TY - JOUR A1 - Schulz, Johannes H. P. A1 - Barkai, Eli A1 - Metzler, Ralf T1 - Aging renewal theory and application to random walks JF - Physical review : X, Expanding access N2 - We discuss a renewal process in which successive events are separated by scale-free waiting time periods. Among other ubiquitous long-time properties, this process exhibits aging: events counted initially in a time interval [0, t] statistically strongly differ from those observed at later times [t(a,) t(a) + t]. The versatility of renewal theory is owed to its abstract formulation. Renewals can be interpreted as steps of a random walk, switching events in two-state models, domain crossings of a random motion, etc. In complex, disordered media, processes with scale-free waiting times play a particularly prominent role. We set up a unified analytical foundation for such anomalous dynamics by discussing in detail the distribution of the aging renewal process. We analyze its half-discrete, half-continuous nature and study its aging time evolution. These results are readily used to discuss a scale-free anomalous diffusion process, the continuous-time random walk. By this, we not only shed light on the profound origins of its characteristic features, such as weak ergodicity breaking, along the way, we also add an extended discussion on aging effects. In particular, we find that the aging behavior of time and ensemble averages is conceptually very distinct, but their time scaling is identical at high ages. Finally, we show how more complex motion models are readily constructed on the basis of aging renewal dynamics. Y1 - 2014 U6 - https://doi.org/10.1103/PhysRevX.4.011028 SN - 2160-3308 VL - 4 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Metzler, Ralf A1 - Jeon, Jae-Hyung A1 - Cherstvy, Andrey G. A1 - Barkai, Eli T1 - Anomalous diffusion models and their properties BT - non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking JF - physical chemistry, chemical physics : PCCP N2 - Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion. KW - intermittent chaotic systems KW - Fokker-Planck equations KW - time random-walks KW - fluorescence photobleaching recovery KW - fluctuation-dissipation theorem KW - fractional dynamics approach KW - photon-counting statistics KW - weak ergodicity breaking KW - flight search patterns KW - levy flights Y1 - 2014 U6 - https://doi.org/10.1039/c4cp03465a SN - 1463-9076 SN - 1463-9084 VL - 2014 IS - 16 SP - 24128 EP - 24164 ER - TY - GEN A1 - Metzler, Ralf A1 - Jeon, Jae-Hyung A1 - Cherstvy, Andrey G. A1 - Barkai, Eli T1 - Anomalous diffusion models and their properties BT - non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking N2 - Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 174 KW - Fokker-Planck equations KW - flight search patterns KW - fluctuation-dissipation theorem KW - fluorescence photobleaching recovery KW - fractional dynamics approach KW - intermittent chaotic systems KW - levy flights KW - photon-counting statistics KW - time random-walks KW - weak ergodicity breaking Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-74448 SP - 24128 EP - 24164 ER - TY - JOUR A1 - Metzler, Ralf A1 - Jeon, Jae-Hyung A1 - Cherstvy, Andrey G. A1 - Barkai, Eli T1 - Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion. Y1 - 2014 U6 - https://doi.org/10.1039/c4cp03465a SN - 1463-9076 SN - 1463-9084 VL - 16 IS - 44 SP - 24128 EP - 24164 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Godec, Aljaz A1 - Chechkin, Aleksei V. A1 - Barkai, Eli A1 - Kantz, Holger A1 - Metzler, Ralf T1 - Localisation and universal fluctuations in ultraslow diffusion processes JF - Journal of physics : A, Mathematical and theoretical N2 - 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. KW - Sinai diffusion KW - anomalous diffusion KW - quenched energy landscape Y1 - 2014 U6 - https://doi.org/10.1088/1751-8113/47/49/492002 SN - 1751-8113 SN - 1751-8121 VL - 47 IS - 49 PB - IOP Publ. Ltd. CY - Bristol ER -