TY - JOUR A1 - Meyer, Philipp A1 - Cherstvy, Andrey G. A1 - Seckler, Henrik A1 - Hering, Robert A1 - Blaum, Niels A1 - Jeltsch, Florian A1 - Metzler, Ralf T1 - Directedeness, correlations, and daily cycles in springbok motion BT - from data via stochastic models to movement prediction JF - Physical review research N2 - 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 land-bound, plant-eating animals in general and ruminants in particular? To answer this question, we here study the movement dynamics of springbok antelopes Antidorcas marsupialis. We propose several 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 result, we are able to predict the springbok movement within the next hour with a certainty of about 20%. The remaining about 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 of springbok activity. The previously known daily affinity of springboks to their water points, as predicted from our machine-learning algorithm, overall accounts for only about 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 in the future. Y1 - 2023 U6 - https://doi.org/10.1103/PhysRevResearch.5.043129 SN - 2643-1564 VL - 5 IS - 4 PB - APS CY - College Park ER - TY - JOUR A1 - Wang, Wei A1 - Balcerek, Michał A1 - Burnecki, Krzysztof A1 - Chechkin, Aleksei A1 - Janušonis, Skirmantas A1 - Ślęzak, Jakub A1 - Vojta, Thomas A1 - Wyłomańska, Agnieszka A1 - Metzler, Ralf T1 - Memory-multi-fractional Brownian motion with continuous correlations JF - Physical review research N2 - We propose a generalization of the widely used fractional Brownian motion (FBM), memory-multi-FBM (MMFBM), to describe viscoelastic or persistent anomalous diffusion with time-dependent memory exponent α(t ) in a changing environment. In MMFBM the built-in, long-range memory is continuously modulated by α(t ). We derive the essential statistical properties of MMFBM such as its response function, mean-squared displacement (MSD), autocovariance function, and Gaussian distribution. In contrast to existing forms of FBM with time-varying memory exponents but a reset memory structure, the instantaneous dynamic of MMFBM is influenced by the process history, e.g., we show that after a steplike change of α(t ) the scaling exponent of the MSD after the α step may be determined by the value of α(t ) before the change. MMFBM is a versatile and useful process for correlated physical systems with nonequilibrium initial conditions in a changing environment. Y1 - 2023 U6 - https://doi.org/10.1103/PhysRevResearch.5.L032025 SN - 2643-1564 VL - 5 IS - 3 PB - APS CY - College Park ER - TY - JOUR A1 - Sarabadani, Jalal A1 - Metzler, Ralf A1 - Ala-Nissila, Tapio T1 - Driven polymer translocation into a channel: Isoflux tension propagation theory and Langevin dynamics simulations JF - Physical Review Research N2 - Isoflux tension propagation (IFTP) theory and Langevin dynamics (LD) simulations are employed to study the dynamics of channel-driven polymer translocation in which a polymer translocates into a narrow channel and the monomers in the channel experience a driving force fc. In the high driving force limit, regardless of the channel width, IFTP theory predicts τ ∝ f βc for the translocation time, where β = −1 is the force scaling exponent. Moreover, LD data show that for a very narrow channel fitting only a single file of monomers, the entropic force due to the subchain inside the channel does not play a significant role in the translocation dynamics and the force exponent β = −1 regardless of the force magnitude. As the channel width increases the number of possible spatial configurations of the subchain inside the channel becomes significant and the resulting entropic force causes the force exponent to drop below unity. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevResearch.4.033003 SN - 2643-1564 VL - 4 SP - 033003-1 EP - 033003-14 PB - American Physical Society CY - College Park, Maryland, USA ET - 3 ER - TY - JOUR 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 JF - Communications Physics 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. Y1 - 2022 U6 - https://doi.org/10.1038/s42005-022-01079-8 SN - 2399-3650 VL - 5 PB - Springer Nature CY - London ER - TY - JOUR A1 - Seckler, Henrik A1 - Metzler, Ralf T1 - Bayesian deep learning for error estimation in the analysis of anomalous diffusion JF - Nature Communnications N2 - 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. KW - random-walk KW - models Y1 - 2022 U6 - https://doi.org/10.1038/s41467-022-34305-6 SN - 2041-1723 VL - 13 PB - Nature Publishing Group UK CY - London ER - TY - JOUR A1 - Seckler, Henrik A1 - Metzler, Ralf T1 - Bayesian deep learning for error estimation in the analysis of anomalous diffusion JF - Nature Communications N2 - 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 diffusion model and the regression of the anomalous diffusion exponent of single-particle-trajectories. Evaluating their performance, we find that these models can achieve a well-calibrated 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.
