arXiv subjects

Ralf Metzler

Publications and source records attributed to Ralf Metzler.

At least 37 records · Page 2Linked to original sources

arXiv2025

Different behaviors of diffusing diffusivity dynamics based on three different definitions of fractional Brownian motion

The effects of a "diffusing diffusivity" (DD), a stochastically time-varying diffusion coefficient, are explored within the frameworks of three different forms of fractional Brownian motion (FBM): (i) the Langevin equation driven by fractional Gaussian noise (LE-FBM), (ii) the Weyl integral representation introduced by Mandelbrot and van Ness (MN-FBM), and (iii) the Riemann-Liouville fractional integral representation (RL-FBM) due to L{é}vy. The statistical properties of the three FBM-generalized DD models are examined, including the mean-squared displacement (MSD), mean-squared increment (MSI), autocovariance function (ACVF) of increments, and the probability density function (PDF). Despite the long-believed equivalence of MN-FBM and LE-FBM, their corresponding FBM-DD models exhibit distinct behavior in terms of the MSD and MSI. In the MN-FBM-DD model, the statistical characteristics directly reflect an effective diffusivity equal to its mean value. In contrast, in LE-FBM-DD, correlations in the random diffusivity give rise to an unexpected crossover behavior in both MSD and MSI. We also find that the MSI and ACVF are nonstationary in RL-FBM-DD but stationary in the other two DD models. All DD models display a crossover from a short-time non-Gaussian PDF to a long-time Gaussian PDF. Our findings offer guidance for experimentalists in selecting appropriate FBM-generalized models to describe viscoelastic yet non-Gaussian dynamics in bio- and soft-matter systems with heterogeneous environments.

cond-mat.stat-mech↗

arXiv2025

Multifractional Brownian motion with telegraphic, stochastically varying exponent

The diversity of diffusive systems exhibiting long-range correlations characterized by a stochastically varying Hurst exponent calls for a generic multifractional model. We present a simple, analytically tractable model which fills the gap between mathematical formulations of multifractional Brownian motion and empirical studies. In our model, called telegraphic multifractional Brownian motion, the Hurst exponent is modelled by a smoothed telegraph process which results in a stationary beta distribution of exponents as observed in biological experiments. We also provide a methodology to identify our model in experimental data and present concrete examples from biology, climate and finance to demonstrate the efficacy of our approach.

cond-mat.stat-mech↗

arXiv2025

Fickian yet non-Gaussian diffusion in an annealed heterogeneous environment

Fickian yet non-Gaussian diffusion is a ubiquitous phenomenon observed in various biological and soft matter systems. This anomalous dynamics is typically attributed to heterogeneous environments inducing spatiotemporal variations in the diffusivity of tracer particles. While previous studies have predominantly focused on systems exhibiting either spatial or temporal heterogeneity, this work bridges the gap by introducing a model based on an annealed extreme landscape to simultaneously account for both types of heterogeneities. Through a combination of computational analyses and analytical derivations, we investigate how the interplay of spatial and temporal heterogeneities in the energy landscape gives rise to Fickian yet non-Gaussian diffusion. Furthermore, we demonstrate that in the presence of temporal environmental fluctuations, the heterogeneous diffusion inevitably converges to classical Brownian motion via a homogenization process. We derive an analytical expression for the homogenization time as a function of key parameters governing the system's spatiotemporal heterogeneities. Additionally, we quantify particle-to-particle diffusion heterogeneity and examine the ergodic properties of this model, providing deeper insights into the dynamics of complex, heterogeneous systems.

cond-mat.stat-mech↗

arXiv2025

Driven Polymer Translocation through a Nanopore from a Confining Channel

We consider the dynamics of pore-driven polymer translocation through a nanopore to semi-infinite space when the chain is initially confined and equilibrated in a narrow channel. To this end, we use Langevin dynamics (LD) simulations and iso-flux tension propagation (IFTP) theory to characterize local and global dynamics of the translocating chain. The dynamics of the process can be described by the IFTP theory in very good agreement with the LD simulations for all values of confinement in the channel. The theory reveals that for channels with size comparable to or less than the end-to-end distance of the unconfined chain, in which the blob theory works, the scaling form of the translocation time depends on both the chain contour length as well as the channel width. %originating from the confinement of the spatial fluctuations of the chain inside the channel. Conversely, for a very narrow channel the translocation time only depends on the chain contour length and is similar to that of a rod due to the absence of spatial chain fluctuations.

