@article{SposiniGrebenkovMetzleretal.2020, author = {Sposini, Vittoria and Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb and Seno, Flavio}, title = {Universal spectral features of different classes of random-diffusivity processes}, series = {New Journal of Physics}, volume = {22}, journal = {New Journal of Physics}, number = {6}, publisher = {Dt. Physikalische Ges.}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab9200}, pages = {26}, year = {2020}, abstract = {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/f²-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.}, language = {en} } @article{SposiniMetzlerOshanin2019, author = {Sposini, Vittoria and Metzler, Ralf and Oshanin, Gleb}, title = {Single-trajectory spectral analysis of scaled Brownian motion}, series = {New Journal of Physics}, volume = {21}, journal = {New Journal of Physics}, publisher = {Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics}, address = {Bad Honnef und London}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab2f52}, pages = {16}, year = {2019}, abstract = {Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.}, language = {en} } @article{ThapaWyłomańskaSikoraetal.2021, author = {Thapa, Samudrajit and Wyłomańska, Agnieszka and Sikora, Grzegorz and Wagner, Caroline E. and Krapf, Diego and Kantz, Holger and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories}, series = {New Journal of Physics}, volume = {23}, journal = {New Journal of Physics}, publisher = {Dt. Physikalische Ges. ; IOP}, address = {Bad Honnef ; London}, issn = {1367-2630}, doi = {10.1088/1367-2630/abd50e}, pages = {22}, year = {2021}, abstract = {Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations.}, language = {en} } @article{VilkAghionNathanetal.2022, author = {Vilk, Ohad and Aghion, Erez and Nathan, Ran and Toledo, Sivan and Metzler, Ralf and Assaf, Michael}, title = {Classification of anomalous diffusion in animal movement data using power spectral analysis}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {55}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {33}, publisher = {IOP Publishing}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac7e8f}, pages = {16}, year = {2022}, abstract = {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.}, language = {en} } @article{WangSenoSokolovetal.2020, author = {Wang, Wei and Seno, Flavio and Sokolov, Igor M. and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Unexpected crossovers in correlated random-diffusivity processes}, series = {New Journal of Physics}, volume = {22}, journal = {New Journal of Physics}, publisher = {Dt. Physikalische Ges.}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/aba390}, pages = {17}, year = {2020}, abstract = {The passive and active motion of micron-sized tracer particles in crowded liquids and inside living biological cells is ubiquitously characterised by 'viscoelastic' anomalous diffusion, in which the increments of the motion feature long-ranged negative and positive correlations. While viscoelastic anomalous diffusion is typically modelled by a Gaussian process with correlated increments, so-called fractional Gaussian noise, an increasing number of systems are reported, in which viscoelastic anomalous diffusion is paired with non-Gaussian displacement distributions. Following recent advances in Brownian yet non-Gaussian diffusion we here introduce and discuss several possible versions of random-diffusivity models with long-ranged correlations. While all these models show a crossover from non-Gaussian to Gaussian distributions beyond some correlation time, their mean squared displacements exhibit strikingly different behaviours: depending on the model crossovers from anomalous to normal diffusion are observed, as well as a priori unexpected dependencies of the effective diffusion coefficient on the correlation exponent. Our observations of the non-universality of random-diffusivity viscoelastic anomalous diffusion are important for the analysis of experiments and a better understanding of the physical origins of 'viscoelastic yet non-Gaussian' diffusion.}, language = {en} } @article{XuZhouMetzleretal.2022, author = {Xu, Pengbo and Zhou, Tian and Metzler, Ralf and Deng, Weihua}, title = {Stochastic harmonic trapping of a L{\´e}vy walk}, series = {New journal of physics : the open-access journal for physics / Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics}, volume = {24}, journal = {New journal of physics : the open-access journal for physics / Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics}, number = {3}, publisher = {Deutsche Physikalische Gesellschaft}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/ac5282}, pages = {1 -- 28}, year = {2022}, abstract = {We introduce and study a L{\´e}vy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic.}, language = {en} } @article{ŚlęzakBurneckiMetzler2019, author = {Ślęzak, Jakub and Burnecki, Krzysztof and Metzler, Ralf}, title = {Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems}, series = {New Journal of Physics}, volume = {21}, journal = {New Journal of Physics}, publisher = {Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics}, address = {Bad Honnef und London}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab3366}, pages = {18}, year = {2019}, abstract = {Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as 'superstatistics' or 'diffusing diffusivity'. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models.Westart from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion.}, language = {en} } @article{ŚlęzakMetzlerMagdziarz2019, author = {Ślęzak, Jakub and Metzler, Ralf and Magdziarz, Marcin}, title = {Codifference can detect ergodicity breaking and non-Gaussianity}, series = {New Journal of Physics}, volume = {21}, journal = {New Journal of Physics}, publisher = {Deutsche Physikalische Gesellschaft}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab13f3}, pages = {25}, year = {2019}, abstract = {We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a nonlinear analogue of the mean squared displacement.}, language = {en} }