechanisms. Here, we propose that structural properties of the cellular environment, specifically the endoplasmic reticulum (ER), can promote molecular transport to the perinucleus without requiring additional energy expenditure. Specifically, physical interaction between proteins and the ER impedes their diffusion and leads to their accumulation near the nucleus. This result explains why larger proteins, more frequently interacting with the ER membrane, tend to accumulate at the perinucleus. Interestingly, such diffusion in a heterogeneous environment follows Chapman's law rather than the popular Fick's law. Our findings suggest a novel protein transport mechanism arising solely from characteristics of the intracellular environment.

Diffusion of active Brownian particles under quenched disorder

The motion of a single active particle in one dimension with quenched disorder under the external force is investigated. Within the tailored parameter range, anomalous diffusion that displays weak ergodicity breaking is observed, i.e., non-ergodic subdiffusion and non-ergodic superdiffusion. This non-ergodic anomalous diffusion is analyzed through the time-dependent probability distributions of the particle's velocities and positions. Its origin is attributed to the relative weights of the locked state (predominant in the subdiffusion state) and running state (predominant in the superdiffusion state). These results may contribute to understanding the dynamical behavior of self-propelled particles in nature and the extraordinary response of nonlinear dynamics to the externally biased force.

Anomalous diffusion, non-Gaussianity, nonergodicity, and confinement in stochastic-scaled Brownian motion with diffusing diffusivity dynamics

Scaled Brownian motions (SBMs) with power-law time-dependent diffusivity have been used to describe various types of anomalous diffusion yet Gaussian observed in granular gases kinetics, turbulent diffusion, and molecules mobility in cells, to name a few. However, some of these systems may exhibit non-Gaussian behavior which can be described by SBM with diffusing diffusivity (DD-SBM). Here, we numerically investigate both free and confined DD-SBM models characterized by fixed or stochastic scaling exponent of time-dependent diffusivity. The effects of distributed scaling exponent, random diffusivity, and confinement are considered. Different regimes of ultraslow diffusion, subdiffusion, normal diffusion, and superdiffusion are observed. In addition, weak ergodic and non-Gaussian behaviors are also detected. These results provide insights into diffusion in time-fluctuating diffusivity landscapes with potential applications to time-dependent temperature systems spreading in heterogeneous environments.

Lévy-walk-like Langevin dynamics with random parameters

Anomalous diffusion phenomena have been widely found in systems within an inhomogeneous complex environment. For Lévy walk in an inhomogeneous complex environment, we characterize the particle's trajectory through an underdamped Langevin system coupled with a subordinator. The influence of the inhomogeneous environment on the particle's motion is parameterized by the random system parameters, relaxation timescale τ, and velocity diffusivity σ. We find that the two random parameters make different effects on the original superdiffusion behavior of the Lévy walk. The random σ contributes to a trivial result after an ensemble average, which is independent of the specific distribution of σ. By contrast, we find that a specific distribution of τ, a modified Lévy distribution with a finite mean, slows down the decaying rate of the velocity correlation function with respect to the lag time. However, the random τ does not promote the diffusion behavior in a direct way, but plays a competition role to the superdiffusion of the original Lévy walk. In addition, the effect of the random τ is also related to the specific subordinator in the coupled Langevin model, which corresponds to the distribution of the flight time of the Lévy walk. The random system parameters are capable of leading to novel dynamics, which needs detailed analyses, rather than an intuitive judgment, especially in complex systems.

