TY - JOUR A1 - Mutothya, Nicholas Mwilu A1 - Xu, Yong A1 - Li, Yongge A1 - Metzler, Ralf T1 - Characterising stochastic motion in heterogeneous media driven by coloured non-Gaussian noise JF - Journal of physics : A, Mathematical and theoretical N2 - We study the stochastic motion of a test particle in a heterogeneous medium in terms of a position dependent diffusion coefficient mimicking measured deterministic diffusivity gradients in biological cells or the inherent heterogeneity of geophysical systems. Compared to previous studies we here investigate the effect of the interplay of anomalous diffusion effected by position dependent diffusion coefficients and coloured non-Gaussian noise. The latter is chosen to be distributed according to Tsallis' q-distribution, representing a popular example for a non-extensive statistic. We obtain the ensemble and time averaged mean squared displacements for this generalised process and establish its non-ergodic properties as well as analyse the non-Gaussian nature of the associated displacement distribution. We consider both non-stratified and stratified environments. KW - diffusion KW - anomalous diffusion KW - non-extensive statistics KW - coloured KW - noise KW - heterogeneous diffusion process Y1 - 2021 U6 - https://doi.org/10.1088/1751-8121/abfba6 SN - 1751-8113 SN - 1751-8121 VL - 54 IS - 29 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Varghese, Alan J. A1 - Chechkin, Aleksei A1 - Metzler, Ralf A1 - Sujith, Raman I. T1 - Capturing multifractality of pressure fluctuations in thermoacoustic systems using fractional-order derivatives JF - Chaos : an interdisciplinary journal of nonlinear science N2 - The stable operation of a turbulent combustor is not completely silent; instead, there is a background of small amplitude aperiodic acoustic fluctuations known as combustion noise. Pressure fluctuations during this state of combustion noise are multifractal due to the presence of multiple temporal scales that contribute to its dynamics. However, existing models are unable to capture the multifractality in the pressure fluctuations. We conjecture an underlying fractional dynamics for the thermoacoustic system and obtain a fractional-order model for pressure fluctuations. The data from this model has remarkable visual similarity to the experimental data and also has a wide multifractal spectrum during the state of combustion noise. Quantitative similarity with the experimental data in terms of the Hurst exponent and the multifractal spectrum is observed during the state of combustion noise. This model is also able to produce pressure fluctuations that are qualitatively similar to the experimental data acquired during intermittency and thermoacoustic instability. Furthermore, we argue that the fractional dynamics vanish as we approach the state of thermoacoustic instability. Y1 - 2021 U6 - https://doi.org/10.1063/5.0032585 SN - 1054-1500 SN - 1089-7682 VL - 31 IS - 3 PB - American Institute of Physics, AIP CY - Melville ER - TY - JOUR A1 - Emanuel, Marc D. A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf A1 - Gompper, Gerhard T1 - Buckling transitions and soft-phase invasion of two-component icosahedral shells JF - Physical review / publ. by The American Physical Society. E, Statistical, nonlinear, and soft matter physics N2 - What is the optimal distribution of two types of crystalline phases on the surface of icosahedral shells, such as of many viral capsids? We here investigate the distribution of a thin layer of soft material on a crystalline convex icosahedral shell. We demonstrate how the shapes of spherical viruses can be understood from the perspective of elasticity theory of thin two-component shells. We develop a theory of shape transformations of an icosahedral shell upon addition of a softer, but still crystalline, material onto its surface. We show how the soft component "invades" the regions with the highest elastic energy and stress imposed by the 12 topological defects on the surface. We explore the phase diagram as a function of the surface fraction of the soft material, the shell size, and the incommensurability of the elastic moduli of the rigid and soft phases. We find that, as expected, progressive filling of the rigid shell by the soft phase starts from the most deformed regions of the icosahedron. With a progressively increasing soft-phase coverage, the spherical segments of domes are filled first (12 vertices of the shell), then the cylindrical segments connecting the domes (30 edges) are invaded, and, ultimately, the 20 flat faces of the icosahedral shell tend to be occupied by the soft material. We present a detailed theoretical investigation of the first two stages of this invasion process and develop a model of morphological