Anomalous Diffusion Regime Research Articles

Multidimensional and Nonlinear Fractional Langevin Equations: Advancing the Theory in of Anomalous Diffusion

This paper presents a comprehensive theoretical analysis of fractional Langevin equations (FLEs) and their applications in modeling anomalous diffusion processes. We extend the classical FLE framework in several significant directions to address the complexities observed in various physical, biological, and social systems. First, we derive a multidimensional extension of the FLE, providing a robust tool for studying anomalous diffusion in higher-dimensional spaces. Second, we incorporate non-Gaussian Lévy α-stable noise into the FLE, enabling the description of systems characterized by heavy-tailed distributions and extreme events. Third, we develop a nonlinear variant of the FLE to model more intricate restoring forces and potential landscapes. Finally, we analyze a generalized FLE with different orders of fractional derivatives for the inertial and friction terms, offering a flexible framework capable of capturing a wider range of anomalous diffusion phenomena. For each extension, we prove the existence and uniqueness of solutions, derive the corresponding probability distributions, and analyze the scaling behavior of the mean squared displacement in the long-time limit. Our results demonstrate how these generalized FLEs can describe various regimes of anomalous diffusion, from subdiffusive to superdiffusive behavior. This work provides a unified theoretical foundation for understanding and modeling complex diffusive processes across diverse disciplines, paving the way for more accurate descriptions of systems exhibiting memory effects, non-Gaussian statistics, and multiscale dynamics.

Fractional Extended Diffusion Theory to capture anomalous relaxation from biased/accelerated molecular simulations.

Biased and accelerated molecular simulations (BAMS) are widely used tools to observe relevant molecular phenomena occurring on time scales inaccessible to standard molecular dynamics, but evaluation of the physical time scales involved in the processes is not directly possible from them. For this reason, the problem of recovering dynamics from such kinds of simulations is the object of very active research due to the relevant theoretical and practical implications of dynamics on the properties of both natural and synthetic molecular systems. In a recent paper [A. Rapallo et al., J. Comput. Chem. 42, 586-599 (2021)], it has been shown how the coupling of BAMS (which destroys the dynamics but allows to calculate average properties) with Extended Diffusion Theory (EDT) (which requires input appropriate equilibrium averages calculated over the BAMS trajectories) allows to effectively use the Smoluchowski equation to calculate the orientational time correlation function of the head-tail unit vector defined over a peptide in water solution. Orientational relaxation of this vector is the result of the coupling of internal molecular motions with overall molecular rotation, and it was very well described by correlation functions expressed in terms of weighted sums of suitable time-exponentially decaying functions, in agreement with a Brownian diffusive regime. However, situations occur where exponentially decaying functions are no longer appropriate to capture the actual dynamical behavior, which exhibits persistent long time correlations, compatible with the so called subdiffusive regimes. In this paper, a generalization of EDT will be given, exploiting a fractional Smoluchowski equation (FEDT) to capture the non-exponential character observed in the relaxation of intramolecular distances and molecular radius of gyration, whose dynamics depend on internal molecular motions only. The calculation methods, proper to EDT, are adapted to implement the generalization of the theory, and the resulting algorithm confirms FEDT as a tool of practical value in recovering dynamics from BAMS, to be used in general situations, involving both regular and anomalous diffusion regimes.

Stochastic model for the transfer of gaseous particles in polymer-carbon-nanotube nanocomposites with interfacial regions.

In this work, a stochastic model of gaseous transfer in polymer-carbon-nanotube (CNT) nanocomposites is presented. The model takes into account interfacial areas, i.e., polymer depletion regions. The local regime of transport is controlled by the density of the polymer. In a dense polymer, this regime corresponds to the ordinary diffusion, while in free volume regions, it corresponds to the ballistic transport. The introduction of a free volume and/or a depleted polymer layer near to a CNT wall leads to the emergence of anomalous diffusion. We have demonstrated how the anomalous diffusion regime changes in the presence of nanotubes for different distributions of polymer density. The presented approach allows us to describe the threshold effect in the diffusion coefficient as a function of CNTs density in polymer-CNT nanocomposites.

