TY - JOUR A1 - Schulz, Johannes H. P. A1 - Barkai, Eli A1 - Metzler, Ralf T1 - Aging effects and population splitting in single-particle trajectoryaverages JF - Physical review letters N2 - We study time averages of single particle trajectories in scale-free anomalous diffusion processes, in which the measurement starts at some time t(a) > 0 after initiation of the process at t = 0. Using aging renewal theory, we show that for such nonstationary processes a large class of observables are affected by a unique aging function, which is independent of boundary conditions or the external forces. Moreover, we discuss the implications of aging induced population splitting: with growing age ta of the process, an increasing fraction of particles remains motionless in a measurement of fixed duration. Consequences for single biomolecule tracking in live cells are discussed. Y1 - 2013 U6 - https://doi.org/10.1103/PhysRevLett.110.020602 SN - 0031-9007 SN - 1079-7114 VL - 110 IS - 2 PB - American Physical Society CY - College Park ER - 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 - Vahabi, Mahsa A1 - Schulz, Johannes H. P. A1 - Shokri, Babak A1 - Metzler, Ralf T1 - Area coverage of radial Levy flights with periodic boundary conditions JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We consider the area coverage of radial Levy flights in a finite square area with periodic boundary conditions. From simulations we show how the fractal path dimension d(f) and thus the degree of area coverage depends on the number of steps of the trajectory, the size of the area, and the resolution of the applied box counting algorithm. For sufficiently long trajectories and not too high resolution, the fractal dimension returned by the box counting method equals two, and in that sense the Levy flight fully covers the area. Otherwise, the determined fractal dimension equals the stable index of the distribution of jump lengths of the Levy flight. We provide mathematical expressions for the turnover between these two scaling regimes. As complementary methods to analyze confined Levy flights we investigate fractional order moments of the position for which we also provide scaling arguments. Finally, we study the time evolution of the probability density function and the first passage time density of Levy flights in a square area. Our findings are of interest for a general understanding of Levy flights as well as for the analysis of recorded trajectories of animals searching for food or for human motion patterns. Y1 - 2013 U6 - https://doi.org/10.1103/PhysRevE.87.042136 SN - 1539-3755 VL - 87 IS - 4 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Schulz, Johannes H. P. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Correlated continuous time random walks - combining scale-invariance with long-range memory for spatial and temporal dynamics JF - Journal of physics : A, Mathematical and theoretical N2 - Standard continuous time random walk (CTRW) models are renewal processes in the sense that at each jump a new, independent pair of jump length and waiting time are chosen. Globally, anomalous diffusion emerges through scale-free forms of the jump length and/or waiting time distributions by virtue of the generalized central limit theorem. Here we present a modified version of recently proposed correlated CTRW processes, where we incorporate a power-law correlated noise on the level of both jump length and waiting time dynamics. We obtain a very general stochastic model, that encompasses key features of several paradigmatic models of anomalous diffusion: discontinuous, scale-free displacements as in Levy flights, scale-free waiting times as in subdiffusive CTRWs, and the long-range temporal correlations of fractional Brownian motion (FBM). We derive the exact solutions for the single-time probability density functions and extract the scaling behaviours. Interestingly, we find that different combinations of the model parameters lead to indistinguishable shapes of the emerging probability density functions and identical scaling laws. Our model will be useful for describing recent experimental single particle tracking data that feature a combination of CTRW and FBM properties. Y1 - 2013 U6 - https://doi.org/10.1088/1751-8113/46/47/475001 SN - 1751-8113 SN - 1751-8121 VL - 46 IS - 47 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Kursawe, Jochen A1 - Schulz, Johannes H. P. A1 - Metzler, Ralf T1 - Transient aging in fractional brownian and langevin-equation motion JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - Stochastic processes driven by stationary fractional Gaussian noise, that is, fractional Brownian motion and fractional Langevin-equation motion, are usually considered to be ergodic in the sense that, after an algebraic relaxation, time and ensemble averages of physical observables coincide. Recently it was demonstrated that fractional Brownian motion and fractional Langevin-equation motion under external confinement are transiently nonergodic-time and ensemble averages behave differently-from the moment when the particle starts to sense the confinement. Here we show that these processes also exhibit transient aging, that is, physical observables such as the time-averaged mean-squared displacement depend on the time lag between the initiation of the system at time t = 0 and the start of the measurement at the aging time t(a). In particular, it turns out that for fractional Langevin-equation motion the aging dependence on ta is different between the cases of free and confined motion. We obtain explicit analytical expressions for the aged moments of the particle position as well as the time-averaged mean-squared displacement and present a numerical analysis of this transient aging phenomenon. Y1 - 2013 U6 - https://doi.org/10.1103/PhysRevE.88.062124 SN - 1539-3755 SN - 1550-2376 VL - 88 IS - 6 PB - American Physical Society CY - College Park ER -