cond-mat.soft↗

arXiv2025

Reversal of tracer advection and Hall drift in an interacting chiral fluid

Chiral fluids are defined by broken mirror or time-reversal symmetry, giving rise to tensorial transport coefficients with antisymmetric components. A key example is the odd mobility tensor, which governs the response of a chiral tracer to an applied force and induces a characteristic transverse drift. While this response is well understood in the infinite dilution limit, the impact of interparticle interactions on the tracer dynamics remains largely unexplored. Here, we conduct an analytical and computational study of a chiral fluid with interparticle interactions and show that, under an external driving force, a chiral tracer can undergo a complete reversal of both its transverse Hall drift and its advection along the force. This reversal emerges from the interplay between odd mobility and interaction-mediated forces, resulting in a phenomenon reminiscent of absolute negative mobility.

cond-mat.stat-mech↗

arXiv2024

Paradoxical non-Gaussian behavior in fractional Laplace motion with drift

We study fractional Laplace motion (FLM) obtained from subordination of fractional Brownian motion to a gamma process, in the presence of an external drift that acts on the composite process or of an internal drift acting solely on the parental process. We derive the statistical properties of this FLM process and find that the external drift does not influence the mean-squared displacement (MSD), whereas the internal drift leads to normal diffusion, dominating at long times in the subdiffusive Hurst exponent regime. We also investigate the intricate properties of the probability density function (PDF), demonstrating that it possesses a central Gaussian region, whose expansion in time is influenced by FBM's Hurst exponent. Outside of this region the PDF follows a non-Gaussian pattern. The kurtosis of this FLM process converges toward the Gaussian limit at long times insensitive to the extreme non-Gaussian tails. Additionally, in the presence of the external drift, the PDF remains symmetric and centered at $x=vt$. In contrast, for the internal drift this symmetry is broken. The results of our computer simulations are fully consistent with the theoretical predictions. The FLM model is suitable for describing stochastic processes with a non-Gaussian PDF and long-ranged correlations of the motion.

cond-mat.stat-mech↗

arXiv2024

Fractional Langevin equation far from equilibrium: Riemann-Liouville fractional Brownian motion, spurious nonergodicity and aging

We consider the fractional Langevin equation far from equilibrium (FLEFE) to describe stochastic dynamics which do not obey the fluctuation-dissipation theorem, unlike the conventional fractional Langevin equation (FLE). The solution of this equation is Riemann-Liouville fractional Brownian motion (RL-FBM), also known in the literature as FBM II. Spurious nonergodicity, stationarity, and aging properties of the solution are explored for all admissible values $α>1/2$ of the order $α$ of the time-fractional Caputo derivative in the FLEFE. The increments of the process are asymptotically stationary. However when $1/2<α<3/2$, the time-averaged mean-squared displacement (TAMSD) does not converge to the mean-squared displacement (MSD). Instead, it converges to the mean-squared increment (MSI) or structure function, leading to the phenomenon of spurious nonergodicity. When $α\ge 3/2$, the increments of FLEFE motion are nonergodic, however the higher order increments are asymptotically ergodic. We also discuss the aging effect in the FLEFE by investigating the influence of an aging time $t_a$ on the mean-squared displacement, time-averaged mean-squared displacement and autocovariance function of the increments. We find that under strong aging conditions the process becomes ergodic, and the increments become stationary in the domain $1/2<α<3/2$.

cond-mat.stat-mech↗

arXiv2024

Pb transport on Si(111)-(7x7) following tip induced imbalance

Pb on Si(111)-(7x7) shows surprising nucleation and mass transport dynamics at odds with standard theories. To create local imbalances on stable Pb islands we use the tip of a scanning force microscope. We enforce a short, local contact between the island and our tip. The subsequent island height growth and the local contact potential difference are studied via scanning force microscopy and Kelvin probe force microscopy. Though the island has a large volume increase after the contact, we observe that its surrounding wetting layer shows the same Pb density decrease as the global wetting layer. This indicates a collective density thinning of the wetting layer.

cond-mat.mes-hall↗

arXiv2024

Self-diffusion anomalies of an odd tracer in soft-core media

Odd-diffusive systems, characterised by broken time-reversal and/or parity symmetry, have recently been shown to display counterintuitive features such as interaction-enhanced dynamics in the dilute limit. Here we we extend the investigation to the high-density limit of an odd tracer embedded in a soft-Gaussian core medium (GCM) using a field-theoretic approach based on the Dean-Kawasaki equation. Our theory reveals that interactions can enhance the dynamics of an odd tracer even in dense systems. We demonstrate that oddness results in a complete reversal of the well-known self-diffusion ($D_\mathrm{s}$) anomaly of the GCM. Ordinarily, $D_\mathrm{s}$ exhibits a non-monotonic trend with increasing density, approaching but remaining below the interaction-free diffusion, $D_0$, ($D_\mathrm{s} < D_0$) so that $D_\mathrm{s} \uparrow D_0$ at high densities. In contrast, for an odd tracer, self-diffusion is enhanced ($D_\mathrm{s}> D_0$) and the GCM anomaly is inverted, displaying $D_\mathrm{s} \downarrow D_0$ at high densities. The transition between the standard and reversed GCM anomaly is governed by the tracer's oddness, with a critical oddness value at which the tracer diffuses as a free particle ($D_\mathrm{s} \approx D_0$) across all densities. We validate our theoretical predictions with Brownian dynamics simulations, finding strong agreement between the two.

