@article{AwadMetzler2020,
  author    = {Awad, Emad and Metzler, Ralf},
  title     = {Crossover dynamics from superdiffusion to subdiffusion},
  series = {Fractional calculus and applied analysis : an international journal for theory and applications},
  volume    = {23},
  journal   = {Fractional calculus and applied analysis : an international journal for theory and applications},
  number    = {1},
  publisher = {De Gruyter},
  address   = {Berlin ; Boston},
  issn      = {1311-0454},
  doi       = {10.1515/fca-2020-0003},
  pages     = {55 -- 102},
  year      = {2020},
  abstract  = {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.},
  language  = {en}
}
@article{WangSenoSokolovetal.2020,
  author    = {Wang, Wei and Seno, Flavio and Sokolov, Igor M. and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Unexpected crossovers in correlated random-diffusivity processes},
  series = {New Journal of Physics},
  volume    = {22},
  journal   = {New Journal of Physics},
  publisher = {Dt. Physikalische Ges.},
  address   = {Bad Honnef},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/aba390},
  pages     = {17},
  year      = {2020},
  abstract  = {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.},
  language  = {en}
}
@misc{WangSenoSokolovetal.2020,
  author    = {Wang, Wei and Seno, Flavio and Sokolov, Igor M. and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Unexpected crossovers in correlated random-diffusivity processes},
  number    = {1006},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-48004},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-480049},
  pages     = {18},
  year      = {2020},
  abstract  = {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.},
  language  = {en}
}
@misc{SposiniMetzlerOshanin2019,
  author    = {Sposini, Vittoria and Metzler, Ralf and Oshanin, Gleb},
  title     = {Single-trajectory spectral analysis of scaled Brownian motion},
  series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  number    = {753},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-43652},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436522},
  pages     = {16},
  year      = {2019},
  abstract  = {Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.},
  language  = {en}
}
@misc{GuggenbergerPagniniVojtaetal.2019,
  author    = {Guggenberger, Tobias and Pagnini, Gianni and Vojta, Thomas and Metzler, Ralf},
  title     = {Fractional Brownian motion in a finite interval},
  series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  number    = {755},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-43666},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436665},
  pages     = {13},
  year      = {2019},
  abstract  = {Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) 〈X² (t)〉 ⋍ tᵅ with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.},
  language  = {en}
}
@article{GuggenbergerPagniniVojtaetal.2019,
  author    = {Guggenberger, Tobias and Pagnini, Gianni and Vojta, Thomas and Metzler, Ralf},
  title     = {Fractional Brownian motion in a finite interval},
  series = {New Journal of Physics},
  volume    = {21},
  journal   = {New Journal of Physics},
  publisher = {Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics},
  address   = {Bad Honnef und London},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ab075f},
  pages     = {13},
  year      = {2019},
  abstract  = {Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) 〈X² (t)〉 ⋍ tᵅ with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.},
  language  = {en}
}
@article{SposiniMetzlerOshanin2019,
  author    = {Sposini, Vittoria and Metzler, Ralf and Oshanin, Gleb},
  title     = {Single-trajectory spectral analysis of scaled Brownian motion},
  series = {New Journal of Physics},
  volume    = {21},
  journal   = {New Journal of Physics},
  publisher = {Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics},
  address   = {Bad Honnef und London},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ab2f52},
  pages     = {16},
  year      = {2019},
  abstract  = {Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.},
  language  = {en}
}
@misc{ŚlęzakMetzlerMagdziarz2018,
  author    = {Ślęzak, Jakub and Metzler, Ralf and Magdziarz, Marcin},
  title     = {Superstatistical generalised Langevin equation},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-409315},
  pages     = {25},
  year      = {2018},
  abstract  = {Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.},
  language  = {en}
}
@article{ŚlęzakMetzlerMagdziarz2018,
  author    = {Ślęzak, Jakub and Metzler, Ralf and Magdziarz, Marcin},
  title     = {Superstatistical generalised Langevin equation},
  series = {New Journal of Physics},
  volume    = {20},
  journal   = {New Journal of Physics},
  number    = {023026},
  publisher = {Deutsche Physikalische Gesellschaft / Institute of Physics},
  address   = {Bad Honnef und London},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/aaa3d4},
  pages     = {1 -- 25},
  year      = {2018},
  abstract  = {Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.},
  language  = {en}
}
@article{SandevMetzlerChechkin2018,
  author    = {Sandev, Trifce and Metzler, Ralf and Chechkin, Aleksei V.},
  title     = {From continuous time random walks to the generalized diffusion equation},
  series = {Fractional calculus and applied analysis : an international journal for theory and applications},
  volume    = {21},
  journal   = {Fractional calculus and applied analysis : an international journal for theory and applications},
  number    = {1},
  publisher = {De Gruyter},
  address   = {Berlin},
  issn      = {1311-0454},
  doi       = {10.1515/fca-2018-0002},
  pages     = {10 -- 28},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}
@article{GhoshCherstvyGrebenkovetal.2016,
  author    = {Ghosh, Surya K. and Cherstvy, Andrey G. and Grebenkov, Denis S. and Metzler, Ralf},
  title     = {Anomalous, non-Gaussian tracer diffusion in crowded two-dimensional environments},
  series = {NEW JOURNAL OF PHYSICS},
  volume    = {18},
  journal   = {NEW JOURNAL OF PHYSICS},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/18/1/013027},
  pages     = {16},
  year      = {2016},
  abstract  = {A topic of intense current investigation pursues the question of how the highly crowded environment of biological cells affects the dynamic properties of passively diffusing particles. Motivated by recent experiments we report results of extensive simulations of the motion of a finite sized tracer particle in a heterogeneously crowded environment made up of quenched distributions of monodisperse crowders of varying sizes in finite circular two-dimensional domains. For given spatial distributions of monodisperse crowders we demonstrate how anomalous diffusion with strongly non-Gaussian features arises in this model system. We investigate both biologically relevant situations of particles released either at the surface of an inner domain or at the outer boundary, exhibiting distinctly different features of the observed anomalous diffusion for heterogeneous distributions of crowders. Specifically we reveal an asymmetric spreading of tracers even at moderate crowding. In addition to the mean squared displacement (MSD) and local diffusion exponent we investigate the magnitude and the amplitude scatter of the time averaged MSD of individual tracer trajectories, the non-Gaussianity parameter, and the van Hove correlation function. We also quantify how the average tracer diffusivity varies with the position in the domain with a heterogeneous radial distribution of crowders and examine the behaviour of the survival probability and the dynamics of the tracer survival probability. Inter alia, the systems we investigate are related to the passive transport of lipid molecules and proteins in two-dimensional crowded membranes or the motion in colloidal solutions or emulsions in effectively two-dimensional geometries, as well as inside supercrowded, surface adhered cells.},
  language  = {en}
}
@article{SandevIominKantzetal.2016,
  author    = {Sandev, Trifce and Iomin, Alexander and Kantz, Holger and Metzler, Ralf and Chechkin, Aleksei V.},
  title     = {Comb Model with Slow and Ultraslow Diffusion},
  series = {Mathematical modelling of natural phenomena},
  volume    = {11},
  journal   = {Mathematical modelling of natural phenomena},
  publisher = {EDP Sciences},
  address   = {Les Ulis},
  issn      = {0973-5348},
  doi       = {10.1051/mmnp/201611302},
  pages     = {18 -- 33},
  year      = {2016},
  abstract  = {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.},
  language  = {en}
}
@article{BodrovaChechkinCherstvyetal.2015,
  author    = {Bodrova, Anna S. and Chechkin, Aleksei V. and Cherstvy, Andrey G. and Metzler, Ralf},
  title     = {Ultraslow scaled Brownian motion},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {17},
  journal   = {New journal of physics : the open-access journal for physics},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/17/6/063038},
  pages     = {16},
  year      = {2015},
  abstract  = {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.},
  language  = {en}
}
@article{KruesemannGodecMetzler2015,
  author    = {Kr{\"u}semann, Henning and Godec, Aljaz and Metzler, Ralf},
  title     = {Ageing first passage time density in continuous time random walks and quenched energy landscapes},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {48},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {28},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8113/48/28/285001},
  pages     = {20},
  year      = {2015},
  abstract  = {We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function Psi (t) similar or equal to t(-1-alpha) (0 <= alpha <= 2). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time t(a), the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs with 0 < alpha < 1 describe systems such as charge carriers in amorphous semiconducters, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range 1 < alpha < 2, for sufficiently long ta. We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the ageing CTRW results. With small bias or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape.},
