Non-Markovian diffusion of excitons in layered perovskites and transition metal dichalcogenides.

The diffusion of excitons in perovskites and transition metal dichalcogenides shows clear anomalous, subdiffusive behaviour in experiments. In this paper we develop a non-Markovian mobile-immobile model which provides an explanation of this behaviour through paired theoretical and simulation approaches. The simulation model is based on a random walk on a 2D lattice with randomly distributed deep traps such that the trapping time distribution involves slowly decaying power-law asymptotics. The theoretical model uses coupled diffusion and rate equations for free and trapped excitons, respectively, with an integral term responsible for trapping. The model provides a good fitting of the experimental data, thus, showing a way for quantifying the exciton diffusion dynamics.

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.

Fluctuations of random walks in critical random environments.

Percolation networks have been widely used in the description of porous media but are now found to be relevant to understand the motion of particles in cellular membranes or the nucleus of biological cells. Random walks on the infinite cluster at criticality of a percolation network are asymptotically ergodic. On any finite size cluster of the network stationarity is reached at finite times, depending on the cluster's size. Despite of this we here demonstrate by combination of analytical calculations and simulations that at criticality the disorder and cluster size average of the ensemble of clusters leads to a non-vanishing variance of the time averaged mean squared displacement, regardless of the measurement time. Fluctuations of this relevant experimental quantity due to the disorder average of such ensembles are thus persistent and non-negligible. The relevance of our results for single particle tracking analysis in complex and biological systems is discussed.

Effects of strain rate and surface cracks on the mechanical behaviour of Balmoral Red granite.

This work presents a systematic study on the effects of strain rate and surface cracks on the mechanical properties and behaviour of Balmoral Red granite. The tensile behaviour of the rock was studied at low and high strain rates using Brazilian disc samples. Heat shocks were used to produce samples with different amounts of surface cracks. The surface crack patterns were analysed using optical microscopy, and the complexity of the patterns was quantified by calculating the fractal dimensions of the patterns. The strength of the rock clearly drops as a function of increasing fractal dimensions in the studied strain rate range. However, the dynamic strength of the rock drops significantly faster than the quasi-static strength, and, because of this, also the strain rate sensitivity of the rock decreases with increasing fractal dimensions. This can be explained by the fracture behaviour and fragmentation during the dynamic loading, which is more strongly affected by the heat shock than the fragmentation at low strain rates.This article is part of the themed issue 'Experimental testing and modelling of brittle materials at high strain rates'.

Geometry controlled anomalous diffusion in random fractal geometries: looking beyond the infinite cluster.

We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law ∼T(-h) with h < 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided.