arXiv subjects

Ralf Metzler

Publications and source records attributed to Ralf Metzler.

At least 91 records · Page 5Linked to original sources

arXiv2020

Diffusion of antibiotics through a biofilm in the presence of diffusion and absorption barriers

We propose a model of antibiotic diffusion through a bacterial biofilm when diffusion and/or absorption barriers develop in the biofilm. The idea of this model is: We deduce details of the diffusion process in a medium in which direct experimental study is difficult, based on probing diffusion in external regions. Since a biofilm has a gel-like consistency, we suppose that subdiffusion of particles in the biofilm may occur. To describe this process we use a fractional subdiffusion-absorption equation with an adjustable anomalous diffusion exponent. The boundary conditions at the boundaries of the biofilm are derived by means of a particle random walk model on a discrete lattice leading to an expression involving a fractional time derivative. We show that the temporal evolution of the total amount of substance that has diffused through the biofilm explicitly depends on whether there is antibiotic absorption in the biofilm. This fact is used to experimentally check for antibiotic absorption in the biofilm and if the biofilm parameters change over time. We propose a four-stage model of antibiotic diffusion in biofilm based on the mentioned above physical characteristics. The biological interpretation of the stages, in particular their relation with the bacterial defence mechanisms, is discussed. Theoretical results are compared with empirical results of ciprofloxacin diffusion through Pseudomonas aeruginosa biofilm, and ciprofloxacin and gentamicin diffusion through Proteus mirabilis biofilm.

physics.bio-ph↗

arXiv2020

Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories

Extensive time-series encoding the position of particles such as viruses, vesicles, or individual proteins are routinely garnered in single-particle tracking experiments or supercomputing studies. They contain vital clues on how viruses spread or drugs may be delivered in biological cells. Similar time-series are being recorded of stock values in financial markets and of climate data. Such time-series are most typically evaluated in terms of time-average mean-squared displacements, which remain random variables for finite measurement times. Their statistical properties are different for different physical stochastic processes, thus allowing us to extract valuable information on the stochastic process itself. To exploit the full potential of the statistical information encoded in measured time-series we here propose an easy-to-implement and computationally inexpensive new methodology, based on deviations of the time-averaged mean-squared displacement from its ensemble average counterpart. Specifically, we use the upper bound of these deviations for Brownian motion to check the applicability of this approach to simulated and real data sets. By comparing the probability of deviations for different data sets, we demonstrate how the theoretical bound for Brownian motion reveals additional information about observed stochastic processes. We apply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracer beads measured in mucin hydrogels, and of geographic surface temperature anomalies. Our analysis shows how the large-deviation properties can be efficiently used as a simple yet effective routine test to reject the Brownian motion hypothesis and unveil crucial information on statistical properties such as ergodicity breaking and short-time correlations.

cond-mat.stat-mech↗

arXiv2020

Unexpected crossovers in correlated random-diffusivity processes

The passive and active motion of micron-sized tracer particles in crowded liquids and inside living biological cells is ubiquitously characterised by "viscoelastic" anomalous diffusion, in which the increments of the motion feature long-ranged negative and positive correlations. While viscoelastic anomalous diffusion is typically modelled by a Gaussian process with correlated increments, so-called fractional Gaussian noise, an increasing number of systems are reported, in which viscoelastic anomalous diffusion is paired with non-Gaussian displacement distributions. Following recent advances in Brownian yet non-Gaussian diffusion we here introduce and discuss several possible versions of random-diffusivity models with long-ranged correlations. While all these models show a crossover from non-Gaussian to Gaussian distributions beyond some correlation time, their mean squared displacements exhibit strikingly different behaviours: depending on the model crossovers from anomalous to normal diffusion are observed, as well as unexpected dependencies of the effective diffusion coefficient on the correlation exponent. Our observations of the strong non-universality of random-diffusivity viscoelastic anomalous diffusion are important for the analysis of experiments and a better understanding of the physical origins of "viscoelastic yet non-Gaussian" diffusion.

cond-mat.stat-mech↗

arXiv2020

Lévy walk dynamics in an external harmonic potential

Lévy walks (LWs) are spatiotemporally coupled random-walk processes describing superdiffusive heat conduction in solids, propagation of light in disordered optical materials, motion of molecular motors in living cells, or motion of animals, humans, robots, and viruses. We here investigate a key feature of LWs, their response to an external harmonic potential. In this generic setting for confined motion we demonstrate that LWs equilibrate exponentially and may assume a bimodal stationary distribution. We also show that the stationary distribution has a horizontal slope next to a reflecting boundary placed at the origin, in contrast to correlated superdiffusive processes. Our results generalize LWs to confining forces and settle some long-standing puzzles around LWs.