  language  = {en}
}
@article{ShinCherstvyMetzler2015,
  author    = {Shin, Jaeoh and Cherstvy, Andrey G. and Metzler, Ralf},
  title     = {Self-subdiffusion in solutions of star-shaped crowders: non-monotonic effects of inter-particle interactions},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {17},
  journal   = {New journal of physics : the open-access journal for physics},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/17/11/113028},
  pages     = {12},
  year      = {2015},
  abstract  = {We examine by extensive computer simulations the self-diffusion of anisotropic star-like particles in crowded two-dimensional solutions. We investigate the implications of the area coverage fraction phi of the crowders and the crowder-crowder adhesion properties on the regime of transient anomalous diffusion. We systematically compute the mean squared displacement (MSD) of the particles, their time averaged MSD, and the effective diffusion coefficient. The diffusion is ergodic in the limit of long traces, such that the mean time averaged MSD converges towards the ensemble averaged MSD, and features a small residual amplitude spread of the time averaged MSD from individual trajectories. At intermediate time scales, we quantify the anomalous diffusion in the system. Also, we show that the translational-but not rotational-diffusivity of the particles Dis a nonmonotonic function of the attraction strength between them. Both diffusion coefficients decrease as the power law D(phi) similar to (1 - phi/phi*)(2 ... 2.4) with the area fraction phi occupied by the crowders and the critical value phi*. Our results might be applicable to rationalising the experimental observations of non-Brownian diffusion for a number of standard macromolecular crowders used in vitro to mimic the cytoplasmic conditions of living cells.},
  language  = {en}
}
@article{MetzlerCherstvyChechkinetal.2015,
  author    = {Metzler, Ralf and Cherstvy, Andrey G. and Chechkin, Aleksei V. and Bodrova, Anna S.},
  title     = {Ultraslow scaled Brownian motion},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {17},
  journal   = {New journal of physics : the open-access journal for physics},
  number    = {063038},
  publisher = {Dt. Physikalische Ges., IOP},
  address   = {Bad Honnef, London},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/17/6/063038},
  year      = {2015},
  abstract  = {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.},
  language  = {en}
}
@misc{MetzlerCherstvyChechkinetal.2015,
  author    = {Metzler, Ralf and Cherstvy, Andrey G. and Chechkin, Aleksei V. and Bodrova, Anna S.},
  title     = {Ultraslow scaled Brownian motion},
  series = {New journal of physics : the open-access journal for physics},
  journal   = {New journal of physics : the open-access journal for physics},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-78618},
  year      = {2015},
  abstract  = {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.},
  language  = {en}
}
@article{SafdariCherstvyChechkinetal.2015,
  author    = {Safdari, Hadiseh and Cherstvy, Andrey G. and Chechkin, Aleksei V. and Thiel, Felix and Sokolov, Igor M. and Metzler, Ralf},
  title     = {Quantifying the non-ergodicity of scaled Brownian motion},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {48},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {37},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8113/48/37/375002},
  pages     = {18},
  year      = {2015},
  abstract  = {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.},
  language  = {en}
}
@article{SandevChechkinKantzetal.2015,
  author    = {Sandev, Trifce and Chechkin, Aleksei V. and Kantz, Holger and Metzler, Ralf},
  title     = {Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel},
  series = {Fractional calculus and applied analysis : an international journal for theory and applications},
  volume    = {18},
  journal   = {Fractional calculus and applied analysis : an international journal for theory and applications},
  number    = {4},
  publisher = {De Gruyter},
  address   = {Berlin},
  issn      = {1311-0454},
  doi       = {10.1515/fca-2015-0059},
  pages     = {1006 -- 1038},
  year      = {2015},
  abstract  = {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.},
  language  = {en}
}
@article{PalyulinAlaNissilaMetzler2014,
  author    = {Palyulin, Vladimir V. and Ala-Nissila, Tapio and Metzler, Ralf},
  title     = {Polymer translocation: the first two decades and the recent diversification},
  series = {Soft matter},
  volume    = {45},
  journal   = {Soft matter},
  number    = {10},
  editor    = {Metzler, Ralf},
  publisher = {the Royal Society of Chemistry},
  address   = {Cambridge},
