TY - JOUR A1 - Thapa, Samudrajit A1 - Wyłomańska, Agnieszka A1 - Sikora, Grzegorz A1 - Wagner, Caroline E. A1 - Krapf, Diego A1 - Kantz, Holger A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories JF - New Journal of Physics N2 - 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. KW - diffusion KW - anomalous diffusion KW - large-deviation statistic KW - time-averaged mean squared displacement KW - Chebyshev inequality Y1 - 2020 U6 - https://doi.org/10.1088/1367-2630/abd50e SN - 1367-2630 VL - 23 PB - Dt. Physikalische Ges. ; IOP CY - Bad Honnef ; London ER - TY - JOUR A1 - Sposini, Vittoria A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - First passage statistics for diffusing diffusivity JF - Journal of physics : A, Mathematical and theoretical N2 - A rapidly increasing number of systems is identified in which the stochastic motion of tracer particles follows the Brownian law < r(2)(t)> similar or equal to Dt yet the distribution of particle displacements is strongly non-Gaussian. A central approach to describe this effect is the diffusing diffusivity (DD) model in which the diffusion coefficient itself is a stochastic quantity, mimicking heterogeneities of the environment encountered by the tracer particle on its path. We here quantify in terms of analytical and numerical approaches the first passage behaviour of the DD model. We observe significant modifications compared to Brownian-Gaussian diffusion, in particular that the DD model may have a faster first passage dynamics. Moreover we find a universal crossover point of the survival probability independent of the initial condition. KW - diffusion KW - superstatistics KW - first passage Y1 - 2018 U6 - https://doi.org/10.1088/1751-8121/aaf6ff SN - 1751-8113 SN - 1751-8121 VL - 52 IS - 4 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Dybiec, Bartlomiej A1 - Capala, Karol A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Conservative random walks in confining potentials JF - Journal of physics : A, Mathematical and theoretical N2 - Levy walks are continuous time random walks with spatio-temporal coupling of jump lengths and waiting times, often used to model superdiffusive spreading processes such as animals searching for food, tracer motion in weakly chaotic systems, or even the dynamics in quantum systems such as cold atoms. In the simplest version Levy walks move with a finite speed. Here, we present an extension of the Levy walk scenario for the case when external force fields influence the motion. The resulting motion is a combination of the response to the deterministic force acting on the particle, changing its velocity according to the principle of total energy conservation, and random velocity reversals governed by the distribution of waiting times. For the fact that the motion stays conservative, that is, on a constant energy surface, our scenario is fundamentally different from thermal motion in the same external potentials. In particular, we present results for the velocity and position distributions for single well potentials of different steepness. The observed dynamics with its continuous velocity changes enriches the theory of Levy walk processes and will be of use in a variety of systems, for which the particles are externally confined. KW - Levy walk KW - conservative random walks KW - Levy flight Y1 - 2018 U6 - https://doi.org/10.1088/1751-8121/aaefc2 SN - 1751-8113 SN - 1751-8121 VL - 52 IS - 1 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Mardoukhi, Yousof A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Spurious ergodicity breaking in normal and fractional Ornstein–Uhlenbeck process JF - New Journal of Physics N2 - The Ornstein–Uhlenbeck process is a stationary and ergodic Gaussian process, that is fully determined by its covariance function and mean. We show here that the generic definitions of the ensemble- and time-averaged mean squared displacements fail to capture these properties consistently, leading to a spurious ergodicity breaking. We propose to remedy this failure by redefining the mean squared displacements such that they reflect unambiguously the statistical properties of any stochastic process. In particular we study the effect of the initial condition in the Ornstein–Uhlenbeck process and its fractional extension. For the fractional Ornstein–Uhlenbeck process representing typical experimental situations in crowded environments such as living biological cells, we show that the stationarity of the process delicately depends on the initial condition. KW - Ornstein–Uhlenbeck process KW - stationary stochastic process KW - ensemble and time averaged mean squared displacement Y1 - 2020 U6 - https://doi.org/10.1088/1367-2630/ab950b SN - 1367-2630 VL - 22 PB - IOP CY - London ER - TY - JOUR A1 - Palyulin, Vladimir V. A1 - Mantsevich, Vladimir N. A1 - Klages, Rainer A1 - Metzler, Ralf A1 - Chechkin, Aleksei V. T1 - Comparison of pure and combined search strategies for single and multiple targets JF - The European physical journal : B, Condensed matter and complex systems N2 - We address the generic problem of random search for a point-like target on a line. Using the measures of search reliability and efficiency to quantify the random search quality, we compare Brownian search with Levy search based on long-tailed jump length distributions. We then compare these results with a search process combined of two different long-tailed jump length distributions. Moreover, we study the case of multiple