arXiv subjects

R. Metzler

Publications and source records attributed to R. Metzler.

At least 19 recordsLinked to original sources

arXiv2026

Universal Loop Statistics from Active Extrusion with Kinetic Barriers

We develop a kinetic theory of cohesin-driven loop extrusion on a disordered chromatin track with transient barriers. In the stationary state, the mean loop size is shown to obey a universal law determined by the bare processivity and a renormalized obstacle density. Beyond the mean, one-sided extrusion always yields a single-exponential loop-length distribution, whereas two-sided extrusion produces a finite sum of exponential modes and, generically, a peaked distribution. Experimental CTCF-anchored loop statistics exhibit such a peak, thereby providing a direct discriminator of extrusion symmetry. The theory therefore establishes a unified framework for disorder-limited loop extrusion and supports a scenario in which both cohesin arms actively operate in living cells.

cond-mat.soft↗

arXiv2024

Competing random searchers under restarts

We study independent searchers competing for a target under restarts and find that introduction of restarts tends to enhance the search efficiency of an already efficient searcher. As a result, the difference between the search probabilities of the individual searchers increases when the system is subject to restarts. This result holds true independent of the identity of individual searchers or the specific details of the distribution of restart times. However, when only one of a pair of searchers is subject to restarts while the other evolves in an unperturbed manner, a concept termed as subsystem restarts, we find that the search probability exhibits a nonmonotonic dependence on the restart rate. We also study the mean search time for a pair of run and tumble and Brownian searchers when only the run and tumble particle is subject to restarts. We find that, analogous to restarting the whole system, the mean search time exhibits a nonmonotonic dependence on restart rates.

cond-mat.stat-mech↗

arXiv2023

Anomalous and ultraslow diffusion of a particle driven by power-law-correlated and distributed-order noises

We study the generalized Langevin equation approach to anomalous diffusion for a harmonic oscillator and a free particle driven by different forms of internal noises, such as power-law-correlated and distributed-order noises that fulfil generalized versions of the fluctuation-dissipation theorem. The mean squared displacement and the normalized displacement correlation function are derived for the different forms of the friction memory kernel. The corresponding overdamped generalized Langevin equations for these cases are also investigated. It is shown that such models can be used to describe anomalous diffusion in complex media, giving rise to subdiffusion, superdiffusion, ultraslow diffusion, strong anomaly, and other complex diffusive behaviors.

cond-mat.stat-mech↗

arXiv2023

Time-dependent probability density function for partial resetting dynamics

Stochastic resetting is a rapidly developing topic in the field of stochastic processes and their applications. It denotes the occasional reset of a diffusing particle to its starting point and effects, inter alia, optimal first-passage times to a target. Recently the concept of partial resetting, in which the particle is reset to a given fraction of the current value of the process, has been established and the associated search behaviour analysed. Here we go one step further and we develop a general technique to determine the time-dependent probability density function (PDF) for Markov processes with partial resetting. We obtain an exact representation of the PDF in the case of general symmetric Lévy flights with stable index $0<α\le2$. For Cauchy and Brownian motions (i.e., $α=1,2$), this PDF can be expressed in terms of elementary functions in position space. We also determine the stationary PDF. Our numerical analysis of the PDF demonstrates intricate crossover behaviours as function of time.

cond-mat.stat-mech↗

arXiv2023

Directedeness, correlations, and daily cycles in springbok motion: from data over stochastic models to movement prediction