  issn      = {1744-683X},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-76266},
  pages     = {9016 -- 9037},
  year      = {2014},
  abstract  = {Probably no other field of statistical physics at the borderline of soft matter and biological physics has caused such a flurry of papers as polymer translocation since the 1994 landmark paper by Bezrukov, Vodyanoy, and Parsegian and the study of Kasianowicz in 1996. Experiments, simulations, and theoretical approaches are still contributing novel insights to date, while no universal consensus on the statistical understanding of polymer translocation has been reached. We here collect the published results, in particular, the famous-infamous debate on the scaling exponents governing the translocation process. We put these results into perspective and discuss where the field is going. In particular, we argue that the phenomenon of polymer translocation is non-universal and highly sensitive to the exact specifications of the models and experiments used towards its analysis.},
  language  = {en}
}
@misc{PalyulinAlaNissilaMetzler2014,
  author    = {Palyulin, Vladimir V. and Ala-Nissila, Tapio and Metzler, Ralf},
  title     = {Polymer translocation: the first two decades and the recent diversification},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-76287},
  pages     = {9016 -- 9037},
  year      = {2014},
  abstract  = {Probably no other field of statistical physics at the borderline of soft matter and biological physics has caused such a flurry of papers as polymer translocation since the 1994 landmark paper by Bezrukov, Vodyanoy, and Parsegian and the study of Kasianowicz in 1996. Experiments, simulations, and theoretical approaches are still contributing novel insights to date, while no universal consensus on the statistical understanding of polymer translocation has been reached. We here collect the published results, in particular, the famous-infamous debate on the scaling exponents governing the translocation process. We put these results into perspective and discuss where the field is going. In particular, we argue that the phenomenon of polymer translocation is non-universal and highly sensitive to the exact specifications of the models and experiments used towards its analysis.},
  language  = {en}
}
@article{GodecBauerMetzler2014,
  author    = {Godec, Aljaz and Bauer, Maximilian and Metzler, Ralf},
  title     = {Collective dynamics effect transient subdiffusion of inert tracers in flexible gel networks},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {16},
  journal   = {New journal of physics : the open-access journal for physics},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/16/9/092002},
  pages     = {13},
  year      = {2014},
  abstract  = {Based on extensive Brownian dynamics simulations we study the thermal motion of a tracer bead in a cross-linked, flexible gel in the limit when the tracer particle size is comparable to or even larger than the equilibrium mesh size of the gel. The analysis of long individual trajectories of the tracer demonstrates the existence of pronounced transient anomalous diffusion. From the time averaged mean squared displacement and the time averaged van Hove correlation functions we elucidate the many-body origin of the non-Brownian tracer bead dynamics. Our results shed new light onto the ongoing debate over the physical origin of steric tracer interactions with structured environments.},
  language  = {en}
}
@article{PalyulinMetzler2014,
  author    = {Palyulin, Vladimir V. and Metzler, Ralf},
  title     = {Speeding up the first-passage for subdiffusion by introducing a finite potential barrier},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {47},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {3},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8113/47/3/032002},
  pages     = {13},
  year      = {2014},
  abstract  = {We show that for a subdiffusive continuous time random walk with scale-free waiting time distribution the first-passage dynamics on a finite interval can be optimized by introduction of a piecewise linear potential barrier. Analytical results for the survival probability and first-passage density based on the fractional Fokker-Planck equation are shown to agree well with Monte Carlo simulations results. As an application we discuss an improved design for efficient translocation of gradient copolymers compared to homopolymer translocation in a quasi-equilibrium approximation.},
  language  = {en}
}
@article{GodecChechkinBarkaietal.2014,
  author    = {Godec, Aljaz and Chechkin, Aleksei V. and Barkai, Eli and Kantz, Holger and Metzler, Ralf},
  title     = {Localisation and universal fluctuations in ultraslow diffusion processes},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {47},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {49},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8113/47/49/492002},
  pages     = {10},
  year      = {2014},
