Anomalous Diffusion Exponent Research Articles

Change-point detection in anomalous-diffusion trajectories utilising machine-learning-based uncertainty estimates

Abstract When recording the movement of individual animals, cells or molecules one will often observe changes in their diffusive behaviour at certain points in time along their trajectory. In order to capture the different diffusive modes assembled in such heterogeneous trajectories it becomes necessary to segment them by determining these change-points. Such a change-point detection can be challenging for conventional statistical methods, especially when the changes are subtle. We here apply Bayesian Deep Learning to obtain point-wise estimates of not only the anomalous diffusion exponent but also the uncertainties in these predictions from a single anomalous diffusion trajectory generated according to four theoretical models of anomalous diffusion. We show that we are able to achieve an accuracy similar to single-mode (without change-points) predictions as well as a well calibrated uncertainty predictions of this accuracy. Additionally, we find that the predicted uncertainties feature interesting behaviour at the change-points leading us to examine the capabilities of these predictions for change-point detection. While the series of predicted uncertainties on their own are not sufficient to improve change-point detection, they do lead to a performance boost when applied in combination with the predicted anomalous diffusion exponents.

Modeling catalyst effectiveness factor with space-fractional derivative using Haar wavelet collocation method

Abstract Mass transfer limitations may considerably affect the rate of a heterogeneous catalytic process. The catalyst effectiveness factor is a quantitative measure of the impact of the diffusion process inside a catalyst particle. The effectiveness factor is derived from the solution of the steady-state reaction-diffusion problem. Herein, we simulate the steady-state reaction-diffusion equation with space-fractional derivative and linear reaction kinetics. The solution to the problem is obtained numerically using the Haar wavelet collocation method. The effect of the anomalous diffusion exponent on the catalyst effectiveness factor and process parameters, e.g. reactor volume and catalyst mass, is demonstrated. We anticipate that the process efficiency will be notably improved by changing the diffusion regime from standard to superdiffusive.

Quantum diffusion induced by small quantum chaos.

It is demonstrated that quantum systems classically exhibiting strong and homogeneous chaos in a bounded region of the phase space can induce a global quantum diffusion. As an ideal model system, a small quantum chaos with finite Hilbert space dimension N weakly coupled with M additional degrees of freedom which is approximated by linear systems is proposed. By twinning the system the diffusion process in the additional modes can be numerically investigated without taking the unbounded diffusion space into account explicitly. Even though N is not very large, diffusion occurs in the additional modes as the coupling strength increases if M≥3. If N is large enough, a definite quantum transition to diffusion takes place through a critical subdiffusion characterized by an anomalous diffusion exponent.

Concentration-Dependent Diffusion of Monoclonal Antibodies: Underlying Mechanisms of Anomalous Diffusion.

The development, storage, transport, and subcutaneous delivery of highly concentrated monoclonal antibody formulations pose significant challenges due to the high solution viscosity and low diffusion of the antibody molecules in crowded environments. These issues often stem from the self-associating behavior of the antibody molecules, potentially leading to aggregation. In this work, we used a dissipative particle dynamics-based coarse-grained model to investigate the diffusion behavior of IgG1 antibody molecules in aqueous solutions with 15 and 32 mM NaCl and antibody concentrations ranging from 10 to 400 mg/mL. We determined the coarse-grained interaction parameters by matching the calculated structure factor with the computational and experimental data from the literature. Our results indicate Fickian diffusion for antibody concentrations of 10 and 25 mg/mL and anomalous diffusion for concentrations exceeding 50 mg/mL. The anomalous diffusion was observed for ∼0.33 to 0.4 μs, followed by Fickian diffusion for all antibody concentrations. We observed a strong linear correlation between the diffusion behavior of the antibody molecules (diffusion coefficient D and anomalous diffusion exponent α) and the amount of aggregates present in the solution and between the amount of aggregates and the Coulomb interaction energy. The investigation of underlying mechanisms for anomalous diffusion revealed that in crowded environments at high antibody concentrations, the attractive interaction between electrostatically complementary regions of the antibody molecules could further bring the neighboring molecules closer to one another, ultimately resulting in aggregate formation. Further, the Coulomb attraction can continue to draw more molecules together, forming larger aggregates.

