TY - JOUR A1 - Thapa, Samudrajit A1 - Park, Seongyu A1 - Kim, Yeongjin A1 - Jeon, Jae-Hyung A1 - Metzler, Ralf A1 - Lomholt, Michael A. T1 - Bayesian inference of scaled versus fractional Brownian motion JF - Journal of physics : A, mathematical and theoretical N2 - We present a Bayesian inference scheme for scaled Brownian motion, and investigate its performance on synthetic data for parameter estimation and model selection in a combined inference with fractional Brownian motion. We include the possibility of measurement noise in both models. We find that for trajectories of a few hundred time points the procedure is able to resolve well the true model and parameters. Using the prior of the synthetic data generation process also for the inference, the approach is optimal based on decision theory. We include a comparison with inference using a prior different from the data generating one. KW - Bayesian inference KW - scaled Brownian motion KW - single particle tracking Y1 - 2022 U6 - https://doi.org/10.1088/1751-8121/ac60e7 SN - 1751-8113 SN - 1751-8121 VL - 55 IS - 19 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Sandev, Trifce A1 - Domazetoski, Viktor A1 - Kocarev, Ljupco A1 - Metzler, Ralf A1 - Chechkin, Aleksei T1 - Heterogeneous diffusion with stochastic resetting JF - Journal of physics : A, Mathematical and theoretical N2 - We study a heterogeneous diffusion process (HDP) with position-dependent diffusion coefficient and Poissonian stochastic resetting. We find exact results for the mean squared displacement and the probability density function. The nonequilibrium steady state reached in the long time limit is studied. We also analyse the transition to the non-equilibrium steady state by finding the large deviation function. We found that similarly to the case of the normal diffusion process where the diffusion length grows like t (1/2) while the length scale xi(t) of the inner core region of the nonequilibrium steady state grows linearly with time t, in the HDP with diffusion length increasing like t ( p/2) the length scale xi(t) grows like t ( p ). The obtained results are verified by numerical solutions of the corresponding Langevin equation. KW - heterogeneous diffusion KW - Fokker-Planck equation KW - Langevin equation KW - stochastic resetting KW - nonequilibrium stationary state KW - large deviation function Y1 - 2022 U6 - https://doi.org/10.1088/1751-8121/ac491c SN - 1751-8113 SN - 1751-8121 VL - 55 IS - 7 PB - IOP Publ. Ltd. CY - Bristol ER - TY - GEN A1 - Seckler, Henrik A1 - Metzler, Ralf T1 - Bayesian deep learning for error estimation in the analysis of anomalous diffusion T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - Sprache Englisch Modern single-particle-tracking techniques produce extensive time-series of diffusive motion in a wide variety of systems, from single-molecule motion in living-cells to movement ecology. The quest is to decipher the physical mechanisms encoded in the data and thus to better understand the probed systems. We here augment recently proposed machine-learning techniques for decoding anomalous-diffusion data to include an uncertainty estimate in addition to the predicted output. To avoid the Black-Box-Problem a Bayesian-Deep-Learning technique named Stochastic-Weight-Averaging-Gaussian is used to train models for both the classification of the diffusionmodel and the regression of the anomalous diffusion exponent of single-particle-trajectories. Evaluating their performance, we find that these models can achieve a wellcalibrated error estimate while maintaining high prediction accuracies. In the analysis of the output uncertainty predictions we relate these to properties of the underlying diffusion models, thus providing insights into the learning process of the machine and the relevance of the output. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1314 KW - random-walk KW - models Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-586025 SN - 1866-8372 IS - 1314 ER - TY - GEN A1 - Sposini, Vittoria A1 - Krapf, Diego A1 - Marinari, Enzo A1 - Sunyer, Raimon A1 - Ritort, Felix A1 - Taheri, Fereydoon A1 - Selhuber-Unkel, Christine A1 - Benelli, Rebecca A1 - Weiss, Matthias A1 - Metzler, Ralf A1 - Oshanin, Gleb T1 - Towards a robust criterion of anomalous diffusion T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - Anomalous-diffusion, the departure of the spreading dynamics of diffusing particles from the traditional law of Brownian-motion, is a signature feature of a large number of complex soft-matter and biological systems. Anomalous-diffusion emerges due to a variety of physical mechanisms, e.g., trapping interactions or the viscoelasticity