changes of the cone structure that permits noncircular cross sections. In conclusion, we discuss the biological relevance of some structures predicted from our calculations, in particular for the shape of viral capsids. Y1 - 2020 U6 - https://doi.org/10.1103/PhysRevE.102.062104 SN - 2470-0045 SN - 2470-0053 SN - 2470-0061 SN - 1538-4519 VL - 102 IS - 6 PB - Woodbury CY - New York ER - TY - JOUR A1 - Metzler, Ralf T1 - Brownian motion and beyond: first-passage, power spectrum, non-Gaussianity, and anomalous diffusion JF - Journal of statistical mechanics: theory and experiment N2 - 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 exists a very large range of applications of stochastic processes in various disciplines. Here we provide a summary of some of the recent developments in the field of stochastic processes, highlighting both the experimental findings and theoretical frameworks. KW - 15 KW - 4 Y1 - 2019 U6 - https://doi.org/10.1088/1742-5468/ab4988 SN - 1742-5468 VL - 2019 IS - 11 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Hou, Ru A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf A1 - Akimoto, Takuma T1 - Biased continuous-time random walks for ordinary and equilibrium cases BT - facilitation of diffusion, ergodicity breaking and ageing JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent alpha are considered for the WTD psi(t) similar to 1/t(1+alpha). We treat continuous-time random walks (CTRWs) with 0 < alpha < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < alpha < 2 and alpha > 2. We demonstrate that in the presence of a drift CTRWs with alpha < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < alpha < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with alpha > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < alpha < 2 and alpha > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with alpha > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent alpha. Y1 - 2018 U6 - https://doi.org/10.1039/c8cp01863d SN - 1463-9076 SN - 1463-9084 VL - 20 IS - 32 SP - 20827 EP - 20848 PB - Royal Society of Chemistry CY - Cambridge 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 - Thapa, Samudrajit A1 - Lomholt, Michael Andersen A1 - Krog, Jens A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Bayesian analysis of single-particle tracking data using the nested-sampling algorithm: maximum-likelihood model selection applied to stochastic-diffusivity data JF - Physical chemistry, chemical physics : PCCP ; a journal of European Chemical Societies N2 - We employ Bayesian statistics using the nested-sampling algorithm to compare and rank multiple models of ergodic diffusion (including anomalous diffusion) as well as to assess their optimal parameters for in silico-generated and real time-series. We focus on the recently-introduced model of Brownian motion with "diffusing diffusivity'-giving rise to widely-observed non-Gaussian displacement statistics-and its comparison to Brownian and fractional Brownian motion, also for the time-series with some measurement noise. We conduct this model-assessment analysis using Bayesian statistics and the nested-sampling algorithm on the level of individual particle trajectories. We evaluate relative model probabilities and compute best-parameter sets for each diffusion model, comparing the estimated parameters to the true ones. We test the performance of the nested-sampling algorithm and its predictive power both for computer-generated (idealised) trajectories as well as for real single-particle-tracking trajectories. Our approach delivers new important insight into the objective selection of the most suitable stochastic model for a given time-series. We also present first model-ranking results in application to experimental data of tracer diffusion in polymer-based hydrogels. Y1 - 2018 U6 - https://doi.org/10.1039/c8cp04043e SN - 1463-9076 SN - 1463-9084 VL - 20 IS - 46 SP - 29018 EP - 29037 PB - Royal Society of Chemistry CY - Cambridge 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 - Doerries, Timo J. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Apparent anomalous diffusion and non-Gaussian distributions in a simple mobile-immobile transport model with Poissonian switching JF - Interface : journal of the Royal Society N2 - 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. Specifically, 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 (MSD) displays a plateau. Moreover, we demonstrate a short-time cubic time dependence of the MSD for immobile tracers when initially all particles are immobile. KW - diffusion KW - mobile-immobile model KW - tau proteins Y1 - 2022 U6 - https://doi.org/10.1098/rsif.2022.0233 SN - 1742-5689 SN - 1742-5662 VL - 19 IS - 192 PB - Royal Society CY - London ER -