TY - JOUR A1 - Metzler, Ralf T1 - Superstatistics and non-Gaussian diffusion JF - The European physical journal special topics N2 - Brownian motion and viscoelastic anomalous diffusion in homogeneous environments are intrinsically Gaussian processes. In a growing number of systems, however, non-Gaussian displacement distributions of these processes are being reported. The physical cause of the non-Gaussianity is typically seen in different forms of disorder. These include, for instance, imperfect "ensembles" of tracer particles, the presence of local variations of the tracer mobility in heteroegenous environments, or cases in which the speed or persistence of moving nematodes or cells are distributed. From a theoretical point of view stochastic descriptions based on distributed ("superstatistical") transport coefficients as well as time-dependent generalisations based on stochastic transport parameters with built-in finite correlation time are invoked. After a brief review of the history of Brownian motion and the famed Gaussian displacement distribution, we here provide a brief introduction to the phenomenon of non-Gaussianity and the stochastic modelling in terms of superstatistical and diffusing-diffusivity approaches. KW - Brownian diffusion KW - anomalous diffusion KW - dynamics KW - kinetic-theory KW - models KW - motion KW - nanoparticles KW - nonergodicity KW - statistics KW - subdiffusion Y1 - 2020 U6 - https://doi.org/10.1140/epjst/e2020-900210-x SN - 1951-6355 SN - 1951-6401 VL - 229 IS - 5 SP - 711 EP - 728 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Wang, Wei A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Thapa, Samudrajit A1 - Seno, Flavio A1 - Liu, Xianbin A1 - Metzler, Ralf T1 - Fractional Brownian motion with random diffusivity BT - emerging residual nonergodicity below the correlation time JF - Journal of physics : A, Mathematical and theoretical N2 - Numerous examples for a priori unexpected non-Gaussian behaviour for normal and anomalous diffusion have recently been reported in single-particle tracking experiments. Here, we address the case of non-Gaussian anomalous diffusion in terms of a random-diffusivity mechanism in the presence of power-law correlated fractional Gaussian noise. We study the ergodic properties of this model via examining the ensemble- and time-averaged mean-squared displacements as well as the ergodicity breaking parameter EB quantifying the trajectory-to-trajectory fluctuations of the latter. For long measurement times, interesting crossover behaviour is found as function of the correlation time tau characterising the diffusivity dynamics. We unveil that at short lag times the EB parameter reaches a universal plateau. The corresponding residual value of EB is shown to depend only on tau and the trajectory length. The EB parameter at long lag times, however, follows the same power-law scaling as for fractional Brownian motion. We also determine a corresponding plateau at short lag times for the discrete representation of fractional Brownian motion, absent in the continuous-time formulation. These analytical predictions are in excellent agreement with results of computer simulations of the underlying stochastic processes. Our findings can help distinguishing and categorising certain nonergodic and non-Gaussian features of particle displacements, as observed in recent single-particle tracking experiments. KW - stochastic processes KW - anomalous diffusion KW - fractional Brownian motion KW - diffusing diffusivity KW - weak ergodicity breaking Y1 - 2020 U6 - https://doi.org/10.1088/1751-8121/aba467 SN - 1751-8113 SN - 1751-8121 VL - 53 IS - 47 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Awad, Emad A1 - Metzler, Ralf T1 - Crossover dynamics from superdiffusion to subdiffusion BT - models and solutions JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - The Cattaneo or telegrapher's equation describes the crossover from initial ballistic to normal diffusion. Here we study and survey time-fractional generalisations of this equation that are shown to produce the crossover of the mean squared displacement from superdiffusion to subdiffusion. Conditional solutions are derived in terms of Fox H-functions and the dth-order moments as well as the diffusive flux of the different models are derived. Moreover, the concept of the distribution-like is proposed as an alternative to the probability density function. KW - Cattaneo equation KW - telegrapher's equation KW - crossover dynamics KW - fractional dynamic equations KW - anomalous diffusion KW - superdiffusion and KW - subdiffusion KW - Fox H-functions Y1 - 2020 U6 - https://doi.org/10.1515/fca-2020-0003 SN - 1311-0454 SN - 1314-2224 VL - 23 IS - 1 SP - 55 EP - 102 PB - De Gruyter CY - Berlin ; Boston ER - TY - JOUR A1 - Xu, Pengbo A1 - Deng, Weihua A1 - Sandev, Trifce T1 - Levy walk with parameter dependent velocity BT - hermite polynomial approach and numerical simulation JF - Journal of physics : A, Mathematical and theoretical N2 - To analyze stochastic processes, one often uses integral transform (Fourier and Laplace) methods. However, for the time-space coupled cases, e.g. the Levy walk, sometimes the integral transform method may fail. Here we provide a Hermite polynomial expansion approach, being complementary to the integral transform method, to the Levy walk. Two approaches are compared for some already known results. We also consider the generalized Levy walk with parameter dependent velocity. Namely, we consider the Levy walk with velocity which depends on the walking length or on the duration of each step. Some interesting features of the generalized Levy walk are observed, including the special shapes of the probability density function, the first passage time distributions, and various diffusive behaviors of the mean squared displacement. KW - Hermite polynomial expansion KW - Levy walk KW - anomalous diffusion KW - parameter KW - dependent velocity Y1 - 2020 U6 - https://doi.org/10.1088/1751-8121/ab7420 SN - 1751-8113 SN - 1751-8121 VL - 53 IS - 11 PB - IOP Publ. Ltd. CY - Bristol 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 - GEN 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 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1006 KW - diffusion KW - anomalous diffusion KW - non-Gaussianity KW - fractional Brownian motion Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-480049 SN - 1866-8372 IS - 1006 ER -