  abstract  = {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.},
  language  = {en}
}
@article{CherstvyChechkinMetzler2014,
  author    = {Cherstvy, Andrey G. and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Ageing and confinement in non-ergodic heterogeneous diffusion processes},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {47},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {48},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8113/47/48/485002},
  pages     = {18},
  year      = {2014},
  abstract  = {We study the effects of ageing-the time delay between initiation of the physical process at t = 0 and start of observation at some time t(a) > 0-and spatial confinement on the properties of heterogeneous diffusion processes (HDPs) with deterministic power-law space-dependent diffusivities, D(x) = D-0 vertical bar x vertical bar(alpha). From analysis of the ensemble and time averaged mean squared displacements and the ergodicity breaking parameter quantifying the inherent degree of irreproducibility of individual realizations of the HDP we obtain striking similarities to ageing subdiffusive continuous time random walks with scale-free waiting time distributions. We also explore how both processes can be distinguished. For confined HDPs we study the long-time saturation of the ensemble and time averaged particle displacements as well as the magnitude of the inherent scatter of time averaged displacements and contrast the outcomes to the results known for other anomalous diffusion processes under confinement.},
  language  = {en}
}
@article{BauerGodecMetzler2014,
  author    = {Bauer, Maximilian and Godec, Aljaž and Metzler, Ralf},
  title     = {Diffusion of finite-size particles in two-dimensional channels with random wall configurations},
  series = {Physical chemistry, chemical physics : PCCP ; a journal of European chemical societies},
  volume    = {16},
  journal   = {Physical chemistry, chemical physics : PCCP ; a journal of European chemical societies},
  number    = {13},
  publisher = {RSC Publications},
  address   = {Cambridge},
  issn      = {1463-9084},
  doi       = {10.1039/C3CP55160A},
  pages     = {6118 -- 6128},
  year      = {2014},
  abstract  = {Diffusion of chemicals or tracer molecules through complex systems containing irregularly shaped channels is important in many applications. Most theoretical studies based on the famed Fick-Jacobs equation focus on the idealised case of infinitely small particles and reflecting boundaries. In this study we use numerical simulations to consider the transport of finite-size particles through asymmetrical two-dimensional channels. Additionally, we examine transient binding of the molecules to the channel walls by applying sticky boundary conditions. We consider an ensemble of particles diffusing in independent channels, which are characterised by common structural parameters. We compare our results for the long-time effective diffusion coefficient with a recent theoretical formula obtained by Dagdug and Pineda [J. Chem. Phys., 2012, 137, 024107].},
  language  = {en}
}
@misc{BauerGodecMetzler2014,
  author    = {Bauer, Maximilian and Godec, Aljaž and Metzler, Ralf},
  title     = {Diffusion of finite-size particles in two-dimensional channels with random wall configurations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-76199},
  year      = {2014},
  abstract  = {Diffusion of chemicals or tracer molecules through complex systems containing irregularly shaped channels is important in many applications. Most theoretical studies based on the famed Fick-Jacobs equation focus on the idealised case of infinitely small particles and reflecting boundaries. In this study we use numerical simulations to consider the transport of finite-size particles through asymmetrical two-dimensional channels. Additionally, we examine transient binding of the molecules to the channel walls by applying sticky boundary conditions. We consider an ensemble of particles diffusing in independent channels, which are characterised by common structural parameters. We compare our results for the long-time effective diffusion coefficient with a recent theoretical formula obtained by Dagdug and Pineda [J. Chem. Phys., 2012, 137, 024107].},
  language  = {en}
}
@article{SandevMetzlerTomovski2012,
  author    = {Sandev, Trifce and Metzler, Ralf and Tomovski, Zivorad},
  title     = {Velocity and displacement correlation functions for fractional generalized Langevin equations},
  series = {Fractional calculus and applied analysis : an international journal for theory and applications},
  volume    = {15},
  journal   = {Fractional calculus and applied analysis : an international journal for theory and applications},
  number    = {3},
  publisher = {Versita},
  address   = {Warsaw},
  issn      = {1311-0454},
  doi       = {10.2478/s13540-012-0031-2},
  pages     = {426 -- 450},
  year      = {2012},
  abstract  = {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.},
  language  = {en}
}