Diffusive motions in complex environments such as living biological cells or soft matter systems can be analyzed with single-particle-tracking approaches, where accuracy of output may vary. The authors involve a machine-learning technique for decoding anomalous-diffusion data and provide an uncertainty estimate together with predicted output. Y1 - 2022 U6 - https://doi.org/10.1038/s41467-022-34305-6 SN - 2041-1723 VL - 13 IS - 1 PB - Nature portfolio CY - Berlin ER - TY - JOUR A1 - Awad, Emad A1 - Metzler, Ralf T1 - Closed-form multi-dimensional solutions and asymptotic behaviours for subdiffusive processes with crossovers: II. Accelerating case JF - Journal of physics : A, Mathematical and theoretical N2 - Anomalous diffusion with a power-law time dependence vertical bar R vertical bar(2)(t) similar or equal to t(alpha i) of the mean squared displacement occurs quite ubiquitously in numerous complex systems. Often, this anomalous diffusion is characterised by crossovers between regimes with different anomalous diffusion exponents alpha(i). Here we consider the case when such a crossover occurs from a first regime with alpha(1) to a second regime with alpha(2) such that alpha(2) > alpha(1), i.e., accelerating anomalous diffusion. A widely used framework to describe such crossovers in a one-dimensional setting is the bi-fractional diffusion equation of the so-called modified type, involving two time-fractional derivatives defined in the Riemann-Liouville sense. We here generalise this bi-fractional diffusion equation to higher dimensions and derive its multidimensional propagator (Green's function) for the general case when also a space fractional derivative is present, taking into consideration long-ranged jumps (Levy flights). We derive the asymptotic behaviours for this propagator in both the short- and long-time as well the short- and long-distance regimes. Finally, we also calculate the mean squared displacement, skewness and kurtosis in all dimensions, demonstrating that in the general case the non-Gaussian shape of the probability density function changes. KW - multidimensional fractional diffusion equation KW - continuous time random KW - walks KW - crossover anomalous diffusion dynamics KW - non-Gaussian probability KW - density Y1 - 2022 U6 - https://doi.org/10.1088/1751-8121/ac5a90 SN - 1751-8113 SN - 1751-8121 VL - 55 IS - 20 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Scott, Shane A1 - Weiss, Matthias A1 - Selhuber-Unkel, Christine A1 - Barooji, Younes F. A1 - Sabri, Adal A1 - Erler, Janine T. A1 - Metzler, Ralf A1 - Oddershede, Lene B. T1 - Extracting, quantifying, and comparing dynamical and biomechanical properties of living matter through single particle tracking JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - A panoply of new tools for tracking single particles and molecules has led to an explosion of experimental data, leading to novel insights into physical properties of living matter governing cellular development and function, health and disease. In this Perspective, we present tools to investigate the dynamics and mechanics of living systems from the molecular to cellular scale via single-particle techniques. In particular, we focus on methods to measure, interpret, and analyse complex data sets that are associated with forces, materials properties, transport, and emergent organisation phenomena within biological and soft-matter systems. Current approaches, challenges, and existing solutions in the associated fields are outlined in order to support the growing community of researchers at the interface of physics and the life sciences. Each section focuses not only on the general physical principles and the potential for understanding living matter, but also on details of practical data extraction and analysis, discussing limitations, interpretation, and comparison across different experimental realisations and theoretical frameworks. Particularly relevant results are introduced as examples. While this Perspective describes living matter from a physical perspective, highlighting experimental and theoretical physics techniques relevant for such systems, it is also meant to serve as a solid starting point for researchers in the life sciences interested in the implementation of biophysical methods. Y1 - 2022 U6 - https://doi.org/10.1039/d2cp01384c SN - 1463-9076 SN - 1463-9084 VL - 25 IS - 3 SP - 1513 EP - 1537 PB - RSC Publ. CY - Cambridge ER - TY - JOUR A1 - Tomovski, Živorad A1 - Metzler, Ralf A1 - Gerhold, Stefan T1 - Fractional characteristic functions, and a fractional calculus approach for moments of random variables JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - In this paper we introduce a fractional variant of the characteristic function of a random variable. It exists on the whole real line, and is uniformly continuous. We show that fractional moments can be expressed in terms of Riemann-Liouville integrals and derivatives of the fractional characteristic function. The fractional moments are of interest in particular for distributions whose integer moments do not exist. Some illustrative examples for particular distributions are also presented. KW - Fractional calculus (primary) KW - Characteristic function KW - Mittag-Leffler KW - function KW - Fractional moments KW - Mellin transform Y1 - 2022 U6 - https://doi.org/10.1007/s13540-022-00047-x SN - 1314-2224 VL - 25 IS - 4 SP - 1307 EP - 1323 PB - De Gruyter CY - Berlin ; Boston ER - TY - JOUR A1 - Xu, Pengbo A1 - Metzler, Ralf A1 - Wang, Wanli T1 - Infinite density and relaxation for Levy walks in an external potential BT - Hermite polynomial approach JF - Physical review N2 - Levy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding qth-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed v(r ) = v0/./r. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevE.105.044118 SN - 2470-0045 SN - 2470-0053 VL - 105 IS - 4 PB - American Physical Society CY - College Park ER -