Apparent anomalous diffusion and non-Gaussian distributions in a simple mobile-immobile transport model with Poissonian switching

Quantifying anomalous chemical diffusion through disordered porous rock materials

Fickian (normal) diffusion models show limitations in quantifying diffusion-controlled migration of solute species through porous rock structures, as observed in experiments. Anomalous diffusion prevails and can be interpreted using a Continuous Time Random Walk (CTRW) framework with a clear mechanistic underpinning. From the associated fractional diffusion equation we derive solutions over a broad range of anomalous diffusion behaviours, from highly anomalous to nearly Fickian, that yield temporal breakthrough curves and spatial concentration profiles of diffusing solutes. We illustrate that these solutions can be tailored to match realistic experimental conditions and resulting measurements that display anomalous diffusion. In particular, our analysis enables clear differentiation between early-time Fickian and anomalous diffusion, which becomes more pronounced over longer durations. It is shown that recent measurements of diffusion in natural rocks display distinct anomalous behaviour, with significant implications for critical assessment of solute migration in diverse geological and engineering applications.

Diffusion of fast and slow excitons with an exchange in quasi-two-dimensional systems

By means of analytical calculations and numerical simulations, we study the diffusion properties in quasi-two-dimensional structures with two exciton subsystems with an exchange between them. The experimental realization is possible in systems where fast and slow exciton subsystems appear. For substantially different diffusion coefficients of the species, the negative diffusion can be observed if one measures the transport properties of only a single subsystem, just as was obtained in experimental studies for quasi-two-dimensional semiconductor systems. The initial transition from a fast subsystem to a slow one results in a delayed release of fast excitons in the area close to the original excitation spot. Hence, the signal from the fast excitons alone includes the delayed replenishment from the transition of slow quasiparticles. This is seen as the narrowing of the exciton density profile or decrease of mean-squared displacement which is then interpreted as a negative diffusion. We show that the analytical theory matches the available experimental data for negative diffusion quite well. The average diffusion coefficients for the combined population are analytically expressed through the diffusion coefficients of fast and slow excitons. Simple analytical expressions for effective diffusion coefficients obtained in limiting cases are of interest both for theoretical and experimental analysis of not only the exciton transport, but also for a variety of systems, where fast and slow moving subsystems are present.

Defect related anomalous mobility of small polarons in dielectric oxides at the example of congruent lithium niobate

Polarons play a major role in the description of optical, electrical and dielectrical properties of several ferroelectric oxides. The motion of those particles occurs by elementary hops among the material lattice sites. In order to compute macroscopic transport parameters such as charge mobility, normal (i.e. Fickian) diffusion laws are generally assumed. In this paper we show that when defect states able to trap the polarons for long times are considered, significant deviations from the normal diffusion behaviour arise. As an example of this behavior, we consider here the case of lithium niobate (LN). This can be considered as a prototypical system, having a rich landscape of interacting polaron types and for which a significant wealth of information is available in literature. Our analysis considers the case of a stoichiometric, defect-free lithium niobate containing a certain concentration of small electron polarons hopping on regular Nb sites, and compares it to the material in congruent composition, which is generally found in real-life applications and which is characterized by a large concentration of antisite NbLi defects. While in the first case the charge carriers are free polarons hopping on a regular Nb sublattice, in the second case a fraction of polarons is trapped on antisite defects. Thus, in the congruent material, a range of different hopping possibilities arises, depending on the type of starting and destination sites. We develop a formalism encompassing all these microscopic processes in the framework of a switching diffusion model which can be well approximated by a mobile-immobile transport model providing explicit expressions for the polaron mobility. Finally, starting from the Marcus-Holstein's model for the polaron hopping frequency we verify by means of a Monte Carlo approach the diffusion/mobility of the different polarons species showing that, while free polarons obey the laws for normal diffusion as expected, bound polarons follow an anomalous diffusion behaviour and that in the case of the congruent crystal where mixed free and bound polaron transport is involved, our expressions indeed provide a satisfactory description.

Hidden Markov modeling of single-particle diffusion with stochastic tethering

The statistics of the diffusive motion of particles often serve as an experimental proxy for their interaction with the environment. However, inferring the physical properties from the observed trajectories is challenging. Inspired by a recent experiment, here we analyze the problem of particles undergoing two-dimensional Brownian motion with transient tethering to the surface. We model the problem as a hidden Markov model where the physical position is observed and the tethering state is hidden. We develop an alternating maximization algorithm to infer the hidden state of the particle and estimate the physical parameters of the system. The crux of our method is a saddle-point-like approximation, which involves finding the most likely sequence of hidden states and estimating the physical parameters from it. Extensive numerical tests demonstrate that our algorithm reliably finds the model parameters and is insensitive to the initial guess. We discuss the different regimes of physical parameters and the algorithm's performance in these regimes. We also provide a free software implementation of our algorithm.

Negative diffusion of excitons in quasi-two-dimensional systems

We show how two different mobile-immobile type models explain the observation of negative diffusion of excitons reported in experimental studies in quasi-two-dimensional semiconductor systems. The main reason for the effect is the initial trapping and a delayed release of free excitons in the area close to the original excitation spot. The density of trapped excitons is not registered experimentally. Hence, the signal from the free excitons alone includes the delayed release of not diffusing trapped particles. This is seen as the narrowing of the exciton density profile or decrease of mean-squared displacement which is then interpreted as a negative diffusion. The effect is enhanced with the increase of recombination intensity as well as the rate of the exciton-exciton binary interactions.

Machine-Learning Solutions for the Analysis of Single-Particle Diffusion Trajectories

Single-particle traces of the diffusive motion of molecules, cells, or animals are by now routinely measured, similar to stochastic records of stock prices or weather data. Deciphering the stochastic mechanism behind the recorded dynamics is vital in understanding the observed systems. Typically, the task is to decipher the exact type of diffusion and/or to determine the system parameters. The tools used in this endeavor are currently being revolutionized by modern machine-learning techniques. In this Perspective we provide an overview of recently introduced methods in machine-learning for diffusive time series, most notably, those successfully competing in the anomalous diffusion challenge. As such methods are often criticized for their lack of interpretability, we focus on means to include uncertainty estimates and feature-based approaches, both improving interpretability and providing concrete insight into the learning process of the machine. We expand the discussion by examining predictions on different out-of-distribution data. We also comment on expected future developments.

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

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