How predictable is the next move of an animal? Specifically, which factors govern the short- and long-term motion patterns and the overall dynamics of landbound, plant-eating animals and ruminants in particular? To answer this question, we here study the movement dynamics of springbok antelopes Antidorcas marsupialis. We propose complementary statistical analysis techniques combined with machine learning approaches to analyze, across multiple time scales, the springbok motion recorded in long-term GPS-tracking of collared springboks at a private wildlife reserve in Namibia. As a new result, we are able to predict the springbok movement within the next hour with a certainty of about 20\%. The remaining 80\% are stochastic in nature and are induced by unaccounted factors in the modeling algorithm and by individual behavioral features of springboks. We find that directedness of motion contributes approximately 17\% to this predicted fraction. We find that the measure for directedeness is strongly dependent on the daily cycle. The previously known daily affinity of springboks to their water points, as predicted from our machine learning algorithm, overall accounts for only 3\% of this predicted deterministic component of springbok motion. Moreover, the resting points are found to affect the motion of springboks at least as much as the formally studied effects of water points. The generality of these statements for the motion patterns and their underlying behavioral reasons for other ruminants can be examined on the basis of our statistical analysis tools.

cond-mat.stat-mech↗

arXiv2022

Non-Gaussianity in stochastic transport: phenomenology and modelling

Non-Gaussian shapes, despite a linear form of the mean-squared displacement, have been observed for the displacement distribution in a large range of diffusive systems. Stochastic models for such "Brownian yet non-Gaussian" diffusion will be introduced and discussed. Systems with non-Gaussian, anomalous diffusion will also be addressed.

cond-mat.stat-mech↗

arXiv2022

Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions

We present a framework for systems in which diffusion-advection transport of a tracer substance in a mobile zone is interrupted by trapping in an immobile zone. Our model unifies different model approaches based on distributed-order diffusion equations, exciton diffusion rate models, and random walk models for multi-rate mobile-immobile mass transport. We study various forms for the trapping time dynamics and their effects on the tracer mass in the mobile zone. Moreover we find the associated breakthrough curves, the tracer density at a fixed point in space as function of time, as well as the mobile and immobile concentration profiles and the respective moments of the transport. Specifically we derive explicit forms for the anomalous transport dynamics and an asymptotic power-law decay of the mobile mass for a Mittag-Leffler trapping time distribution. In our analysis we point out that even for exponential trapping time densities transient anomalous transport is observed. Our results have direct applications in geophysical contexts but also in biological, soft matter, and solid state systems.

cond-mat.stat-mech↗

arXiv2022

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

We analyse mobile-immobile transport of particles that switch between the mobile and immobile phases with finite rates. Despite this seemingly simple assumption of Poissonian switching we unveil a rich transport dynamics including significant transient anomalous diffusion and non-Gaussian displacement distributions. Our discussion is based on experimental parameters for tau proteins in neuronal cells, but the results obtained here are expected to be of relevance for a broad class of processes in complex systems. Concretely, we obtain that when the mean binding time is significantly longer than the mean mobile time, transient anomalous diffusion is observed at short and intermediate time scales, with a strong dependence on the fraction of initially mobile and immobile particles. We unveil a Laplace distribution of particle displacements at relevant intermediate time scales. For any initial fraction of mobile particles, the respective mean squared displacement displays a plateau. Moreover, we demonstrate a short-time cubic time dependence of the mean squared displacement for immobile tracers when initially all particles are immobile.

cond-mat.stat-mech↗

arXiv2021

Backbone diffusion and first-passage dynamics in a comb structure with confining branches under stochastic resetting

We study the diffusive motion of a test particle in a two-dimensional comb structure consisting of a main backbone channel with continuously distributed side branches, in the presence of stochastic Markovian resetting to the initial position of the particle. We assume that the motion along the infinitely long branches is biased by a confining potential. The crossover to the steady state is quantified in terms of a large deviation function, which is derived for the first time for comb structures in present paper. We show that the relaxation region is demarcated by a nonlinear "light-cone" beyond which the system is evolving in time. We also investigate the first-passage times along the backbone and calculate the mean first-passage time and optimal resetting rate.

cond-mat.stat-mech↗

arXiv2021

Stochastic resetting by a random amplitude

Stochastic resetting, a diffusive process whose amplitude is "reset" to the origin at random times, is a vividly studied strategy to optimize encounter dynamics, e.g., in chemical reactions. We here generalize the resetting step by introducing a random resetting amplitude, such that the diffusing particle may be only partially reset towards the trajectory origin, or even overshoot the origin in a resetting step. We introduce different scenarios for the random-amplitude stochastic resetting process and discuss the resulting dynamics. Direct applications are geophysical layering (stratigraphy) as well as population dynamics or financial markets, as well as generic search processes.