Langevin picture of subdiffusion in nonuniformly expanding medium

Anomalous diffusion phenomena have been observed in many complex physical and biological systems. One significant advance recently is the physical extension of particle's motion in a static medium to a uniformly and even nonuniformly expanding medium. The dynamic mechanism of the anomalous diffusion in the nonuniformly expanding medium has only been investigated by the approach of continuous-time random walk. To study more physical observables and to supplement the physical models of the anomalous diffusion in the expanding mediums, we characterize the nonuniformly expanding medium with a spatiotemporal dependent scale factor a(x,t) and build the Langevin picture describing the particle's motion in the nonuniformly expanding medium. Besides the existing comoving and physical coordinates, by introducing a new coordinate and assuming that a(x,t) is separable at a long-time limit, we build the relation between the nonuniformly expanding medium and the uniformly expanding one and further obtain the moments of the comoving and physical coordinates. Different forms of the scale factor a(x,t) are considered to uncover the combined effects of the particle's intrinsic diffusion and the nonuniform expansion of medium. The theoretical analyses and simulations provide the foundation for studying more anomalous diffusion phenomena in the expanding mediums.

First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion

We study the statistics of random functionals Z=∫_{0}^{T}[x(t)]^{γ-2}dt, where x(t) is the trajectory of a one-dimensional Brownian motion with diffusion constant D under the effect of a logarithmic potential V(x)=V_{0}ln(x). The trajectory starts from a point x_{0} inside an interval entirely contained in the positive real axis, and the motion is evolved up to the first-exit time T from the interval. We compute explicitly the PDF of Z for γ=0, and its Laplace transform for γ≠0, which can be inverted for particular combinations of γ and V_{0}. Then we consider the dynamics in (0,∞) up to the first-passage time to the origin and obtain the exact distribution for γ>0 and V_{0}>-D. By using a mapping between Brownian motion in logarithmic potentials and heterogeneous diffusion, we extend this result to functionals measured over trajectories generated by x[over ̇](t)=sqrt[2D][x(t)]^{θ}η(t), where θ<1 and η(t) is a Gaussian white noise. We also emphasize how the different interpretations that can be given to the Langevin equation affect the results. Our findings are illustrated by numerical simulations, with good agreement between data and theory.

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.

On heterogeneous diffusion processes and the formation of spatial-temporal nonlocality

Heterogeneous diffusion processes defined as a solution to the overdamped Langevin equation with multiplicative noise, the amplitude of which has a power-law space-dependent form, are studied. Particular emphasis is on discrete analogs of these processes, for which, in particular, an asymptotic estimate of their variance behavior in time is obtained. In addition, a class of processes formed by deformation of the discrete analog of the fractional Brownian motion using the Cantor ladder and its inverse transformation is considered. It is found that such a class turns out to be close in structure to discrete analogs of heterogeneous processes. This class of processes allows us to illustrate geometrically the emergence of sub- and superdiffusion transport regimes. On the basis of discrete analogs of heterogeneous processes and memory flow phenomenology, we construct a class of random processes that allows us to model nonlocality in time and space taking into account spatial heterogeneity.

Anomalous diffusion and long-range memory in the scaled voter model

We analyze the scaled voter model, which is a generalization of the noisy voter model with time-dependent herding behavior. We consider the case when the intensity of herding behavior grows as a power-law function of time. In this case, the scaled voter model reduces to the usual noisy voter model, but it is driven by the scaled Brownian motion. We derive analytical expressions for the time evolution of the first and second moments of the scaled voter model. In addition, we have derived an analytical approximation of the first passage time distribution. By numerical simulation, we confirm our analytical results as well as showing that the model exhibits long-range memory indicators despite being a Markov model. The proposed model has steady-state distribution consistent with the bounded fractional Brownian motion, thus we expect it to be a good substitute model for the bounded fractional Brownian motion.

Weak ergodicity breaking and anomalous diffusion in collective motion of active particles under spatiotemporal disorder

The effects of spatiotemporal disorder, i.e., both the noise and quenched disorder, on the dynamics of active particles in two dimensions are investigated. We demonstrate that within the tailored parameter regime, nonergodic superdiffusion and nonergodic subdiffusion occur in the system, identified by the observable quantities (the mean squared displacement and ergodicity-breaking parameter) averaged over both the noise and realizations of quenched disorder. Their origins are attributed to the competition effects between the neighbor alignment and spatiotemporal disorder on the collective motion of active particles. These results may be helpful for further understanding the nonequilibrium transport process of active particles, as well as for detection of the transport of self-propelled particles in complex and crowded environments.