Identifying potential imaging markers for diffusion property changes in a mouse model of amyotrophic lateral sclerosis: Application of the continuous time random walk model to ultrahigh b-value diffusion-weighted MR images of spinal cord tissue.

Diffusion MRI (dMRI) explores tissue microstructures by analyzing diffusion-weighted signal decay measured at different b-values. While relatively low b-values are used for most dMRI models, high b-value diffusion-weighted imaging (DWI) techniques have gained interest given that the non-Gaussian water diffusion behavior observed at high b-values can yield potentially valuable information. In this study, we investigated anomalous diffusion behaviors associated with degeneration of spinal cord tissue using a continuous time random walk (CTRW) model for DWI data acquired across an extensive range of ultrahigh b-values. The diffusion data were acquired in situ from the lumbar level of spinal cords of wild-type and age-matched transgenic SOD1G93A mice, a well-established animal model of amyotrophic lateral sclerosis (ALS) featuring progressive degeneration of axonal tracts in this tissue. Based on the diffusion decay behaviors at low and ultrahigh b-values, we applied the CTRW model using various combinations of b-values and compared diffusion metrics calculated from the CTRW model between the experimental groups. We found that diffusion-weighted signal decay curves measured with ultrahigh b-values (up to 858,022 s/mm2 in this study) were well represented by the CTRW model. The anomalous diffusion coefficient obtained from lumbar spinal cords was significantly higher in SOD1G93A mice compared with control mice (14.7 × 10-5 ± 5.54 × 10-5 vs. 7.87 × 10-5 ± 2.48 × 10-5 mm2 /s, p = 0.01). We believe this is the first study to illustrate the efficacy of the CTRW model for analyzing anomalous diffusion regimes at ultrahigh b-values. The CTRW modeling of ultrahigh b-value dMRI can potentially present a novel approach for noninvasively evaluating alterations in spinal cord tissue associated with ALS pathology.

Thermodynamic bounds for diffusion in nonequilibrium systems with multiple timescales.

We derive a thermodynamic uncertainty relation bounding the mean squared displacement of a Gaussian process with memory, driven out of equilibrium by unbalanced thermal baths and/or by external forces. Our bound is tighter with respect to previous results and also holds at finite time. We apply our findings to experimental and numerical data for a vibrofluidized granular medium, characterized by regimes of anomalous diffusion. In some cases our relation can distinguish between equilibrium and nonequilibrium behavior, a nontrivial inference task, particularly for Gaussian processes.

Dissecting diffusion and oligomerization dynamics of GPCRs in live cells using single-particle tracking.

Crucial insights about the signaling and trafficking mechanisms of G Protein Coupled Receptors (GPCRs) can be derived from their membrane diffusion properties. Previous studies in our lab (Shivnaraine et al., JACS 2016, Li at al., Biophys. J. 2018) have shown that the M2 muscarinic receptor (M2R) can be purified as oligomers, yet the oligomerization dynamics and the implications for signalling in live cells remain elusive. Here we used single-molecule fluorescence techniques, such as single-particle tracking (SPT) and single-molecule photobleaching (smPB), to characterize the dynamic distribution of M2R in HEK293 cells. To that end, receptors have been co-expressed at their N-terminus either with a green fluorescent protein (eGFP), or with a HaloTag for labelling with HaloTag ligand (HTL) dyes, such as JF549 HTL. For comparison, we applied the same techniques to μ-opioid receptors (MORs) in live U2OS cells. For both cases, measured diffusion maps are spatially and temporally heterogeneous, with receptors transitioning between normal and anomalous diffusion regimes. The analysis of experimental data was reinforced by simulations and by control measurements on monomeric (CD86) and dimeric (CD28) membrane proteins. Intensity traces of immobile, single receptor complexes and of monomeric/dimeric controls in the membrane of fixed cells were analyzed using an in-house smPB code based on a Bayesian algorithm (Garry et al., J. Chem. Phys. 2020). The results confirm the presence, correlated with expression levels, of higher order M2R oligomers in cells, and lay out the foundation for subsequent studies of their functional role.

