TY - JOUR A1 - Chechkin, Aleksei V. A1 - Sokolov, Igor M. T1 - Relation between generalized diffusion equations and subordination schemes JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - Generalized (non-Markovian) diffusion equations with different memory kernels and subordination schemes based on random time change in the Brownian diffusion process are popular mathematical tools for description of a variety of non-Fickian diffusion processes in physics, biology, and earth sciences. Some of such processes (notably, the fluid limits of continuous time random walks) allow for either kind of description, but other ones do not. In the present work we discuss the conditions under which a generalized diffusion equation does correspond to a subordination scheme, and the conditions under which a subordination scheme does possess the corresponding generalized diffusion equation. Moreover, we discuss examples of random processes for which only one, or both kinds of description are applicable. Y1 - 2021 U6 - https://doi.org/10.1103/PhysRevE.103.032133 SN - 2470-0045 SN - 2470-0053 VL - 103 IS - 3 PB - American Physical Society CY - College Park 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 - TY - GEN A1 - Chechkin, Aleksei V. A1 - Zaid, Irwin M. A1 - Lomholt, Michael A. A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Bulk-mediated surface diffusion on a cylinder in the fast exchange limit T2 - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe N2 - In various biological systems and small scale technological applications particles transiently bind to a cylindrical surface. Upon unbinding the particles diffuse in the vicinal bulk before rebinding to the surface. Such bulk-mediated excursions give rise to an effective surface translation, for which we here derive and discuss the dynamic equations, including additional surface diffusion. We discuss the time evolution of the number of surface-bound particles, the effective surface mean squared displacement, and the surface propagator. In particular, we observe sub- and superdiffusive regimes. A plateau of the surface mean-squared displacement reflects a stalling of the surface diffusion at longer times. Finally, the corresponding first passage problem for the cylindrical geometry is analysed. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 593 KW - Bulk-mediated diffusion; KW - anomalous diffusion KW - Levy flights KW - stochastic processes Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-415480 SN - 1866-8372 IS - 593 SP - 114 EP - 126 ER - TY - JOUR A1 - Doerries, Timo J. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Apparent anomalous diffusion and non-Gaussian distributions in a simple mobile-immobile transport model with Poissonian switching JF - Interface : journal of the Royal Society N2 - We analyse mobile-immobile transport of particles that switch between the mobile and immobile phases with finite rates. Despite this seemingly simple assumption of Poissonian switching, we unveil a rich transport dynamics including significant transient anomalous diffusion and non-Gaussian displacement distributions. Our discussion is based on experimental parameters for tau proteins in neuronal cells, but the results obtained here are expected to be of relevance for a broad class of processes in complex systems. Specifically, we obtain that, when the mean binding time is significantly longer than the mean mobile time, transient anomalous diffusion is observed at short and intermediate time scales, with a strong dependence on the fraction of initially mobile and immobile particles. We unveil a Laplace distribution of particle displacements at relevant intermediate time scales. For any initial fraction of mobile particles, the respective mean squared displacement (MSD) displays a plateau. Moreover, we demonstrate a short-time cubic time dependence of the MSD for immobile tracers when initially all particles are immobile. KW - diffusion KW - mobile-immobile model KW - tau proteins Y1 - 2022 U6 - https://doi.org/10.1098/rsif.2022.0233 SN - 1742-5689 SN - 1742-5662 VL - 19 IS - 192 PB - Royal Society CY - London ER - TY - GEN A1 - Bodrova, Anna S. A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Safdari, Hadiseh A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Underdamped scaled Brownian motion BT - (non-)existence of the overdamped limit in anomalous diffusion N2 - It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 267 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-97158 ER - TY - JOUR A1 - Bodrova, Anna S. A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Safdari, Hadiseh A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Underdamped scaled Brownian motion BT - (non-)existence of the overdamped limit in anomalous diffusion JF - Scientific reports N2 - It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases. Y1 - 2016 U6 - https://doi.org/10.1038/srep30520 SN - 2045-2322 VL - 6 PB - Nature Publishing Group CY - London ER - TY - JOUR A1 - Safdari, Hadiseh A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Thiel, Felix A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Quantifying the non-ergodicity of scaled Brownian motion JF - Journal of physics : A, Mathematical and theoretical N2 - We examine the non-ergodic properties of scaled Brownian motion (SBM), a non-stationary stochastic process with a time dependent diffusivity of the form D(t) similar or equal to t(alpha-1). We compute the ergodicity breaking parameter EB in the entire range of scaling exponents a, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths T and short lag times Delta the EB parameter as function of the scaling exponent a has no divergence at alpha - 1/2 and present the asymptotes for EB in different limits. We generalize the analytical and simulations results for the time averaged and ergodic properties of SBM in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion. KW - scaled Brownian motion KW - anomalous diffusion KW - ageing Y1 - 2015 U6 - https://doi.org/10.1088/1751-8113/48/37/375002 SN - 1751-8113 SN - 1751-8121 VL - 48 IS - 37 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Safdari, Hadiseh A1 - Chechkin, Aleksei V. A1 - Jafari, Gholamreza R. A1 - Metzler, Ralf T1 - Aging scaled Brownian motion JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time. Y1 - 2015 U6 - https://doi.org/10.1103/PhysRevE.91.042107 SN - 1539-3755 SN - 1550-2376 VL - 91 IS - 4 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Bodrova, Anna S. A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Safdari, Hadiseh A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion JF - Scientific reports N2 - It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases. Y1 - 2016 U6 - https://doi.org/10.1038/srep30520 SN - 2045-2322 VL - 6 PB - Nature Publ. Group CY - London ER - TY - JOUR A1 - Safdari, Hadiseh A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Bodrova, Anna A1 - Metzler, Ralf T1 - Aging underdamped scaled Brownian motion BT - Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble-and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected. Y1 - 2017 U6 - https://doi.org/10.1103/PhysRevE.95.012120 SN - 2470-0045 SN - 2470-0053 VL - 95 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Sandev, Trifce A1 - Chechkin, Aleksei V. A1 - Korabel, Nickolay A1 - Kantz, Holger A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Distributed-order diffusion equations and multifractality: Models and solutions JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided. Y1 - 2015 U6 - https://doi.org/10.1103/PhysRevE.92.042117 SN - 1539-3755 SN - 1550-2376 VL - 92 IS - 4 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Molina-Garcia, Daniel A1 - Sandev, Trifce A1 - Safdari, Hadiseh A1 - Pagnini, Gianni A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Crossover from anomalous to normal diffusion BT - truncated power-law noise correlations and applications to dynamics in lipid bilayers JF - New Journal of Physics N2 - Abstract The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive–diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive–diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion–diffusion and subdiffusion–subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling. KW - anomalous diffusion KW - truncated power-law correlated noise KW - lipid bilayer membrane dynamics Y1 - 2018 U6 - https://doi.org/10.1088/1367-2630/aae4b2 SN - 1367-2630 VL - 20 PB - IOP Publishing Ltd CY - London und Bad Honnef ER - TY - GEN A1 - Molina-Garcia, Daniel A1 - Sandev, Trifce A1 - Safdari, Hadiseh A1 - Pagnini, Gianni A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Crossover from anomalous to normal diffusion BT - truncated power-law noise correlations and applications to dynamics in lipid bilayers T2 - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe N2 - Abstract The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive–diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive–diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion–diffusion and subdiffusion–subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 507 KW - anomalous diffusion KW - truncated power-law correlated noise KW - lipid bilayer membrane dynamics Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-422590 SN - 1866-8372 IS - 507 ER - TY - JOUR A1 - Sandev, Trifce A1 - Tomovski, Zivorad A1 - Dubbeldam, Johan L. A. A1 - Chechkin, Aleksei V. T1 - Generalized diffusion-wave equation with memory kernel JF - Journal of physics : A, Mathematical and theoretical N2 - We study generalized diffusion-wave equation in which the second