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.

Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models

How does a systematic time-dependence of the diffusion coefficient $D (t)$ affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we examine how the behavior of the...

Ergodic property of random diffusivity system with trapping events

A Brownian yet non-Gaussian phenomenon has recently been observed in many biological and active matter systems. The main idea of explaining this phenomenon is to introduce a random diffusivity for particles moving in inhomogeneous environment. This paper considers a Langevin system containing a random diffusivity and an α-stable subordinator with α<1. This model describes the particle's motion in complex media where both the long trapping events and random diffusivity exist. We derive the general expressions of ensemble- and time-averaged mean-squared displacements which only contain the values of the inverse subordinator and diffusivity. Further taking specific time-dependent diffusivity, we obtain the analytic expressions of ergodicity breaking parameter and probability density function of the time-averaged mean-squared displacement. The results imply the nonergodicity of the random diffusivity model with any kind of diffusivity, including the critical case where the model presents normal diffusion.

Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x)=D_{0}|x|^{γ} and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity-breaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB∼(1/r)-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.

Local Analysis of Heterogeneous Intracellular Transport: Slow and Fast Moving Endosomes

Trajectories of endosomes inside living eukaryotic cells are highly heterogeneous in space and time and diffuse anomalously due to a combination of viscoelasticity, caging, aggregation and active transport. Some of the trajectories display switching between persistent and anti-persistent motion, while others jiggle around in one position for the whole measurement time. By splitting the ensemble of endosome trajectories into slow moving subdiffusive and fast moving superdiffusive endosomes, we analyzed them separately. The mean squared displacements and velocity auto-correlation functions confirm the effectiveness of the splitting methods. Applying the local analysis, we show that both ensembles are characterized by a spectrum of local anomalous exponents and local generalized diffusion coefficients. Slow and fast endosomes have exponential distributions of local anomalous exponents and power law distributions of generalized diffusion coefficients. This suggests that heterogeneous fractional Brownian motion is an appropriate model for both fast and slow moving endosomes. This article is part of a Special Issue entitled: "Recent Advances In Single-Particle Tracking: Experiment and Analysis" edited by Janusz Szwabiński and Aleksander Weron.

Anomalous diffusion in nonlinear transformations of the noisy voter model

Voter models are well known in the interdisciplinary community, yet they have not been studied from the perspective of anomalous diffusion. In this paper, we show that the original voter model exhibits a ballistic regime. Nonlinear transformations of the observation variable and time scale allow us to observe other regimes of anomalous diffusion as well as normal diffusion. We show that numerical simulation results coincide with derived analytical approximations describing the temporal evolution of the raw moments.

Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes

Heterogeneous diffusion processes (HDPs) with space-dependent diffusion coefficients D(x) are found in a number of real-world systems, such as for diffusion of macromolecules or submicron tracers in biological cells. Here, we examine HDPs in quenched-disorder systems with Gaussian colored noise (GCN) characterized by a diffusion coefficient with a power-law dependence on the particle position and with a spatially random scaling exponent. Typically, D(x) is considered to be centerd at the origin and the entire x axis is characterized by a single scaling exponent α. In this work we consider a spatially random scenario: in periodic intervals ("layers") in space D(x) is centerd to the midpoint of each interval. In each interval the scaling exponent α is randomly chosen from a Gaussian distribution. The effects of the variation of the scaling exponents, the periodicity of the domains ("layer thickness") of the diffusion coefficient in this stratified system, and the correlation time of the GCN are analyzed numerically in detail. We discuss the regimes of superdiffusion, subdiffusion, and normal diffusion realisable in this system. We observe and quantify the domains where nonergodic and non-Gaussian behaviors emerge in this system. Our results provide new insights into the understanding of weak ergodicity breaking for HDPs driven by colored noise, with potential applications in quenched layered systems, typical model systems for diffusion in biological cells and tissues, as well as for diffusion in geophysical systems.

Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise

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}}/sqrt[π]. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.