Semi-Markovian Discrete-Time Telegraph Process with Generalized Sibuya Waiting Times

In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred to this random walk as the ‘squirrel random walk’ (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal process (GSP) which only recently appeared in the literature. The waiting time density of the GSP, the ‘generalized Sibuya distribution’ (GSD), is such that the moments are finite up to a certain order r≤m−1 (m≥1) and diverging for orders r≥m capturing all behaviors from broad to narrow and containing the standard Sibuya distribution as a special case (m=1). We also derive some new representations for the generating functions related to the GSD. We show that the generalized Sibuya SRW exhibits several regimes of anomalous diffusion depending on the lowest order m of diverging GSD moment. The generalized Sibuya SRW opens various new directions in anomalous physics.

Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the ‘aging effect’ as a hallmark of non-Markovianity where the discrete-time version of the ‘aging renewal process’ comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a t2-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (‘narrow’ with finite mean and variance) the walk exhibits normal diffusion and for ‘broad’ waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.

A Peridynamic Differential Operator-Based Model for Quantifying Spatial Non-Local Transport Behavior of Pollutants in Heterogeneous Media

Modeling pollutant transport in heterogeneous media is an important task of hydrology. Pollutant transport in a non-homogeneous environment typically exhibits non-local transport dynamics, whose efficient characterization requires a parsimonious model with the non-local feature. This study encapsulates the non-local transport characteristic of pollutants into the peridynamic differential operator (PDDO) and develops a PDDO-based model for quantifying the observed pollutant non-local transport behavior. The simulation results show that the proposed model can describe pollutant non-local transport behavior in various heterogeneous media. The non-local nature of pollutant transport can be adjusted by pre-defined weight function w(|ξ|) and horizon Hx. Applications show that the PDDO-based model can better capture pollutant non-local transport behavior than the classical advection–diffusion equation (ADE) model, especially for quantifying the tail of the experimental data late. Analyses further reveal that the PDDO-based model can characterize both normal (Fickian) and anomalous (Lévy) diffusion regimes.

Active particles in reactive disordered media: How does adsorption affect diffusion?

In this work we study analytically and numerically the transport properties of non-interacting active particles moving on a d-dimensional disordered medium. The disorder is modeled by means of a set of non-overlapping spherical obstacles. We assume that obstacles are reactive in the sense that they react in the presence of the particles in an attractive manner: when the particle collides with an obstacle, it is attached during a random time (adsorption time), i.e., it gets adsorbed by an obstacle; thereafter the particle is detached from the obstacle and continues its motion in a random direction. We give an approximate analytical expression for the effective diffusion coefficient when the mean adsorption time is finite. If the mean adsorption time is infinite the system undergoes a transition from a normal to anomalous diffusion regime. We also show that another transition takes place in the mean number of adsorbed particles: in the anomalous diffusion phase all the particles become adsorbed in the average. We show that the fraction of adsorbed particles, seen as an order parameter of the system, undergoes a second-order-like phase transition, because the fraction of adsorbed particles is not differentiable but changes continuously as a function of a parameter of the model.

Fractional Diffusion with Geometric Constraints: Application to Signal Decay in Magnetic Resonance Imaging (MRI)

We investigate diffusion in three dimensions on a comb-like structure in which the particles move freely in a plane, but, out of this plane, are constrained to move only in the perpendicular direction. This model is an extension of the two-dimensional version of the comb model, which allows diffusion along the backbone when the particles are not in the branches. We also consider memory effects, which may be handled with different fractional derivative operators involving singular and non-singular kernels. We find exact solutions for the particle distributions in this model that display normal and anomalous diffusion regimes when the mean-squared displacement is determined. As an application, we use this model to fit the anisotropic diffusion of water along and across the axons in the optic nerve using magnetic resonance imaging. The results for the observed diffusion times (8 to 30 milliseconds) show an anomalous diffusion both along and across the fibers.

Time-averaged mean squared displacement ratio test for Gaussian processes with unknown diffusion coefficient.