Distinguishing between fractional Brownian motion with random and constant Hurst exponent using sample autocovariance-based statistics.

Fractional Brownian motion (FBM) is a canonical model for describing dynamics in various complex systems. It is characterized by the Hurst exponent, which is responsible for the correlation between FBM increments, its self-similarity property, and anomalous diffusion behavior. However, recent research indicates that the classical model may be insufficient in describing experimental observations when the anomalous diffusion exponent varies from trajectory to trajectory. As a result, modifications of the classical FBM have been considered in the literature, with a natural extension being the FBM with a random Hurst exponent. In this paper, we discuss the problem of distinguishing between two models: (i) FBM with the constant Hurst exponent and (ii) FBM with random Hurst exponent, by analyzing the probabilistic properties of statistics represented by the quadratic forms. These statistics have recently found application in Gaussian processes and have proven to serve as efficient tools for hypothesis testing. Here, we examine two statistics-the sample autocovariance function and the empirical anomaly measure-utilizing the correlation properties of the considered models. Based on these statistics, we introduce a testing procedure to differentiate between the two models. We present analytical and simulation results considering the two-point and beta distributions as exemplary distributions of the random Hurst exponent. Finally, to demonstrate the utility of the presented methodology, we analyze real-world datasets from the financial market and single particle tracking experiment in biological gels.

Anomalous diffusion analysis of semantic evolution in major Indo-European languages.

How do words change their meaning? Although semantic evolution is driven by a variety of distinct factors, including linguistic, societal, and technological ones, we find that there is one law that holds universally across five major Indo-European languages: that semantic evolution is subdiffusive. Using an automated pipeline of diachronic distributional semantic embedding that controls for underlying symmetries, we show that words follow stochastic trajectories in meaning space with an anomalous diffusion exponent α = 0.45 ± 0.05 across languages, in contrast with diffusing particles that follow α = 1. Randomization methods indicate that preserving temporal correlations in semantic change directions is necessary to recover strongly subdiffusive behavior; however, correlations in change sizes play an important role too. We furthermore show that strong subdiffusion is a robust phenomenon under a wide variety of choices in data analysis and interpretation, such as the choice of fitting an ensemble average of displacements or averaging best-fit exponents of individual word trajectories.

Inferring pointwise diffusion properties of single trajectories with deep learning

To characterize the mechanisms governing the diffusion of particles in biological scenarios, it is essential to accurately determine their diffusive properties. To do so, we propose a machine-learning method to characterize diffusion processes with time-dependent properties at the experimental time resolution. Our approach operates at the single-trajectory level predicting the properties of interest, such as the diffusion coefficient or the anomalous diffusion exponent, at every time step of the trajectory. In this way, changes in the diffusive properties occurring along the trajectory emerge naturally in the prediction and thus allow the characterization without any prior knowledge or assumption about the system. We first benchmark the method on synthetic trajectories simulated under several conditions. We show that our approach can successfully characterize both abrupt and continuous changes in the diffusion coefficient or the anomalous diffusion exponent. Finally, we leverage the method to analyze experiments of single-molecule diffusion of two membrane proteins in living cells: the pathogen-recognition receptor DC-SIGN and the integrin α5β1. The analysis allows us to characterize physical parameters and diffusive states with unprecedented accuracy, shedding new light on the underlying mechanisms.

Electrostatic hindrance to diffusion in flexible crosslinked gels: A coarse-grained simulation study.

In this work, we study how electrostatic forces slow down the diffusion of solute in flexible gels through coarse-grained simulations. The model used explicitly considers the movement of solute particles and polyelectrolyte chains. These movements are performed by following a Brownian dynamics algorithm. The effect of three electrostatic parameters characterizing the system (solute charge, polyelectrolyte chain charge, and ionic strength) is analyzed. Our results show that the behavior of both the diffusion coefficient and the anomalous diffusion exponent changes upon the reversal of the electric charge of one of the species. In addition, the diffusion coefficient in flexible gels differs significantly from that in rigid gels if the ionic strength is low enough. However, the effect of chain flexibility on the exponent of anomalous diffusion is significant even at high ionic strength (100 mM). Our simulations also prove that varying the polyelectrolyte chain charge does not have exactly the same effect as varying the solute particle charge.