Anomalous and heterogeneous DNA transport in biomimetic cytoskeleton networks

The cytoskeleton, a complex network of protein filaments and crosslinking proteins, dictates diverse cellular processes ranging from division to cargo transport. Yet, the role the cytoskeleton plays in the intracellular transport of DNA and other macromolecules remains poorly understood. Here, using single-molecule conformational tracking, we measure the transport and conformational dynamics of linear and relaxed circular (ring) DNA in composite networks of actin and microtubules with variable types of crosslinking. While both linear and ring DNA undergo anomalous, non-Gaussian, and non-ergodic subdiffusion, the detailed dynamics are controlled by both DNA topology (linear vs. ring) and crosslinking motif. Ring DNA swells, exhibiting heterogeneous subdiffusion controlled via threading by cytoskeleton filaments, while linear DNA compacts, exhibiting transport via caging and hopping. Importantly, while the crosslinking motif has little effect on ring DNA, linear DNA in networks with actin-microtubule crosslinking is significantly less ergodic and shows more heterogeneous transport than with actin-actin or microtubule-microtubule crosslinking.

Run-and-tumble motion in one dimension with space-dependent speed

We consider a particle performing run-and-tumble dynamics with space-dependent speed. The model has biological relevance as it describes motile bacteria or cells in heterogeneous environments. We give exact expression for the probability density function in the case of free motion in unbounded space. We then analyze the case of a particle moving in a confined interval in the presence of partially absorbing boundaries, reporting the probability density in the Laplace (time) domain and the mean time to absorption. We also discuss the relaxation to the steady state in the case of confinement with reflecting boundaries and drift effects due to direction-dependent tumbling rates, modeling taxis phenomena of cells. The case of diffusive particles with spatially variable diffusivity is obtained as a limiting case.

Nonergodicity in silo unclogging: Broken and unbroken arches

We report an experiment on the unclogging dynamics in a two-dimensional silo submitted to a sustained gentle vibration. We find that arches present a jerking motion where rearrangements in the positions of their beads are interspersed with quiescent periods. This behavior occurs for both arches that break down and those that withstand the external perturbation: Arches evolve until they either collapse or get trapped in a stable configuration. This evolution is described in terms of a scalar variable characterizing the arch shape that can be modeled as a continuous-time random walk. By studying the diffusivity of this variable, we show that the unclogging is a weakly nonergodic process. Remarkably, arches that do not collapse explore different configurations before settling in one of them and break ergodicity much in the same way than arches that break down.

Infinite ergodic theory for heterogeneous diffusion processes

We show the relation between processes which are modeled by a Langevin equation with multiplicative noise and infinite ergodic theory. We concentrate on a spatially dependent diffusion coefficient that behaves as D(x)∼|x-x[over ̃]|^{2-2/α} in the vicinity of a point x[over ̃], where α can be either positive or negative. We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system. For processes under investigation, the time averages of a wide class of observables are obtained using an ensemble average with respect to the nonnormalized density. A Langevin equation which involves multiplicative noise may take different interpretation, Itô, Stratonovich, or Hänggi-Klimontovich, so the existence of an infinite density and the density's shape are both related to the considered interpretation and the structure of D(x).

Non-Gaussian, non-ergodic, and non-Fickian diffusion of tracers in mucin hydrogels

Native mucus is polymer-based soft-matter material of paramount biological importance. How non-Gaussian and non-ergodic is the diffusive spreading of pathogens in mucus? We study the passive, thermally driven motion of micron-sized tracers in hydrogels of mucins, the main polymeric component of mucus. We report the results of the Bayesian analysis for ranking several diffusion models for a set of tracer trajectories [C. E. Wagner et al., Biomacromolecules, 2017, 18, 3654]. The models with "diffusing diffusivity", fractional and standard Brownian motion are used. The likelihood functions and evidences of each model are computed, ranking the significance of each model for individual traces. We find that viscoelastic anomalous diffusion is often most probable, followed by Brownian motion, while the model with a diffusing diffusion coefficient is only realised rarely. Our analysis also clarifies the distribution of time-averaged displacements, correlations of scaling exponents and diffusion coefficients, and the degree of non-Gaussianity of displacements at varying pH levels. Weak ergodicity breaking is also quantified. We conclude that-consistent with the original study-diffusion of tracers in the mucin gels is most non-Gaussian and non-ergodic at low pH that corresponds to the most heterogeneous networks. Using the Bayesian approach with the nested-sampling algorithm, together with the quantitative analysis of multiple statistical measures, we report new insights into possible physical mechanisms of diffusion in mucin gels.

