@article{GodecChechkinBarkaietal.2014, author = {Godec, Aljaz and Chechkin, Aleksei V. and Barkai, Eli and Kantz, Holger and Metzler, Ralf}, title = {Localisation and universal fluctuations in ultraslow diffusion processes}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {47}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {49}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/47/49/492002}, pages = {10}, year = {2014}, abstract = {We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) < x(2)(t)> similar or equal to log(gamma)t. Comparison of annealed (renewal) continuous time random walks (CTRWs) with logarithmic waiting time distribution psi(tau) similar or equal to 1/(tau log(1+gamma)tau) and Sinai diffusion in quenched random landscapes reveals striking similarities, despite the great differences in their physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time-averaged and ensemble-averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal, with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble-averaged MSD and time-averaged MSD.}, language = {en} } @article{SandevChechkinKorabeletal.2015, author = {Sandev, Trifce and Chechkin, Aleksei V. and Korabel, Nickolay and Kantz, Holger and Sokolov, Igor M. and Metzler, Ralf}, title = {Distributed-order diffusion equations and multifractality: Models and solutions}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {92}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.92.042117}, pages = {19}, year = {2015}, abstract = {We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.}, language = {en} } @article{SandevChechkinKantzetal.2015, author = {Sandev, Trifce and Chechkin, Aleksei V. and Kantz, Holger and Metzler, Ralf}, title = {Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel}, series = {Fractional calculus and applied analysis : an international journal for theory and applications}, volume = {18}, journal = {Fractional calculus and applied analysis : an international journal for theory and applications}, number = {4}, publisher = {De Gruyter}, address = {Berlin}, issn = {1311-0454}, doi = {10.1515/fca-2015-0059}, pages = {1006 -- 1038}, year = {2015}, abstract = {We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck-Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.}, language = {en} } @article{SandevIominKantzetal.2016, author = {Sandev, Trifce and Iomin, Alexander and Kantz, Holger and Metzler, Ralf and Chechkin, Aleksei V.}, title = {Comb Model with Slow and Ultraslow Diffusion}, series = {Mathematical modelling of natural phenomena}, volume = {11}, journal = {Mathematical modelling of natural phenomena}, publisher = {EDP Sciences}, address = {Les Ulis}, issn = {0973-5348}, doi = {10.1051/mmnp/201611302}, pages = {18 -- 33}, year = {2016}, abstract = {We consider a generalized diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyze the probability distribution functions and we derive the mean squared displacement in x and y directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both x and y directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.}, language = {en} } @article{ChechkinKantzMetzler2017, author = {Chechkin, Aleksei V. and Kantz, Holger and Metzler, Ralf}, title = {Ageing effects in ultraslow continuous time random walks}, series = {The European physical journal : B, Condensed matter and complex systems}, volume = {90}, journal = {The European physical journal : B, Condensed matter and complex systems}, publisher = {Springer}, address = {New York}, issn = {1434-6028}, doi = {10.1140/epjb/e2017-80270-9}, pages = {12}, year = {2017}, abstract = {In ageing systems physical observables explicitly depend on the time span elapsing between the original initiation of the system and the actual start of the recording of the particle motion. We here study the signatures of ageing in the framework of ultraslow continuous time random walk processes with super-heavy tailed waiting time densities. We derive the density for the forward or recurrent waiting time of the motion as function of the ageing time, generalise the Montroll-Weiss equation for this process, and analyse the ageing behaviour of the ensemble and time averaged mean squared displacements.}, language = {en} } @article{GajdaWylomanskaKantzetal.2018, author = {Gajda, J. and Wylomanska, Agnieszka and Kantz, Holger and Chechkin, Aleksei V. and Sikora, Grzegorz}, title = {Large deviations of time-averaged statistics for Gaussian processes}, series = {Statistics \& Probability Letters}, volume = {143}, journal = {Statistics \& Probability Letters}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-7152}, doi = {10.1016/j.spl.2018.07.013}, pages = {47 -- 55}, year = {2018}, abstract = {In this paper we study the large deviations of time averaged mean square displacement (TAMSD) for Gaussian processes. The theory of large deviations is related to the exponential decay of probabilities of large fluctuations in random systems. From the mathematical point of view a given statistics satisfies the large deviation principle, if the probability that it belongs to a certain range decreases exponentially. The TAMSD is one of the main statistics used in the problem of anomalous diffusion detection. Applying the theory of generalized chi-squared distribution and sub-gamma random variables we prove the upper bound for large deviations of TAMSD for Gaussian processes. As a special case we consider fractional Brownian motion, one of the most popular models of anomalous diffusion. Moreover, we derive the upper bound for large deviations of the estimator for the anomalous diffusion exponent. (C) 2018 Elsevier B.V. All rights reserved.}, 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} } @misc{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 = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1118}, issn = {1866-8372}, doi = {10.25932/publishup-49349}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-493494}, pages = {24}, 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{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{MeyerAghionKantz2022, author = {Meyer, Philipp and Aghion, Erez and Kantz, Holger}, title = {Decomposing the effect of anomalous diffusion enables direct calculation of the Hurst exponent and model classification for single random paths}, series = {Journal of physics / Institute of Physics. A, Mathematical, nuclear and general}, volume = {55}, journal = {Journal of physics / Institute of Physics. A, Mathematical, nuclear and general}, number = {27}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac72d4}, pages = {22}, year = {2022}, abstract = {Recently, a large number of research teams from around the world collaborated in the so-called 'anomalous diffusion challenge'. Its aim: to develop and compare new techniques for inferring stochastic models from given unknown time series, and estimate the anomalous diffusion exponent in data. We use various numerical methods to directly obtain this exponent using the path increments, and develop a questionnaire for model selection based on feature analysis of a set of known stochastic processes given as candidates. Here, we present the theoretical background of the automated algorithm which we put for these tasks in the diffusion challenge, as a counter to other pure data-driven approaches.}, language = {en} }