Gramian angular fields for leveraging pretrained computer vision models with anomalous diffusion trajectories.

Anomalous diffusion is present at all scales, from atomic to large ones. Some exemplary systems are ultracold atoms, telomeres in the nucleus of cells, moisture transport in cement-based materials, arthropods' free movement, and birds' migration patterns. The characterization of the diffusion gives critical information about the dynamics of these systems and provides an interdisciplinary framework with which to study diffusive transport. Thus, the problem of identifying underlying diffusive regimes and inferring the anomalous diffusion exponent α with high confidence is critical to physics, chemistry, biology, and ecology. Classification and analysis of raw trajectories combining machine learning techniques with statistics extracted from them have widely been studied in the Anomalous Diffusion Challenge [Muñoz-Gil etal., Nat. Commun. 12, 6253 (2021)2041-172310.1038/s41467-021-26320-w]. Here we present a new data-driven method for working with diffusive trajectories. This method utilizes Gramian angular fields (GAF) to encode one-dimensional trajectories as images (Gramian matrices), while preserving their spatiotemporal structure for input to computer-vision models. This allows us to leverage two well-established pretrained computer-vision models, ResNet and MobileNet, to characterize the underlying diffusive regime and infer the anomalous diffusion exponent α. Short raw trajectories of lengths between 10 and 50 are commonly encountered in single-particle tracking experiments and are the most difficult ones to characterize. We show that GAF images can outperform the current state-of-the-art while increasing accessibility to machine learning methods in an applied setting.

Characterization of anomalous diffusion through convolutional transformers

The results of the Anomalous Diffusion Challenge (AnDi Challenge) (Muñoz-Gil G et al 2021 Nat. Commun. 12 6253) have shown that machine learning methods can outperform classical statistical methodology at the characterization of anomalous diffusion in both the inference of the anomalous diffusion exponent α associated with each trajectory (Task 1), and the determination of the underlying diffusive regime which produced such trajectories (Task 2). Furthermore, of the five teams that finished in the top three across both tasks of the AnDi Challenge, three of those teams used recurrent neural networks (RNNs). While RNNs, like the long short-term memory network, are effective at learning long-term dependencies in sequential data, their key disadvantage is that they must be trained sequentially. In order to facilitate training with larger data sets, by training in parallel, we propose a new transformer based neural network architecture for the characterization of anomalous diffusion. Our new architecture, the Convolutional Transformer (ConvTransformer) uses a bi-layered convolutional neural network to extract features from our diffusive trajectories that can be thought of as being words in a sentence. These features are then fed to two transformer encoding blocks that perform either regression (Task 1 1D) or classification (Task 2 1D). To our knowledge, this is the first time transformers have been used for characterizing anomalous diffusion. Moreover, this may be the first time that a transformer encoding block has been used with a convolutional neural network and without the need for a transformer decoding block or positional encoding. Apart from being able to train in parallel, we show that the ConvTransformer is able to outperform the previous state of the art at determining the underlying diffusive regime (Task 2 1D) in short trajectories (length 10–50 steps), which are the most important for experimental researchers.

Water adsorption and diffusion in phosphoric acid-based geopolymer using molecular modeling

Phosphoric acid-based geopolymer (PABG) is an environmentally friendly cementitious material. The adsorption and diffusion of water in PABG are investigated at the atomic scale. The adsorption isotherms are fitted by the Dubinin-Astakhov (DA) equation, and the adsorption isotherm is Type-V, qualitatively consistent with porous materials' adsorption characteristics. Adsorption isotherms and isosteric heats indicate that high temperatures weaken the attraction between water molecules and PABG, decreasing the adsorption amount with temperature. Water MSD suggests that the diffusion type is subdiffusion (one type of anomalous diffusion), verified by monitoring water molecules' diffusion trajectories, where the water molecule mainly fluctuates inside the pore and rarely jumps between pores. Effects of temperature and concentration on diffusion are analyzed. The anomalous diffusion exponent α, which reflects the diffusion rate, rises with temperature. Higher temperature enhances water molecule movement, and diffusion is accelerated. Higher water loading reduces α due to steric hindrance.