targets located by a Levy searcher. Y1 - 2017 U6 - https://doi.org/10.1140/epjb/e2017-80372-4 SN - 1434-6028 SN - 1434-6036 VL - 90 SP - 20 EP - 37 PB - Springer CY - New York ER - TY - JOUR A1 - Palyulin, Vladimir V. A1 - Blackburn, George A1 - Lomholt, Michael A. A1 - Watkins, Nicholas W. A1 - Metzler, Ralf A1 - Klages, Rainer A1 - Chechkin, Aleksei V. T1 - First passage and first hitting times of Levy flights and Levy walks JF - New journal of physics : the open-access journal for physics N2 - For both Lévy flight and Lévy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Lévy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Lévy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms. KW - Levy flights KW - Levy walks KW - first-passage time KW - first-hitting time Y1 - 2019 U6 - https://doi.org/10.1088/1367-2630/ab41bb SN - 1367-2630 VL - 21 IS - 10 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Cherstvy, Andrey G. A1 - Thapa, Samudrajit A1 - Mardoukhi, Yousof A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Time averages and their statistical variation for the Ornstein-Uhlenbeck process BT - Role of initial particle distributions and relaxation to stationarity JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and outof-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed. Y1 - 2018 U6 - https://doi.org/10.1103/PhysRevE.98.022134 SN - 2470-0045 SN - 2470-0053 VL - 98 IS - 2 PB - American Physical Society CY - College Park ER - TY - GEN A1 - Palyulin, Vladimir V A1 - Blackburn, George A1 - Lomholt, Michael A A1 - Watkins, Nicholas W A1 - Metzler, Ralf A1 - Klages, Rainer A1 - Chechkin, Aleksei V. T1 - First passage and first hitting times of Lévy flights and Lévy walks T2 - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe N2 - For both Lévy flight and Lévy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Lévy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Lévy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 785 KW - Lévy flights KW - Lévy walks KW - first-passage time KW - first-hitting time Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-439832 SN - 1866-8372 IS - 785 ER - TY - JOUR A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity JF - Soft matter N2 - We study the thermal Markovian diffusion of tracer particles in a 2D medium with spatially varying diffusivity D(r), mimicking recently measured, heterogeneous maps of the apparent diffusion coefficient in biological cells. For this heterogeneous diffusion process (HDP) we analyse the mean squared displacement (MSD) of the tracer particles, the time averaged MSD, the spatial probability density function, and the first passage time dynamics from the cell boundary to the nucleus. Moreover we examine the non-ergodic properties of this process which are important for the correct physical interpretation of time averages of observables obtained from single particle tracking experiments. From extensive computer simulations of the 2D stochastic Langevin equation we present an in-depth study of this HDP. In particular, we find that the MSDs along the radial and azimuthal directions in a circular domain obey anomalous and Brownian scaling, respectively. We demonstrate that the time averaged MSD stays linear as a function of the lag time and the system thus reveals a weak ergodicity breaking. Our results will enable one to rationalise the diffusive motion of larger tracer particles such as viruses or submicron beads in biological cells. KW - anomalous diffusion KW - intracellular-transport KW - adenoassociated virus KW - infection pathway KW - escherichia-coli KW - endosomal escape KW - living cells KW - trafficking KW - cytoplasm KW - models Y1 - 2014 U6 - https://doi.org/10.1039/c3sm52846d SN - 2046-2069 VL - 2014 IS - 10 SP - 1591 EP - 1601 PB - Royal Society of Chemistry ER - TY - GEN A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity N2 - We study the thermal Markovian diffusion of tracer particles in a 2D medium with spatially varying diffusivity D(r), mimicking recently measured, heterogeneous maps of the apparent diffusion coefficient in biological cells. For this heterogeneous diffusion process (HDP) we analyse the mean squared displacement (MSD) of the tracer particles, the time averaged MSD, the spatial probability density function, and the first passage time dynamics from the cell boundary to the nucleus. Moreover we examine the non-ergodic properties of this process which are important for the correct physical interpretation of time averages of observables obtained from single particle tracking experiments. From extensive computer simulations of the 2D stochastic Langevin equation we present an in-depth study of this HDP. In particular, we find that the MSDs along the radial and azimuthal directions in a circular domain obey anomalous and Brownian scaling, respectively. We demonstrate that the time averaged MSD stays linear as a function of the lag time and the system thus reveals a weak ergodicity breaking. Our results will enable one to rationalise the diffusive motion of larger tracer particles such as viruses or submicron beads in biological cells. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 168 KW - adenoassociated virus KW - anomalous diffusion KW - cytoplasm KW - endosomal escape KW - escherichia-coli KW - infection pathway KW - intracellular-transport KW - living cells KW - models KW - trafficking Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-74021 IS - 168 SP - 1591 EP - 1601 ER -