cond-mat.stat-mech↗

arXiv2020

AnDi: The Anomalous Diffusion Challenge

The deviation from pure Brownian motion generally referred to as anomalous diffusion has received large attention in the scientific literature to describe many physical scenarios. Several methods, based on classical statistics and machine learning approaches, have been developed to characterize anomalous diffusion from experimental data, which are usually acquired as particle trajectories. With the aim to assess and compare the available methods to characterize anomalous diffusion, we have organized the Anomalous Diffusion (AnDi) Challenge (\url{http://www.andi-challenge.org/}). Specifically, the AnDi Challenge will address three different aspects of anomalous diffusion characterization, namely: (i) Inference of the anomalous diffusion exponent. (ii) Identification of the underlying diffusion model. (iii) Segmentation of trajectories. Each problem includes sub-tasks for different number of dimensions (1D, 2D and 3D). In order to compare the various methods, we have developed a dedicated open-source framework for the simulation of the anomalous diffusion trajectories that are used for the training and test datasets. The challenge was launched on March 1, 2020, and consists of three phases. Currently, the participation to the first phase is open. Submissions will be automatically evaluated and the performance of the top-scoring methods will be thoroughly analyzed and compared in an upcoming article.

cond-mat.stat-mech↗

arXiv2019

Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control

Text-book concepts of diffusion- versus kinetic-control are well-defined for reaction-kinetics involving macroscopic concentrations of diffusive reactants that are adequately described by rate-constants -- the inverse of the mean-first-passage-time to the reaction-event. In contradistinction, an open important question is whether the mean-first-passage-time alone is a sufficient measure for biochemical reactions that involve nanomolar reactant concentrations. Here, using a simple yet generic, exactly-solvable model we study the conspiratory effect of diffusion and chemical reaction-limitations on the full reaction-time distribution. We show that it has a complex structure with four distinct regimes delimited by three characteristic time scales spanning a window of several decades. Consequently, the reaction-times are defocused: no unique time-scale characterises the reaction-process, diffusion- and kinetic-control can no longer be disentangled, and it is imperative to know the full reaction-time distribution. We introduce the concepts of geometry- and reaction-control, and also quantify each regime by calculating the corresponding reaction depth.

physics.chem-ph↗

arXiv2019

Probability density of the fractional Langevin equation with reflecting walls

We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while anti-persistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for non-thermal fractional Brownian motion with reflecting walls, and we discuss broader implications.

cond-mat.stat-mech↗

arXiv2019

First passage properties of asymmetric Lévy flights

Lévy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments---the "jump lengths"---are drawn from an $α$-stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy Flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy Flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy Flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index $α$ and the skewness (asymmetry) parameter $β$. The other approach is based on the stochastic Langevin equation with $α$-stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.

cond-mat.stat-mech↗

arXiv2019

Brownian motion and beyond: first-passage, power spectrum, non-Gaussianity, and anomalous diffusion

Brownian motion is a ubiquitous physical phenomenon across the sciences. After its discovery by Brown and intensive study since the first half of the 20th century, many different aspects of Brownian motion and stochastic processes in general have been addressed in Statistical Physics. In particular, there now exist a very large range of applications of stochastic processes in various disciplines. Here, we highlight some of the advances in stochastic processes prompted by novel experimental methods such as superresolution microscopy. Here we provide a summary of some of the recent developments highlighting both the experimental findings and theoretical frameworks.

cond-mat.stat-mech↗

arXiv2019

Gumbel Central Limit Theorem for Max-Min and Min-Max

The Max-Min and Min-Max of matrices arise prevalently in science and engineering. However, in many real-world situations the computation of the Max-Min and Min-Max is challenging as matrices are large and full information about their entries is lacking. Here we take a statistical-physics approach and establish limit-laws -- akin to the Central Limit Theorem -- for the Max-Min and Min-Max of large random matrices. The limit-laws intertwine random-matrix theory and extreme-value theory, couple the matrix-dimensions geometrically, and assert that Gumbel statistics emerge irrespective of the matrix-entries' distribution. Due to their generality and universality, as well as their practicality, these novel results are expected to have a host of applications in the physical sciences and beyond.

cond-mat.stat-mech↗

arXiv2019

Poisson-process limit-laws yield Gumbel Max-Min and Min-Max

"A chain is only as strong as its weakest link" says the proverb. But what about a collection of statistically identical chains: How long till all chains fail? The answer to this question is given by the Max-Min of a matrix whose $\left(i,j\right)$ entry is the failure time of link $j$ of chain $i$: take the minimum of each row, and then the maximum of the rows' minima. The corresponding Min-Max is obtained by taking the maximum of each column, and then the minimum of the columns' maxima. The Min-Max applies to the storage of critical data. Indeed, consider multiple backup copies of a set of critical data items, and consider the $\left(i,j\right)$ matrix entry to be the time at which item $j$ on copy $i$ is lost; then, the Min-Max is the time at which the first critical data item is lost. In this paper, we address random matrices whose entries are independent and identically distributed random variables. We establish Poisson-process limit-laws for the row's minima and for the columns' maxima. Then, we further establish Gumbel limit-laws for the Max-Min and for the Min-Max. The limit-laws hold whenever the entries' distribution has a density, and the Gumbel limit-laws yield highly applicable approximation tools and design tools for large random matrices.