Fractional Brownian gyrator

When a physical system evolves in a thermal bath kept at a constant temperature, it eventually reaches an equilibrium state which properties are independent of the kinetic parameters and of the precise evolution scenario. This is generically not the case for a system driven out of equilibrium which, on the contrary, reaches a steady-state with properties that depend on the full details of the dynamics such as the driving noise and the energy dissipation. How the steady state depends on such parameters is in general a non-trivial question. Here, we approach this broad problem using a minimal model of a two-dimensional nano-machine, the Brownian gyrator, that consists of a trapped particle driven by fractional Gaussian noises—a family of noises with long-ranged correlations in time and characterized by an anomalous diffusion exponent α. When the noise is different in the different spatial directions, our fractional Brownian gyrator persistently rotates. Even if the noise is non-trivial, with long-ranged time correlations, thanks to its Gaussian nature we are able to characterize analytically the resulting nonequilibrium steady state by computing the probability density function, the probability current, its curl and the angular velocity and complement our study by numerical results.

Stochastic dynamics with multiplicative dichotomic noise: Heterogeneous telegrapher’s equation, anomalous crossovers and resetting

We analyze diffusion processes with finite propagation speed in a non-homogeneous medium in terms of the heterogeneous telegrapher’s equation. In the diffusion limit of infinite-velocity propagation we recover the results for the heterogeneous diffusion process. The heterogeneous telegrapher’s process exhibits a rich variety of diffusion regimes including hyperdiffusion, ballistic motion, superdiffusion, normal diffusion and subdiffusion, and different crossover dynamics characteristic for complex systems in which anomalous diffusion is observed. The anomalous diffusion exponent in the short time limit is twice the exponent in the long time limit, in accordance to the crossover dynamics from ballistic diffusion to normal diffusion in the standard telegrapher’s process. We also analyze the finite-velocity heterogeneous diffusion process in presence of stochastic Poissonian resetting. We show that the system reaches a non-equilibrium stationary state. The transition to this non-equilibrium steady state is analyzed in terms of the large deviation function.

Production Decline Analysis of Tight Conglomerate Reservoirs with Small Well Spacing, Based on the Fractal Characteristics of Fracture Networks

The conglomerate matrix and fracture propagation are special in tight conglomerate reservoir with small well spacing. In this article, the fractal propagation characteristics of the fracture network in conglomerate reservoirs are described by experiment and a micro-mathematical model. According to the core slice, the conglomerate reservoir matrix presents the multi-modal pore structure, described as the “pseudo-dual-media” model. Given the above, the unsteady seepage mathematical model, comprehensively considering the fractal fracture network, stress sensitivity of main fractures, and threshold pressure gradient of the reservoir matrix, was developed and analytically solved. The Blasingame type curves for production decline analysis were plotted, and the sensitive parameters were analyzed. The field application was performed for validation. The research results show that the fractal dimension decides the complexity of the fracture network distribution. As it increases, the unsteady flow occurs earlier, and the boundary flow is delayed. The anomalous diffusion exponent represents the smoothness of crude oil migration and a higher value leads to higher resistance to oil migration and larger pressure drawdown for the same production rate. The growth of the threshold pressure gradient within a certain range can result in a localized downward shift of the type curves. The field application in a conglomerate oil reservoir showed that the presented model presents a fitting accuracy 10% higher than that of the conventional SRV model and has high reliability and precision for the production performance evaluation of the small-well-spacing development of tight conglomerate reservoirs.

Bayesian deep learning for error estimation in the analysis of anomalous diffusion

Modern single-particle-tracking techniques produce extensive time-series of diffusive motion in a wide variety of systems, from single-molecule motion in living-cells to movement ecology. The quest is to decipher the physical mechanisms encoded in the data and thus to better understand the probed systems. We here augment recently proposed machine-learning techniques for decoding anomalous-diffusion data to include an uncertainty estimate in addition to the predicted output. To avoid the Black-Box-Problem a Bayesian-Deep-Learning technique named Stochastic-Weight-Averaging-Gaussian is used to train models for both the classification of the diffusion model and the regression of the anomalous diffusion exponent of single-particle-trajectories. Evaluating their performance, we find that these models can achieve a well-calibrated error estimate while maintaining high prediction accuracies. In the analysis of the output uncertainty predictions we relate these to properties of the underlying diffusion models, thus providing insights into the learning process of the machine and the relevance of the output.