The time-averaged mean squared displacement (TAMSD) is one of the most common statistics used for the analysis of anomalous diffusion processes. Anomalous diffusion is manifested by non-linear (mostly power-law) characteristics of the process in contrast to normal diffusion where linear characteristics are expected. One can distinguish between sub- and super-diffusive processes. We consider Gaussian anomalous diffusion models and propose a new approach used for their testing. This approach is based on the TAMSD ratio statistic for different time lags. Similar to the TAMSD, this statistic exhibits a specific behavior in the anomalous diffusion regime. Through its structure, it is independent of the diffusion coefficient, which, in general, does not influence anomalous diffusion behavior. Thus, the TAMSD ratio-based approach does not require preliminary knowledge of the diffusion coefficient's value, in contrast to the TAMSD-approach, where this value is crucial in the testing procedure. Based on the quadratic form representation of the TAMSD ratio, we calculate its main characteristics and propose a step-by-step testing procedure that can be applied for any Gaussian process. For the anomalous diffusion model used here, namely, the fractional Brownian motion, we demonstrate the effectiveness of the proposed methodology. We show that the new approach outperforms the TAMSD-based one, especially for small sample sizes. Finally, the methodology is applied to the real data from the financial market.

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.

Measurement instrumentation and selected signal processing techniques for anomalous diffusion analysis

Anomalous diffusion phenomena are ubiquitous in many areas, including physics, biology, medicine, climate and also financial markets. The description and analysis of this phenomena are challenging tasks. To fully define the anomalous diffusion and properly understand its nature there is need to combine the skills in advanced sensing techniques, mathematical and statistical knowledge of the corresponding theoretical processes and the experience in physical interpretation of this phenomena. The paper reviews on the measurement instrumentation and signal processing used for detection and description of anomalous diffusion regimes. What is more, estimation of the anomalous diffusion exponent is commented. We present here four methods based on the statistics commonly used in the analyzed problem. Finally the fractional Brownian motion (FBM) is proposed as an analog proper for reliable depiction of properties valid for observed anomalous diffusion phenomena. Simulation studies exhibit the scenario of Hurst exponent estimation in the FBM model when the additive noise of various amplitudes disturbance is added to a simulated trajectory of individual particle. Proposed algorithms have different efficiency depending on the model parameters. This implicates a need to provide more advanced analysis and introduce new methods that could be less sensitive to the measurement noise and/or external forces of noise character. This paper is the starting point for the complete description of the anomalous diffusive processes. It presents the areas for prospective applications of the original sensing techniques but also highlights the needs in a range of statistical and intelligent signal processing suitable for description of the small scale complex systems during the SPT experiment.

Hydrodynamic memory can boost enormously driven nonlinear diffusion and transport.

Hydrodynamic memory force or Basset force has been known since the 19th century. Its influence on Brownian motion remains, however, mostly unexplored. Here we investigate its role in nonlinear transport and diffusion within a paradigmatic model of tilted washboard potential. In this model, a giant enhancement of driven diffusion over its potential-free limit [Phys. Rev. Lett. 87, 010602 (2001)PRLTAO0031-900710.1103/PhysRevLett.87.010602] presents a well-established paradoxical phenomenon. In the overdamped limit, it occurs at a critical tilt of vanishing potential barriers. However, for weak damping, it takes place surprisingly at another critical tilt, where the potential barriers are clearly expressed. Recently we showed [Phys. Rev. Lett. 123, 180603 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.180603] that Basset force could make such a diffusion enhancement enormously large. In this paper, we discover that even for moderately strong damping, where the overdamped theory works very well when the memory effects are negligible, substantial hydrodynamic memory unexpectedly makes a strong impact. First, the diffusion boost occurs at nonvanishing potential barriers and can be orders of magnitude larger. Second, transient anomalous diffusion regimes emerge over many time decades and potential periods. Third, particles' mobility can also be dramatically enhanced, and a long transient supertransport regime emerges.

Ordered motion of active colloids and effective temperature

Using recent developments of non-equilibrium thermodynamics, we introduce an effective temperature for active colloidal systems, which is a product of two factors. The first one arises due to the motor activity and describes the randomized motion of the active particles, whereas the second one is a consequence of the ordered (collective) motion due to the alignment effects or initially ordered configuration. These factors depend on the corresponding characteristic velocities: the first one is expressed in terms of the propelled (swimming) velocity of an individual active particle, while the second one — in terms of the local drift velocity, which characterizes the collective motion of the active particles and plays a role analogous to that of an order parameter. The behavior of the active systems is governed by the interplay between these two factors, which leads to a hierarchy of anomalous diffusion regimes with different characteristic time scales and corresponding effective diffusion coefficients.