cond-mat.stat-mech↗

arXiv2020

Exact first-passage time distributions for three random diffusivity models

We study the extremal properties of a stochastic process $x_t$ defined by a Langevin equation $\dot{x}_t=\sqrt{2 D_0 V(B_t)}\,ξ_t$, where $ξ_t$ is a Gaussian white noise with zero mean, $D_0$ is a constant scale factor, and $V(B_t)$ is a stochastic "diffusivity" (noise strength), which itself is a functional of independent Brownian motion $B_t$. We derive exact, compact expressions for the probability density functions (PDFs) of the first passage time (FPT) $t$ from a fixed location $x_0$ to the origin for three different realisations of the stochastic diffusivity: a cut-off case $V(B_t) =Θ(B_t)$ (Model I), where $Θ(x)$ is the Heaviside theta function; a Geometric Brownian Motion $V(B_t)=\exp(B_t)$ (Model II); and a case with $V(B_t)=B_t^2$ (Model III). We realise that, rather surprisingly, the FPT PDF has exactly the Lévy-Smirnov form (specific for standard Brownian motion) for Model II, which concurrently exhibits a strongly anomalous diffusion. For Models I and III either the left or right tails (or both) have a different functional dependence on time as compared to the Lévy-Smirnov density. In all cases, the PDFs are broad such that already the first moment does not exist. Similar results are obtained in three dimensions for the FPT PDF to an absorbing spherical target.

cond-mat.stat-mech↗

arXiv2020

Continuous Time Random Walk in a velocity field: Role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixing

We consider the dynamics of a separable Continuous Time Random Walk (CTRW) when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding universe. The considered CTRW can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a Lévy flight. We first consider the case of zero velocity field. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of Lévy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: In one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.

cond-mat.stat-mech↗

arXiv2020

Universal spectral features of different classes of random diffusivity processes

.Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic $1/f^2$-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations.

cond-mat.stat-mech↗

arXiv2019

Full distribution of first exit times in the narrow escape problem

In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small "escape window" in the otherwise impermeable boundary, once it arrives to this window and over-passes an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the container's boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems.

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

First-passage and first-hitting times of Levy flights and Levy walks

For both Levy flight and Levy 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 Levy 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 Levy 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.

cond-mat.stat-mech↗

arXiv2019

Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries

Fractional Brownian motion is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically fractional Brownian motion confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected fractional Brownian motion 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 $\langle X^2(t)\rangle\simeq t^α$ 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 mean squared displacement 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 fractional Brownian motion for processes such as molecule or tracer diffusion in the confined of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.

cond-mat.stat-mech↗

arXiv2019

Spectral content of a single non-Brownian trajectory

Time-dependent processes are often analysed using the power spectral density (PSD), calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble-average. Frequently, the available experimental data sets are too small for such ensemble averages, and hence it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from $S(f,T)$, the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable, parametrized by frequency $f$ and observation-time $T$, for a broad family of anomalous diffusions---fractional Brownian motion (fBm) with Hurst-index $H$---and derive exactly its probability density function. We show that $S(f,T)$ is proportional---up to a random numerical factor whose universal distribution we determine---to the ensemble-averaged PSD. For subdiffusion ($H<1/2$) we find that $S(f,T)\sim A/f^{2H+1}$ with random-amplitude $A$. In sharp contrast, for superdiffusion $(H>1/2)$ $S(f,T)\sim BT^{2H-1}/f^2$ with random amplitude $B$. Remarkably, for $H>1/2$ the PSD exhibits the same frequency-dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably, for $H>1/2$ the PSD is ageing and is dependent on $T$. Our predictions for both sub- and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels, and by extensive simulations.

cond-mat.stat-mech↗