TY - JOUR A1 - Bodrova, Anna S. A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Ultraslow scaled Brownian motion JF - New journal of physics : the open-access journal for physics N2 - We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form D(t) similar or equal to 1/t. For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations. KW - anomalous diffusion KW - stochastic processes KW - ageing Y1 - 2015 U6 - https://doi.org/10.1088/1367-2630/17/6/063038 SN - 1367-2630 VL - 17 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Metzler, Ralf A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Bodrova, Anna S. T1 - Ultraslow scaled Brownian motion JF - New journal of physics : the open-access journal for physics N2 - We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form . For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations. KW - anomalous diffusion KW - stochastic processes KW - ageing Y1 - 2015 U6 - https://doi.org/10.1088/1367-2630/17/6/063038 SN - 1367-2630 VL - 17 IS - 063038 PB - Dt. Physikalische Ges., IOP CY - Bad Honnef, London ER - TY - JOUR A1 - Wang, Wei A1 - Seno, Flavio A1 - Sokolov, Igor M. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Unexpected crossovers in correlated random-diffusivity processes JF - New Journal of Physics N2 - The passive and active motion of micron-sized tracer particles in crowded liquids and inside living biological cells is ubiquitously characterised by 'viscoelastic' anomalous diffusion, in which the increments of the motion feature long-ranged negative and positive correlations. While viscoelastic anomalous diffusion is typically modelled by a Gaussian process with correlated increments, so-called fractional Gaussian noise, an increasing number of systems are reported, in which viscoelastic anomalous diffusion is paired with non-Gaussian displacement distributions. Following recent advances in Brownian yet non-Gaussian diffusion we here introduce and discuss several possible versions of random-diffusivity models with long-ranged correlations. While all these models show a crossover from non-Gaussian to Gaussian distributions beyond some correlation time, their mean squared displacements exhibit strikingly different behaviours: depending on the model crossovers from anomalous to normal diffusion are observed, as well as a priori unexpected dependencies of the effective diffusion coefficient on the correlation exponent. Our observations of the non-universality of random-diffusivity viscoelastic anomalous diffusion are important for the analysis of experiments and a better understanding of the physical origins of 'viscoelastic yet non-Gaussian' diffusion. KW - diffusion KW - anomalous diffusion KW - non-Gaussianity KW - fractional Brownian motion Y1 - 2020 U6 - https://doi.org/10.1088/1367-2630/aba390 SN - 1367-2630 VL - 22 PB - Dt. Physikalische Ges. CY - Bad Honnef ER - TY - JOUR A1 - Sandev, Trifce A1 - Metzler, Ralf A1 - Tomovski, Zivorad T1 - Velocity and displacement correlation functions for fractional generalized Langevin equations JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle. KW - fractional generalized Langevin equation KW - frictional memory kernel KW - variances KW - mean square displacement KW - anomalous diffusion Y1 - 2012 U6 - https://doi.org/10.2478/s13540-012-0031-2 SN - 1311-0454 VL - 15 IS - 3 SP - 426 EP - 450 PB - Versita CY - Warsaw ER -