cond-mat.stat-mech↗

arXiv2024

Change-point detection in anomalous-diffusion trajectories utilising machine-learning-based uncertainty estimates

When recording the movement of individual animals, cells or molecules one will often observe changes in their diffusive behaviour at certain points in time along their trajectory. In order to capture the different diffusive modes assembled in such heterogeneous trajectories it becomes necessary to segment them by determining these change-points. Such a change-point detection can be challenging for conventional statistical methods, especially when the changes are subtle. We here apply Bayesian Deep Learning to obtain point-wise estimates of not only the anomalous diffusion exponent but also the uncertainties in these predictions from a single anomalous diffusion trajectory generated according to four theoretical models of anomalous diffusion. We show that we are able to achieve an accuracy similar to single-mode (without change-points) predictions as well as a well calibrated uncertainty predictions of this accuracy. Additionally, we find that the predicted uncertainties feature interesting behaviour at the change-points leading us to examine the capabilities of these predictions for change-point detection. While the series of predicted uncertainties on their own are not sufficient to improve change-point detection, they do lead to a performance boost when applied in combination with the predicted anomalous diffusion exponents.

cond-mat.stat-mech↗

arXiv2024

The Dance of Odd-Diffusive Particles: A Fourier Approach

Odd-diffusive systems are characterized by transverse responses and exhibit unconventional behaviors in interacting systems. To address the dynamical interparticle rearrangements in a minimal system, we here exactly solve the problem of two hard disk-like interacting odd-diffusing particles. We calculate the probability density function (PDF) of the interacting particles in the Fourier-Laplace domain and find that oddness rotates all modes except the zeroth, resembling a ``mutual rolling'' of interacting odd particles. We show that only the first Fourier mode of the PDF, the polarization, enters the calculation of the force autocorrelation function (FACF) for generic systems with central-force interactions. An analysis of the polarization as a function of time reveals that the relative rotation angle between interacting particles overshoots before relaxation, thereby rationalizing the recently observed oscillating FACF in odd-diffusive systems.

cond-mat.stat-mech↗

arXiv2024

Field-Theory of Active Chiral Hard Disks: A First-Principles Approach to Steric Interactions

A first-principles approach for active chiral hard disks is presented, that explicitly accounts for steric interactions on the two-body level. We derive an effective one-body equation for the joint probability distribution of ositions and angles of the particles. By projecting onto the angular modes, we write a hierarchy for the lowest hydrodynamic modes, i.e. particle density, polarisation, and nematic tensor. Introducing dimensionless variables in the equations, we highlight the assumptions, which - though inherent - are often included implicit in typical closure schemes of the hierarchy. By considering different regimes of the P{é}clet number, the well-known models in active matter can be obtained through our consideration. Explicitly, we derive an effective diffusive description and by going to higher orders in the closure scheme, we show that this first-principles approach results in the recently introduced Active Model B +, a natural extension of the Model B for active processes. Remarkably, here we find that chirality can change the sign of the phenomenological activity parameters.

cond-mat.stat-mech↗

arXiv2024

Non-Markovian gene expression

We study two non-Markovian gene-expression models in which protein production is a stochastic process with a fat-tailed non-exponential waiting time distribution (WTD). For both models, we find two distinct scaling regimes separated by an exponentially long time, proportional to the mean first passage time (MFPT) to a ground state (with zero proteins) of the dynamics, from which the system can only exit via a non-exponential reaction. At times shorter than the MFPT the dynamics are stationary and ergodic, entailing similarity across different realizations of the same process, with an increased Fano factor of the protein distribution, even when the WTD has a finite cutoff. Notably, at times longer than the MFPT the dynamics are nonstationary and nonergodic, entailing significant variability across different realizations. The MFPT to the ground state is shown to directly affect the average population sizes and we postulate that the transition to nonergodicity is universal in such non-Markovian models.

cond-mat.stat-mech↗

arXiv2024

Target search on DNA -- effect of coiling

Some proteins can find their targets on DNA faster than by pure diffusion in the three-dimensional cytoplasm, through the process of facilitated diffusion: They can loosely bind to DNA and temporarily slide along it, thus being guided by the DNA molecule itself to the target. This chapter examines this process in mathematical detail with a focus on including the effect of DNA coiling on the search process.