Bayesian analysis of single-particle tracking data using the nested-sampling algorithm: maximum-likelihood model selection applied to stochastic-diffusivity data

We employ Bayesian statistics using the nested-sampling algorithm to compare and rank multiple models of ergodic diffusion (including anomalous diffusion) as well as to assess their optimal parameters for in silico-generated and real time-series. We focus on the recently-introduced model of Brownian motion with "diffusing diffusivity"-giving rise to widely-observed non-Gaussian displacement statistics-and its comparison to Brownian and fractional Brownian motion, also for the time-series with some measurement noise. We conduct this model-assessment analysis using Bayesian statistics and the nested-sampling algorithm on the level of individual particle trajectories. We evaluate relative model probabilities and compute best-parameter sets for each diffusion model, comparing the estimated parameters to the true ones. We test the performance of the nested-sampling algorithm and its predictive power both for computer-generated (idealised) trajectories as well as for real single-particle-tracking trajectories. Our approach delivers new important insight into the objective selection of the most suitable stochastic model for a given time-series. We also present first model-ranking results in application to experimental data of tracer diffusion in polymer-based hydrogels.

Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing

We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.

Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity

How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and out-of-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.

Non-Gaussian diffusion in static disordered media

Non-Gaussian diffusion is commonly considered as a result of fluctuating diffusivity, which is correlated in time or in space or both. In this work, we investigate the non-Gaussian diffusion in static disordered media via a quenched trap model, where the diffusivity is spatially correlated. Several unique effects due to quenched disorder are reported. We analytically estimate the diffusion coefficient D_{dis} and its fluctuation over samples of finite size. We show a mechanism of population splitting in the non-Gaussian diffusion. It results in a sharp peak in the distribution of displacement P(x,t) around x=0, that has frequently been observed in experiments. We examine the fidelity of the coarse-grained diffusion map, which is reconstructed from particle trajectories. Finally, we propose a procedure to estimate the correlation length in static disordered environments, where the information stored in the sample-to-sample fluctuation has been utilized.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

Elucidating fluctuating diffusivity in center-of-mass motion of polymer models with time-averaged mean-square-displacement tensor

There have been increasing reports that the diffusion coefficient of macromolecules depends on time and fluctuates randomly. Here a method is developed to elucidate this fluctuating diffusivity from trajectory data. Time-averaged mean-square displacement (MSD), a common tool in single-particle-tracking (SPT) experiments, is generalized to a second-order tensor with which both magnitude and orientation fluctuations of the diffusivity can be clearly detected. This method is used to analyze the center-of-mass motion of four fundamental polymer models: the Rouse model, the Zimm model, a reptation model, and a rigid rodlike polymer. It is found that these models exhibit distinctly different types of magnitude and orientation fluctuations of diffusivity. This is an advantage of the present method over previous ones, such as the ergodicity-breaking parameter and a non-Gaussian parameter, because with either of these parameters it is difficult to distinguish the dynamics of the four polymer models. Also, the present method of a time-averaged MSD tensor could be used to analyze trajectory data obtained in SPT experiments.

Quantifying non-ergodicity of anomalous diffusion with higher order moments

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.

