### Refine

#### Year of publication

#### Keywords

- anomalous diffusion (21)
- diffusion (10)
- stochastic processes (7)
- ageing (6)
- nonergodicity (4)
- superstatistics (4)
- transport (4)
- polymers (3)
- Brownian yet non-Gaussian diffusion (2)
- Debye screening (2)
- Langevin equation (2)
- Lévy flights (2)
- Lévy walks (2)
- Mittag-Leffler functions (2)
- aspect ratio (2)
- autoregressive models (2)
- behavior (2)
- biological physics (2)
- cambridge cb4 0wf (2)
- cambs (2)
- codifference (2)
- coefficient (2)
- coefficients (2)
- critical phenomena (2)
- cylindrical geometry (2)
- diffusing diffusivity (2)
- dynamics (2)
- dynamics simulation (2)
- electrostatic interactions (2)
- england (2)
- equation approach (2)
- financial time series (2)
- first passage (2)
- first passage time (2)
- first-hitting time (2)
- first-passage time (2)
- fractional Brownian motion (2)
- fractional dynamics (2)
- gene regulatory networks (2)
- generalised langevin equation (2)
- geometric Brownian motion (2)
- living cells (2)
- membrane (2)
- membrane channel (2)
- milton rd (2)
- monte-carlo (2)
- non-Gaussian diffusion (2)
- osmotic-pressure (2)
- polyelectrolyte adsorption (2)
- posttranslational protein translocation (2)
- protein search (2)
- random-walks (2)
- reflecting boundary conditions (2)
- royal soc chemistry (2)
- science park (2)
- single-particle tracking (2)
- single-stranded-dna (2)
- solid-state nanopores (2)
- stochastic processes (theory) (2)
- structured polynucleotides (2)
- subdiffusion (2)
- thomas graham house (2)
- time averaging (2)
- time series analysis (2)
- Asymptotic expansions (1)
- Composite fractional derivative (1)
- Fokker-Planck-Smoluchowski equation (1)
- Fox H-function (1)
- Fractional diffusion equation (1)
- Fractional moments (1)
- Grunwald-Letnikov approximation (1)
- Levy foraging hypothesis (1)
- Pareto analysis (1)
- Riesz-Feller fractional derivative (1)
- Sinai diffusion (1)
- active transport (1)
- chemical relaxation (1)
- confinement (1)
- conformational properties (1)
- continuous time random walk (CTRW) (1)
- continuous time random walks (1)
- crowded fluids (1)
- crowding (1)
- driven diffusive systems (theory) (1)
- fluctuations (theory) (1)
- fractional generalized Langevin equation (1)
- frictional memory kernel (1)
- gel network (1)
- mean square displacement (1)
- multi-scaling (1)
- path integration (1)
- polymer translocation (1)
- potential landscape (1)
- power spectral analysis (1)
- quenched energy landscape (1)
- random walks (1)
- reaction kinetics theory (1)
- reaction rate constants (1)
- scaled Brownian motion (1)
- search optimization (1)
- single trajectory analysis (1)
- single-file diffusion (1)
- van Hove correlation (1)
- variances (1)
- weak ergodicity breaking (1)

#### Institute

- Institut für Physik und Astronomie (120) (remove)

Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.

Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as ‘superstatistics’ or ‘diffusing diffusivity’. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models.Westart from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion.

Quorum-sensing bacteria in a growing colony of cells send out signalling molecules (so-called “autoinducers”) and themselves sense the autoinducer concentration in their vicinity. Once—due to increased local cell density inside a “cluster” of the growing colony—the concentration of autoinducers exceeds a threshold value, cells in this clusters get “induced” into a communal, multi-cell biofilm-forming mode in a cluster-wide burst event. We analyse quantitatively the influence of spatial disorder, the local heterogeneity of the spatial distribution of cells in the colony, and additional physical parameters such as the autoinducer signal range on the induction dynamics of the cell colony. Spatial inhomogeneity with higher local cell concentrations in clusters leads to earlier but more localised induction events, while homogeneous distributions lead to comparatively delayed but more concerted induction of the cell colony, and, thus, a behaviour close to the mean-field dynamics. We quantify the induction dynamics with quantifiers such as the time series of induction events and burst sizes, the grouping into induction families, and the mean autoinducer concentration levels. Consequences for different scenarios of biofilm growth are discussed, providing possible cues for biofilm control in both health care and biotechnology.

Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) 〈X² (t)〉 ⋍ tᵅ with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.

For both Lévy flight and Lévy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Lévy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Lévy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms.

Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.