q-bio.BM↗

arXiv2024

A general mechanism for enhancer-insulator pairing reveals heterogeneous dynamics in long-distant 3D gene regulation

Cells regulate fates and complex body plans using spatiotemporal signaling cascades that alter gene expression. Enhancers, short DNA sequences (50-150 base pairs), help coordinate these cascades by attracting regulatory proteins to enhance the transcription of distal genes by binding to promoters. In humans, there are hundreds of thousands of enhancers dispersed across the genome, which poses a challenging coordination task to prevent unintended gene activation. To mitigate this problem, the genome contains additional DNA elements, insulators, that block enhancer-promoter interactions. However, there is an open problem with how the insulation works, especially as enhancer-insulator pairs may be separated by millions of base pairs. Based on recent empirical data from Hi-C experiments, this paper proposes a new mechanism that challenges the common paradigm that rests on specific insulator-insulator interactions. Instead, this paper introduces a stochastic looping model where enhancers bind weakly to surrounding chromatin. After calibrating the model to experimental data, we use simulations to study the broad distribution of hitting times between an enhancer and a promoter when there are blocking insulators. In some cases, there is a large difference between average and most probable hitting times, making it difficult to assign a typical time scale, hinting at highly defocused regulation times. We also map our computational model onto a resetting problem that allows us to derive several analytical results. Besides offering new insights into enhancer-insulator interactions, our paper advances the understanding of gene regulatory networks and causal connections between genome folding and gene activation.

q-bio.MN↗

arXiv2024

Diffusion of intrinsically disordered proteins within viscoelastic membraneless droplets

In living cells, intrinsically disordered proteins (IDPs), such as FUS and DDX4, undergo phase separation, forming biomolecular condensates. Using molecular dynamics simulations, we investigate their behavior in their respective homogenous droplets. We find that the proteins exhibit transient subdiffusion due to the viscoelastic nature and confinement effects in the droplets. The conformation and the instantaneous diffusivity of the proteins significantly vary between the interior and the interface of the droplet, resulting in non-Gaussianity in the displacement distributions. This study highlights key aspects of IDP behavior in biomolecular condensates.

cond-mat.soft↗

arXiv2024

Multifractal-spectral features enhance classification of anomalous diffusion

Anomalous diffusion processes pose a unique challenge in classification and characterization. Previously (Mangalam et al., 2023, Physical Review Research 5, 023144), we established a framework for understanding anomalous diffusion using multifractal formalism. The present study delves into the potential of multifractal spectral features for effectively distinguishing anomalous diffusion trajectories from five widely used models: fractional Brownian motion, scaled Brownian motion, continuous time random walk, annealed transient time motion, and Lévy walk. To accomplish this, we generate extensive datasets comprising $10^6$ trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. Furthermore, we explore the integration of multifractal spectra into traditional feature datasets, enabling us to assess their impact comprehensively. To ensure a statistically meaningful comparison, we categorize features into concept groups and train neural networks using features from each designated group. Notably, several feature groups demonstrate similar levels of accuracy, with the highest performance observed in groups utilizing moving-window characteristics and $p$-variation features. Multifractal spectral features, particularly those derived from three spectra involving different timescales and cutoffs, closely follow, highlighting their robust discriminatory potential. Remarkably, a neural network exclusively trained on features from a single multifractal spectrum exhibits commendable performance, surpassing other feature groups. Our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing classification of anomalous diffusion.

nlin.AO↗

arXiv2023

Brownian particle in a Poisson-shot-noise active bath: exact statistics, effective temperature, and inference

We study the dynamics of an overdamped Brownian particle in a thermal bath that contains a dilute solution of active particles. The particle moves in a harmonic potential and experiences Poisson shot-noise kicks with specified amplitude distribution due to moving active particles in the bath. From the Fokker-Planck equation for the particle dynamics we derive the stationary solution for the displacement distribution along with the moments characterizing mean, variance, skewness, and kurtosis, as well as finite time first and second moments. We also compute an effective temperature through the fluctuation-dissipation theorem and show that equipartition theorem holds for all zero-mean kick distributions, including those leading to non-Gaussian stationary statistics. For the case of Gaussian-distributed active kicks we find a re-entrant behaviour from non-Gaussian to Gaussian stationary states and a heavy-tailed leptokurtic distribution across a wide range of parameters as seen in recent experimental studies. Further analysis reveals statistical signatures of the irreversible dynamics of the particle displacement in terms of the time asymmetry of cross-correlation functions. Fruits of our work is the development of a compact inference scheme that may allow experimentalists to extract the rate and moments of underlying shot-noise solely from the statistics of the particle position.

cond-mat.stat-mech↗