of the environment. However, sometimes systems dynamics are erroneously claimed to be anomalous, despite the fact that the true motion is Brownian—or vice versa. This ambiguity in establishing whether the dynamics as normal or anomalous can have far-reaching consequences, e.g., in predictions for reaction- or relaxation-laws. Demonstrating that a system exhibits normal- or anomalous-diffusion is highly desirable for a vast host of applications. Here, we present a criterion for anomalous-diffusion based on the method of power-spectral analysis of single trajectories. The robustness of this criterion is studied for trajectories of fractional-Brownian-motion, a ubiquitous stochastic process for the description of anomalous-diffusion, in the presence of two types of measurement errors. In particular, we find that our criterion is very robust for subdiffusion. Various tests on surrogate data in absence or presence of additional positional noise demonstrate the efficacy of this method in practical contexts. Finally, we provide a proof-of-concept based on diverse experiments exhibiting both normal and anomalous-diffusion. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1313 Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-585967 SN - 1866-8372 IS - 1313 ER - TY - JOUR A1 - Grebenkov, Denis S. A1 - Metzler, Ralf A1 - Oshanin, Gleb T1 - Search efficiency in the Adam-Delbruck reduction-of-dimensionality scenario versus direct diffusive search JF - New journal of physics : the open-access journal for physics N2 - The time instant-the first-passage time (FPT)-when a diffusive particle (e.g., a ligand such as oxygen or a signalling protein) for the first time reaches an immobile target located on the surface of a bounded three-dimensional domain (e.g., a hemoglobin molecule or the cellular nucleus) is a decisive characteristic time-scale in diverse biophysical and biochemical processes, as well as in intermediate stages of various inter- and intra-cellular signal transduction pathways. Adam and Delbruck put forth the reduction-of-dimensionality concept, according to which a ligand first binds non-specifically to any point of the surface on which the target is placed and then diffuses along this surface until it locates the target. In this work, we analyse the efficiency of such a scenario and confront it with the efficiency of a direct search process, in which the target is approached directly from the bulk and not aided by surface diffusion. We consider two situations: (i) a single ligand is launched from a fixed or a random position and searches for the target, and (ii) the case of 'amplified' signals when N ligands start either from the same point or from random positions, and the search terminates when the fastest of them arrives to the target. For such settings, we go beyond the conventional analyses, which compare only the mean values of the corresponding FPTs. Instead, we calculate the full probability density function of FPTs for both scenarios and study its integral characteristic-the 'survival' probability of a target up to time t. On this basis, we examine how the efficiencies of both scenarios are controlled by a variety of parameters and single out realistic conditions in which the reduction-of-dimensionality scenario outperforms the direct search. KW - first-passage times KW - Adam-Delbruck scenario KW - dimensional reduction KW - bulk KW - and surface diffusion Y1 - 2022 U6 - https://doi.org/10.1088/1367-2630/ac8824 SN - 1367-2630 VL - 24 IS - 8 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Padash, Amin A1 - Sandev, Trifce A1 - Kantz, Holger A1 - Metzler, Ralf A1 - Chechkin, Aleksei T1 - Asymmetric Levy flights are more efficient in random search JF - Fractal and fractional N2 - We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Levy flights by leveraging the Fokker-Planck equation with a delta-sink and an asymmetric space-fractional derivative operator with stable index alpha and asymmetry (skewness) parameter beta. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Levy flights with beta <= 0 (with a rightward bias) for short initial distances, while for beta>0 (with a leftward bias) Levy flights with alpha -> 1 are more efficient. When increasing the initial distance of the searcher to the target, Levy flight search (except for alpha=1 with beta=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Levy search compared to symmetric Levy flights at both short and long distances, and the effect is more pronounced for stable indices alpha close to unity. KW - asymmetric Levy flights KW - first-arrival density KW - search efficiency Y1 - 2022 U6 - https://doi.org/10.3390/fractalfract6050260 SN - 2504-3110 VL - 6 IS - 5 PB - MDPI CY - Basel ER - TY - JOUR A1 - Vilk, Ohad A1 - Aghion, Erez A1 - Nathan, Ran A1 - Toledo, Sivan A1 - Metzler, Ralf A1 - Assaf, Michael T1 - Classification of anomalous diffusion in animal movement data using power spectral analysis JF - Journal of physics : A, Mathematical and theoretical N2 - The field of movement ecology has seen a rapid increase in high-resolution data in recent years, leading to the development of numerous statistical and numerical methods to analyse relocation trajectories. Data are often collected at the level of the individual and for long periods that may encompass a range of behaviours. Here, we use the power spectral density (PSD) to characterise the random movement patterns of a black-winged kite (Elanus caeruleus) and a white stork (Ciconia ciconia). The tracks are first segmented and clustered into different behaviours (movement modes), and for each mode we measure the PSD and the ageing properties of the process. For the foraging kite we find 1/f noise, previously reported in ecological systems mainly in the context of population dynamics, but not for movement data. We further suggest plausible models for each of the behavioural modes by comparing both the measured PSD exponents and the distribution of the single-trajectory PSD to known theoretical results and simulations. KW - diffusion KW - anomalous diffusion KW - power spectral analysis KW - ecological KW - movement data Y1 - 2022 U6 - https://doi.org/10.1088/1751-8121/ac7e8f SN - 1751-8113 SN - 1751-8121 VL - 55 IS - 33 PB - IOP Publishing CY - Bristol ER - TY - JOUR A1 - Stojkoski, Viktor A1 - Jolakoski, Petar A1 - Pal, Arnab A1 - Sandev, Trifce A1 - Kocarev, Ljupco A1 - Metzler, Ralf T1 - Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity JF - Philosophical transactions of the Royal Society A: Mathematical, physical and engineering sciences N2 - We explore the role of non-ergodicity in the relationship between income inequality, the extent of concentration in the income distribution, and income mobility, the feasibility of an individual to change their position in the income rankings. For this purpose, we use the properties of an established model for income growth that includes 'resetting' as a stabilizing force to ensure stationary dynamics. We find that the dynamics of inequality is regime-dependent: it may range from a strictly non-ergodic state where this phenomenon has an increasing trend, up to a stable regime where inequality is steady and the system efficiently mimics ergodicity. Mobility measures, conversely, are always stable over time, but suggest that economies become less mobile in non-ergodic regimes. By fitting the model to empirical data for the income share of the top earners in the USA, we provide evidence that the income dynamics in this country is consistently in a regime in which non-ergodicity characterizes inequality and immobility. Our results can serve as a simple rationale for the observed real-world income dynamics and as such aid in addressing non-ergodicity in various empirical settings across the globe.This article is part of the theme issue 'Kinetic exchange models of societies and economies'. KW - income inequality KW - income mobility KW - geometric Brownian motion KW - non-ergodicity KW - stochastic resetting Y1 - 2022 U6 - https://doi.org/10.1098/rsta.2021.0157 SN - 1364-503X SN - 1471-2962 VL - 380 IS - 2224 PB - Royal Society CY - London ER - TY - JOUR A1 - Padash, Amin A1 - Aghion, Erez A1 - Schulz, Alexander A1 - Barkai, Eli A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf A1 - Kantz, Holger T1 - Local equilibrium properties of ultraslow diffusion in the Sinai model JF - New journal of physics N2 - We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a finite system of growing size. This is due to time scale separation: inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top 10(17) steps and thereby also study finite-time crossover phenomena. KW - Sinai diffusion KW - clustering KW - local equilibrium Y1 - 2022 U6 - https://doi.org/10.1088/1367-2630/ac7df8 SN - 1367-2630 VL - 24 IS - 7 PB - IOP Publishing CY - Bristol ER - TY - JOUR A1 - Doerries, Timo J. A1 - Chechkin, Aleksei A1 - Schumer, Rina A1 - Metzler, Ralf T1 - Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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 multirate 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 a function of time, and 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. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevE.105.014105 SN - 2470-0045 SN - 2470-0053 VL - 105 IS - 1 PB - The American Institute of Physics CY - Woodbury, NY ER -