cond-mat.stat-mech↗

arXiv2019

Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems

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. We start 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.

cond-mat.stat-mech↗

arXiv2019

Correlations and transport in exclusion processes with general finite memory

We consider the correlations and the hydrodynamic description of random walkers with a general finite memory moving on a $d$ dimensional hypercubic lattice. We derive a drift-diffusion equation and identify a memory-dependent critical density. Above the critical density, the effective diffusion coefficient decreases with the particles' propensity to move forward and below the critical density it increases with their propensity to move forward. If the correlations are neglected the critical density is exactly $1/2$. We also derive a low-density approximation for the same time correlations between different sites. We perform simulations on a one-dimensional system with one-step memory and find good agreement between our analytical derivation and the numerical results. We also consider the previously unexplored special case of totally anti-persistent particles. Generally, the correlation length converges to a finite value. However in the special case of totally anti-persistent particles and density $1/2$, the correlation length diverges with time. Furthermore, connecting a system of totally anti-persistent particles to external particle reservoirs creates a new phenomenon: In almost all systems, regardless of the precise details of the microscopic dynamics, when a system is connected to a reservoir, the mean density of particle at the edge is the same as the reservoir following the zeroth law of thermodynamics. In a totally anti-persistent system, however, the density at the edge is always higher than in the reservoir. We find a qualitative description of this phenomenon which agrees reasonably well with the numerics.

cond-mat.stat-mech↗

arXiv2019

Transport in exclusion processes with one-step memory: density dependence and optimal acceleration

We study a lattice gas of persistent walkers, in which each site is occupied by at most one particle and the direction each particle attempts to move to depends on its last step. We analyse the mean squared displacement (MSD) of the particles as a function of the particle density and their persistence (the tendency to continue moving in the same direction). For positive persistence the MSD behaves as expected: it increases with the persistence and decreases with the density. However, for strong anti-persistence we find two different regimes, in which the dependence of the MSD on the density is non-monotonic. For very strong anti-persistence there is an optimal density at which the MSD reaches a maximum. In an intermediate regime, the MSD as a function of the density exhibits both a minimum and a maximum, a phenomenon which has not been observed before. We derive a mean-field theory which qualitatively explains this behaviour.

cond-mat.stat-mech↗

arXiv2019

Codifference can detect ergodicity breaking and non-Gaussianity

We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a non-linear analogue of the mean squared displacement.

cond-mat.stat-mech↗

arXiv2019

Single-trajectory spectral analysis of scaled Brownian motion

A standard 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, $T\to\infty$. 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 $T\to\infty$ 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. We demonstrate 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. We also 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 single-trajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.

cond-mat.stat-mech↗

arXiv2019

Generalised Diffusion and Wave Equations: Recent Advances

We present a short overview of the recent results in the theory of diffusion and wave equations with generalised derivative operators. We give generic examples of such generalised diffusion and wave equations, which include time-fractional, distributed order, and tempered time-fractional diffusion and wave equations. Such equations exhibit multi-scaling time behaviour, which makes them suitable for the description of different diffusive regimes and characteristic crossover dynamics in complex systems.

cond-mat.stat-mech↗

arXiv2018

Towards a full quantitative description of single-molecule reaction kinetics in biological cells

The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. In particular, it quantifies the statistics of instances when biomolecules in a biological cell reach their specific binding sites and trigger cellular regulation. Typically, the first-passage properties are given in terms of mean first-passage times. However, modern experiments now monitor single-molecular binding-processes in living cells and thus provide access to the full statistics of the underlying first-passage events, in particular, inherent cell-to-cell fluctuations. We here present a robust explicit approach for obtaining the distribution of FPTs to a small partially-reactive target in cylindrical-annulus domains, which represent typical bacterial and neuronal cell shapes. We investigate various asymptotic behaviours of this FPT distribution and show that it typically is very broad in many biological situations: thus, the mean FPT can differ from the most probable FPT by orders of magnitude. The most probable FPT is shown to strongly depend only on the starting position within the geometry and to be almost independent of the target size and reactivity. These findings demonstrate the dramatic relevance of knowing the full distribution of FPTs and thus open new perspectives for a more reliable description of many intracellular processes initiated by the arrival of one or few biomolecules to a small, spatially localised region inside the cell.

q-bio.SC↗