Superstatistical approach of the anomalous exponent for scaled Brownian motion

Anomalous diffusion phenomenon is an intriguing process that tracer diffusion presents in numerous complex systems. Current experimental and theoretical investigations have reported the emergence of random diffusivity scenarios accompanied by the heterogeneity of the α-anomalous diffusion exponents. In this framework, we investigate a heterogeneous ensemble of tracers governed by scaled Brownian motion (sBm). The heterogeneous features are considered on anomalous diffusion exponent and diffusivity. We introduce two superstatistics of anomalous exponent, the truncated-Gaussian and truncated χ2-Gamma distributions. In this way, we show how the statistics of anomalous exponent affect the spreading of a mixture of particles governed by sBm. We also analyse the role of different temporal scales of sBm in superstatistics. Furthermore, we investigate the effects of coupling between diffusivity and anomalous exponent on superstatistics of sBm. The investigation provides a thorough analysis of simulation and analytical approach. The results imply rich classes of anomalous diffusion processes accompanied by non-Gaussian diffusion.

Absence of stationary states and non-Boltzmann distributions of fractional Brownian motion in shallow external potentials

We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x| c (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise ξ α (t) with 0 < α ⩽ 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent α. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 − 1/α) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Lévy flights in shallow external potentials.

Decomposing the effect of anomalous diffusion enables direct calculation of the Hurst exponent and model classification for single random paths

Recently, a large number of research teams from around the world collaborated in the so-called ‘anomalous diffusion challenge’. Its aim: to develop and compare new techniques for inferring stochastic models from given unknown time series, and estimate the anomalous diffusion exponent in data. We use various numerical methods to directly obtain this exponent using the path increments, and develop a questionnaire for model selection based on feature analysis of a set of known stochastic processes given as candidates. Here, we present the theoretical background of the automated algorithm which we put for these tasks in the diffusion challenge, as a counter to other pure data-driven approaches.

Dynamic scaling of out-of-plane fluctuations in freestanding graphene

We study the dynamical behavior of the mean-square displacement of height fluctuations of freestanding graphene using a phase-field-crystal model introduced recently. We find that the dynamic scaling behavior obtained numerically at long times is well described by the scaling theory of polymerized membranes. The critical exponent characterizing the power-law increase with time depends only on the equilibrium roughening exponent $\ensuremath{\xi}$ as $\ensuremath{\alpha}=\ensuremath{\xi}/(1+\ensuremath{\xi})$. For sufficiently long times it crosses over to linear behavior for finite-size systems. The critical exponent $\ensuremath{\alpha}$ is in good agreement with the anomalous diffusion exponent observed experimentally in graphene, suggesting this is a property that could also be observable in other two-dimensional crystalline materials.

Closed-form multi-dimensional solutions and asymptotic behaviours for subdiffusive processes with crossovers: II. Accelerating case

Anomalous diffusion with a power-law time dependence of the mean squared displacement occurs quite ubiquitously in numerous complex systems. Often, this anomalous diffusion is characterised by crossovers between regimes with different anomalous diffusion exponents α i . Here we consider the case when such a crossover occurs from a first regime with α 1 to a second regime with α 2 such that α 2 > α 1, i.e., accelerating anomalous diffusion. A widely used framework to describe such crossovers in a one-dimensional setting is the bi-fractional diffusion equation of the so-called modified type, involving two time-fractional derivatives defined in the Riemann–Liouville sense. We here generalise this bi-fractional diffusion equation to higher dimensions and derive its multidimensional propagator (Green’s function) for the general case when also a space fractional derivative is present, taking into consideration long-ranged jumps (Lévy flights). We derive the asymptotic behaviours for this propagator in both the short- and long-time as well the short- and long-distance regimes. Finally, we also calculate the mean squared displacement, skewness and kurtosis in all dimensions, demonstrating that in the general case the non-Gaussian shape of the probability density function changes.