@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{PadashSandevKantzetal.2022, author = {Padash, Amin and Sandev, Trifce and Kantz, Holger and Metzler, Ralf and Chechkin, Aleksei}, title = {Asymmetric Levy flights are more efficient in random search}, series = {Fractal and fractional}, volume = {6}, journal = {Fractal and fractional}, number = {5}, publisher = {MDPI}, address = {Basel}, issn = {2504-3110}, doi = {10.3390/fractalfract6050260}, pages = {23}, year = {2022}, abstract = {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.}, language = {en} } @article{WangCherstvyKantzetal.2021, author = {Wang, Wei and Cherstvy, Andrey G. and Kantz, Holger and Metzler, Ralf and Sokolov, Igor M.}, title = {Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {104}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Institute of Physics}, address = {Woodbury, NY}, issn = {2470-0045}, doi = {10.1103/PhysRevE.104.024105}, pages = {27}, year = {2021}, abstract = {How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x) = D0|x|gamma and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicitybreaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB similar to(1/r )-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.}, language = {en} } @article{DoerriesChechkinSchumeretal.2022, author = {Doerries, Timo J. and Chechkin, Aleksei and Schumer, Rina and Metzler, Ralf}, title = {Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {105}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {1}, publisher = {The American Institute of Physics}, address = {Woodbury, NY}, issn = {2470-0045}, doi = {10.1103/PhysRevE.105.014105}, pages = {24}, year = {2022}, abstract = {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.}, language = {en} } @article{SandevDomazetoskiKocarevetal.2022, author = {Sandev, Trifce and Domazetoski, Viktor and Kocarev, Ljupco and Metzler, Ralf and Chechkin, Aleksei}, title = {Heterogeneous diffusion with stochastic resetting}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {55}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {7}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac491c}, pages = {26}, year = {2022}, abstract = {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.}, language = {en} } @article{PetreskaPejovSandevetal.2022, author = {Petreska, Irina and Pejov, Ljupco and Sandev, Trifce and Kocarev, Ljupčo and Metzler, Ralf}, title = {Tuning of the dielectric relaxation and complex susceptibility in a system of polar molecules: a generalised model based on rotational diffusion with resetting}, series = {Fractal and fractional}, volume = {6}, journal = {Fractal and fractional}, number = {2}, publisher = {MDPI AG, Fractal Fract Editorial Office}, address = {Basel}, issn = {2504-3110}, doi = {10.3390/fractalfract6020088}, pages = {23}, year = {2022}, abstract = {The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.}, language = {en} } @article{ThapaParkKimetal.2022, author = {Thapa, Samudrajit and Park, Seongyu and Kim, Yeongjin and Jeon, Jae-Hyung and Metzler, Ralf and Lomholt, Michael A.}, title = {Bayesian inference of scaled versus fractional Brownian motion}, series = {Journal of physics : A, mathematical and theoretical}, volume = {55}, journal = {Journal of physics : A, mathematical and theoretical}, number = {19}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac60e7}, pages = {21}, year = {2022}, abstract = {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.}, language = {en} } @article{CherstvySafdariMetzler2021, author = {Cherstvy, Andrey G. and Safdari, Hadiseh and Metzler, Ralf}, title = {Anomalous diffusion, nonergodicity, and ageing for exponentially and logarithmically time-dependent diffusivity}, series = {Journal of physics. D, Applied physics}, volume = {54}, journal = {Journal of physics. D, Applied physics}, number = {19}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {0022-3727}, doi = {10.1088/1361-6463/abdff0}, pages = {18}, year = {2021}, abstract = {We investigate a diffusion process with a time-dependent diffusion coefficient, both exponentially increasing and decreasing in time, D(t)=D-0(e +/- 2 alpha t). For this (hypothetical) nonstationary diffusion process we compute-both analytically and from extensive stochastic simulations-the behavior of the ensemble- and time-averaged mean-squared displacements (MSDs) of the particles, both in the over- and underdamped limits. Simple asymptotic relations derived for the short- and long-time behaviors are shown to be in excellent agreement with the results of simulations. The diffusive characteristics in the presence of ageing are also considered, with dramatic differences of the over- versus underdamped regime. Our results for D(t)=D-0(e +/- 2 alpha t) extend and generalize the class of diffusive systems obeying scaled Brownian motion featuring a power-law-like variation of the diffusivity with time, D(t) similar to t(alpha-1). We also examine the logarithmically increasing diffusivity, D(t)=D(0)log[t/tau(0)], as another fundamental functional dependence (in addition to the power-law and exponential) and as an example of diffusivity slowly varying in time. One of the main conclusions is that the behavior of the massive particles is predominantly ergodic, while weak ergodicity breaking is repeatedly found for the time-dependent diffusion of the massless particles at short times. The latter manifests itself in the nonequivalence of the (both nonaged and aged) MSD and the mean time-averaged MSD. The current findings are potentially applicable to a class of physical systems out of thermal equilibrium where a rapid increase or decrease of the particles' diffusivity is inherently realized. One biological system potentially featuring all three types of time-dependent diffusion (power-law-like, exponential, and logarithmic) is water diffusion in the brain tissues, as we thoroughly discuss in the end.}, language = {en} } @article{XuMetzlerWang2022, author = {Xu, Pengbo and Metzler, Ralf and Wang, Wanli}, title = {Infinite density and relaxation for Levy walks in an external potential}, series = {Physical review}, volume = {105}, journal = {Physical review}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.105.044118}, pages = {15}, year = {2022}, abstract = {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.}, language = {en} } @article{CherstvyVinodAghionetal.2021, author = {Cherstvy, Andrey G. and Vinod, Deepak and Aghion, Erez and Sokolov, Igor M. and Metzler, Ralf}, title = {Scaled geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {103}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {6}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.103.062127}, pages = {11}, year = {2021}, abstract = {Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic stochastic process of geometric Brownian motion (GBM). The innate property of such models and of real stock-market prices is the roughly exponential growth of prices with time [on average, in crisis-free times]. We here explore the ensemble- and time averages of a multiplicative-noise stochastic process with power-law-like time-dependent volatility, sigma(t) similar to t(alpha), named scaled GBM (SGBM). For SGBM, the mean-squared displacement (MSD) computed for an ensemble of statistically equivalent trajectories can grow faster than exponentially in time, while the time-averaged MSD (TAMSD)-based on a sliding-window averaging along a single trajectory-is always linear at short lag times Delta. The proportionality factor between these the two averages of the time series is Delta/T at short lag times, where T is the trajectory length, similarly to GBM. This discrepancy of the scaling relations and pronounced nonequivalence of the MSD and TAMSD at Delta/T << 1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with s (t)-modulation, the main focus of our analysis. The analytical predictions for the MSD and mean TAMSD for SGBM are in quantitative agreement with the results of stochastic computer simulations.}, language = {en} }