Investigation of the anomalous diffusion in the porous media: a spatiotemporal scaling

The methanol and the methane transport in the mesoporous zeolite-alumina pellet was studied. Varying the pellet’s size and the diffusant amount, the values of the diffusion coefficients were estimated. Previously, it had been demonstrated that the methane transport through the zeolite-containing pellet is described by the Fickian diffusion equation, whereas the methanol transport is described by the non-Fickian diffusion equation with either space-time-fractional or time-fractional derivative. In this respect, for calculating the diffusion coefficients, the standard diffusion and the time-fractional diffusion models were used for the methane and the methanol respectively. The relations between the obtained values of the methanol non-Fickian diffusion coefficients (0.0068–0.0276 cm2/sα) measured for unequal pellet sizes and various diffusant amounts were found to follow the temporal scaling with a fractional exponent equal to 1.17 ± 0.03, which corresponds to the time-fractional diffusion equation, in a concise manner. It supported a conclusion that the anomalous diffusion of the methanol is time-fractional. An essential applicability of the approach based on the analysis of the temporal diffusion coefficient scaling was additionally verified using the standard Fickian diffusion coefficients of the methane (0.00094–0.00376 cm2/s). In addition, we demonstrate that the investigation of the anomalous diffusion regime using the spatial scaling of the diffusion coefficient is restricted by the measurement of the diffusion length of a certain diffusant in a porous material.

The modeling of anomalous transport in the diffusion-ballistic regime

The anomalous transport of a point-like particle within the disordered heterogeneous medium with accelerating traps was investigated using Monte Carlo simulation within the CTRW model. Outside the traps a particle moves according to the diffusion regime of motion. Inside the traps the ballistic regime takes place. Results of the Monte Carlo simulation shows that the final distribution of particle jumps belongs to the class of the stable distributions which corresponds to the anomalous diffusion regime. It is shown that the characteristic exponent of the stable distribution depends on the trap density and particle velocity inside the traps.

Recurrence statistics for anomalous diffusion regime change detection

For many real-time series, specific behaviors are observed where the character of the time series changes over time. This temporal evolution may indicate that some properties of the data evolve or fluctuate. One can find such problems in many different applications including physical and biological experiments as well as in technical diagnostics. From the mathematical point of view, this complexity can be considered as a segmentation problem, i.e. extraction of the homogeneous parts from the original data. Most segmentation methods assume that a simple characteristic of the time series changes, for example the mean or the variance. However, many physical applications involve a more complex situation dealing with transient statistics. Here, a new technique of the critical change point detection is introduced for the case when the data consist of anomalous diffusion processes with transient anomalous diffusion exponents. The precise mathematical formulation of a new statistics based on recurrence statistics is provided. The proposed recurrence analysis counts the number of data points falling into the appropriate circle built from consecutive observations. This approach proves to be helpful in recognizing subdiffusive and superdiffusive regions, which characterize anomalous diffusion behaviors. The main characteristics of the recurrence statistics are presented and the application to the segmentation problem is described. The effectiveness of the proposed technique is validated for a family of classical anomalous diffusive models, namely fractional Brownian motion. Finally, the methodology is applied to biological data exhibiting anomalous diffusion behavior with transient anomalous diffusion exponents.

Random walks exhibiting anomalous diffusion: elephants, urns and the limits of normality

A random walk model is presented which exhibits a transition from standard to anomalous diffusion as a parameter is varied. The model is a variant on the elephant random walk and differs in respect of the treatment of the initial state, which in the present work consists of a given number of fixed steps. This also links the elephant random walk to other types of history dependent random walk. As well as being amenable to direct analysis, the model is shown to be asymptotically equivalent to a non-linear urn process. This provides fresh insights into the limiting form of the distribution of the walker’s position at large times. Although the distribution is intrinsically non-Gaussian in the anomalous diffusion regime, it gradually reverts to normal form when is large under quite general conditions.