Memory-induced diffusive-superdiffusive transition: Ensemble and time-averaged observables

The ensemble properties and time-averaged observables of a memory-induced diffusive-superdiffusive transition are studied. The model consists in a random walker whose transitions in a given direction depend on a weighted linear combination of the number of both right and left previous transitions. The diffusion process is nonstationary, and its probability develops the phenomenon of aging. Depending on the characteristic memory parameters, the ensemble behavior may be normal, superdiffusive, or ballistic. In contrast, the time-averaged mean squared displacement is equal to that of a normal undriven random walk, which renders the process nonergodic. In addition, and similarly to Lévy walks [Godec and Metzler, Phys. Rev. Lett. 110, 020603 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.020603], for trajectories of finite duration the time-averaged displacement apparently become random with properties that depend on the measurement time and also on the memory properties. These features are related to the nonstationary power-law decay of the transition probabilities to their stationary values. Time-averaged response to a bias is also calculated. In contrast with Lévy walks [Froemberg and Barkai, Phys. Rev. E 87, 030104(R) (2013)PLEEE81539-375510.1103/PhysRevE.87.030104], the response always vanishes asymptotically.

Visual information and expert's idea in Hurst index estimation of the fractional Brownian motion using a diffusion type approximation

An approximation of the fractional Brownian motion based on the Ornstein-Uhlenbeck process is used to obtain an asymptotic likelihood function. Two estimators of the Hurst index are then presented in the likelihood approach. The first estimator is produced according to the observed values of the sample path; while the second one employs the likelihood function of the incremental process. We also employ visual roughness of realization to restrict the parameter space and to obtain prior information in Bayesian approach. The methods are then compared with three contemporary estimators and an experimental data set is studied.

Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble- and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.

Rad4 recognition-at-a-distance: Physical basis of conformation-specific anomalous diffusion of DNA repair proteins

Since Robert Brown's first observations of random walks by pollen particles suspended in solution, the concept of diffusion has been subject to countless theoretical and experimental studies in diverse fields from finance and social sciences, to physics and biology. Diffusive transport of macromolecules in cells is intimately linked to essential cellular functions including nutrient uptake, signal transduction, gene expression, as well as DNA replication and repair. Advancement in experimental techniques has allowed precise measurements of these diffusion processes. Mathematical and physical descriptions and computer simulations have been applied to model complicated biological systems in which anomalous diffusion, in addition to simple Brownian motion, was observed. The purpose of this review is to provide an overview of the major physical models of anomalous diffusion and corresponding experimental evidence on the target search problem faced by DNA-binding proteins, with an emphasis on DNA repair proteins and the role of anomalous diffusion in DNA target recognition.

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

We introduce a class of discrete random-walk model driven by global memory effects. At any time, the right-left transitions depend on the whole previous history of the walker, being defined by an urnlike memory mechanism. The characteristic function is calculated in an exact way, which allows us to demonstrate that the ensemble of realizations is ballistic. Asymptotically, each realization is equivalent to that of a biased Markovian diffusion process with transition rates that strongly differs from one trajectory to another. Using this "inhomogeneous diffusion" feature, the ergodic properties of the dynamics are analytically studied through the time-averaged moments. Even in the long-time regime, they remain random objects. While their average over realizations recovers the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second time-averaged moment, an ergodic limit and the limit of infinite lag times do not commutate. All these effects are induced by the memory effects. A generalized Einstein fluctuation-dissipation relation is also obtained for the time-averaged moments.

Influence of external potentials on heterogeneous diffusion processes

In this paper we consider heterogeneous diffusion processes with the power-law dependence of the diffusion coefficient on the position and investigate the influence of external forces on the resulting anomalous diffusion. The heterogeneous diffusion processes can yield subdiffusion as well as superdiffusion, depending on the behavior of the diffusion coefficient. We assume that not only the diffusion coefficient but also the external force has a power-law dependence on the position. We obtain analytic expressions for the transition probability in two cases: when the power-law exponent in the external force is equal to 2η-1, where 2η is the power-law exponent in the dependence of the diffusion coefficient on the position, and when the external force has a linear dependence on the position. We found that the power-law exponent in the dependence of the mean square displacement on time does not depend on the external force; this force changes only the anomalous diffusion coefficient. In addition, the external force having the power-law exponent different from 2η-1 limits the time interval where the anomalous diffusion occurs. We expect that the results obtained in this paper may be relevant for a more complete understanding of anomalous diffusion processes.