order time derivative is replaced by an integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular cases. We consider different memory kernels of the integro-differential operator, derive corresponding fundamental solutions, specify the conditions of their non-negativity and calculate the mean squared displacement for all cases. In particular, we introduce and study generalized diffusion-wave equations with a regularized Prabhakar derivative of single and distributed orders. The equations considered can be used for modeling the broad spectrum of anomalous diffusion processes and various transitions between different diffusion regimes. KW - diffusion-wave equation KW - Mittag-Leffler function KW - anomalous diffusion Y1 - 2018 U6 - https://doi.org/10.1088/1751-8121/aaefa3 SN - 1751-8113 SN - 1751-8121 VL - 52 IS - 1 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Sandev, Trifce A1 - Sokolov, Igor M. A1 - Metzler, Ralf A1 - Chechkin, Aleksei V. T1 - Beyond monofractional kinetics JF - Chaos, solitons & fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science N2 - We discuss generalized integro-differential diffusion equations whose integral kernels are not of a simple power law form, and thus these equations themselves do not belong to the family of fractional diffusion equations exhibiting a monoscaling behavior. They instead generate a broad class of anomalous nonscaling patterns, which correspond either to crossovers between different power laws, or to a non-power-law behavior as exemplified by the logarithmic growth of the width of the distribution. We consider normal and modified forms of these generalized diffusion equations and provide a brief discussion of three generic types of integral kernels for each form, namely, distributed order, truncated power law and truncated distributed order kernels. For each of the cases considered we prove the non-negativity of the solution of the corresponding generalized diffusion equation and calculate the mean squared displacement. (C) 2017 Elsevier Ltd. All rights reserved. KW - Distributed order diffusion-wave equations KW - Complete Bernstein function KW - Completely monotone function Y1 - 2017 U6 - https://doi.org/10.1016/j.chaos.2017.05.001 SN - 0960-0779 SN - 1873-2887 VL - 102 SP - 210 EP - 217 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Sandev, Trifce A1 - Chechkin, Aleksei V. A1 - Kantz, Holger A1 - Metzler, Ralf T1 - Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck-Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed. KW - continuous time random walk (CTRW) KW - Fokker-Planck-Smoluchowski equation KW - Mittag-Leffler functions KW - anomalous diffusion KW - multi-scaling Y1 - 2015 U6 - https://doi.org/10.1515/fca-2015-0059 SN - 1311-0454 SN - 1314-2224 VL - 18 IS - 4 SP - 1006 EP - 1038 PB - De Gruyter CY - Berlin ER - TY - JOUR A1 - Sandev, Trifce A1 - Metzler, Ralf A1 - Chechkin, Aleksei V. T1 - From continuous time random walks to the generalized diffusion equation JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases. KW - continuous time random walk (CTRW) KW - generalized diffusion equation KW - Mittag-Leffler functions KW - anomalous diffusion Y1 - 2018 U6 - https://doi.org/10.1515/fca-2018-0002 SN - 1311-0454 SN - 1314-2224 VL - 21 IS - 1 SP - 10 EP - 28 PB - De Gruyter CY - Berlin ER - TY - JOUR A1 - Sandev, Trifce A1 - Iomin, Alexander A1 - Kantz, Holger A1 - Metzler, Ralf A1 - Chechkin, Aleksei V. T1 - Comb Model with Slow and Ultraslow Diffusion JF - Mathematical modelling of natural phenomena N2 - We consider a generalized diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyze the probability distribution functions and we derive the mean squared displacement in x and y directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both x and y directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process. KW - comb-like model KW - anomalous diffusion KW - mean squared displacement KW - probability density function Y1 - 2016 U6 - https://doi.org/10.1051/mmnp/201611302 SN - 0973-5348 SN - 1760-6101 VL - 11 SP - 18 EP - 33 PB - EDP Sciences CY - Les Ulis ER - TY - JOUR A1 - Padash, Amin A1 - Aghion, Erez A1 - Schulz, Alexander A1 - Barkai, Eli A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf A1 - Kantz, Holger T1 - Local equilibrium properties of ultraslow diffusion in the Sinai model JF - New journal of physics N2 - We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a finite system of growing size. This is due to time scale separation: inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top 10(17) steps and thereby also study finite-time crossover phenomena. KW - Sinai diffusion KW - clustering KW - local equilibrium Y1 - 2022 U6 - https://doi.org/10.1088/1367-2630/ac7df8 SN - 1367-2630 VL - 24 IS - 7 PB - IOP Publishing CY - Bristol ER -