Weak ergodicity breaking induced by global memory effects

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.

Langevin equation with fluctuating diffusivity: A two-state model

Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though the origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. Namely, the diffusivity follows a dichotomous stochastic process. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a nonequilibrium ensemble, the ensemble-averaged mean-square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time and converges to a Gaussian distribution in a long-time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterizing the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool.

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

We study noisy heterogeneous diffusion processes with a position dependent diffusivity of the form D(x) ∼ D0|x|(α0) in the presence of annealed and quenched disorder of the environment, corresponding to an effective variation of the exponent α in time and space. In the case of annealed disorder, for which effectively α0 = α0(t), we show how the long time scaling of the ensemble mean squared displacement (MSD) and the amplitude variation of individual realizations of the time averaged MSD are affected by the disorder strength. For the case of quenched disorder, the long time behavior becomes effectively Brownian after a number of jumps between the domains of a stratified medium. In the latter situation, the averages are taken over both an ensemble of particles and different realizations of the disorder. As physical observables, we analyze in detail the ensemble and time averaged MSDs, the ergodicity breaking parameter, and higher order moments of the time averages.

First passage time distribution in heterogeneity controlled kinetics: going beyond the mean first passage time

The first passage is a generic concept for quantifying when a random quantity such as the position of a diffusing molecule or the value of a stock crosses a preset threshold (target) for the first time. The last decade saw an enlightening series of new results focusing mostly on the so-called mean and global first passage time (MFPT and GFPT, respectively) of such processes. Here we push the understanding of first passage processes one step further. For a simple heterogeneous system we derive rigorously the complete distribution of first passage times (FPTs). Our results demonstrate that the typical FPT significantly differs from the MFPT, which corresponds to the long time behaviour of the FPT distribution. Conversely, the short time behaviour is shown to correspond to trajectories connecting directly from the initial value to the target. Remarkably, we reveal a previously overlooked third characteristic time scale of the first passage dynamics mirroring brief excursion away from the target.

Non-Brownian diffusion in lipid membranes: Experiments and simulations

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.

Crowding induces complex ergodic diffusion and dynamic elongation of large DNA molecules

Despite the ubiquity of molecular crowding in living cells, the effects of crowding on the dynamics of genome-sized DNA are poorly understood. Here, we track single, fluorescent-labeled large DNA molecules (11, 115 kbp) diffusing in dextran solutions that mimic intracellular crowding conditions (0-40%), and determine the effects of crowding on both DNA mobility and conformation. Both DNAs exhibit ergodic Brownian motion and comparable mobility reduction in all conditions; however, crowder size (10 vs. 500 kDa) plays a critical role in the underlying diffusive mechanisms and dependence on crowder concentration. Surprisingly, in 10-kDa dextran, crowder influence saturates at ∼20% with an ∼5× drop in DNA diffusion, in stark contrast to exponentially retarded mobility, coupled to weak anomalous subdiffusion, with increasing concentration of 500-kDa dextran. Both DNAs elongate into lower-entropy states (compared to random coil conformations) when crowded, with elongation states that are gamma distributed and fluctuate in time. However, the broadness of the distribution of states and the time-dependence and length scale of elongation length fluctuations depend on both DNA and crowder size with concentration having surprisingly little impact. Results collectively show that mobility reduction and coil elongation of large crowded DNAs are due to a complex interplay between entropic effects and crowder mobility. Although elongation and initial mobility retardation are driven by depletion interactions, subdiffusive dynamics, and the drastic exponential slowing of DNA, up to ∼300×, arise from the reduced mobility of larger crowders. Our results elucidate the highly important and widely debated effects of cellular crowding on genome-sized DNA.

Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes

We investigate the diffusive and ergodic properties of massive and confined particles in a model disordered medium, in which the local diffusivity fluctuates in time while its mean has a power law dependence on the diffusion time.