@article{ThapaWyłomańskaSikoraetal.2021,
  author    = {Thapa, Samudrajit and Wyłomańska, Agnieszka and Sikora, Grzegorz and Wagner, Caroline E. and Krapf, Diego and Kantz, Holger and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories},
  series = {New Journal of Physics},
  volume    = {23},
  journal   = {New Journal of Physics},
  publisher = {Dt. Physikalische Ges. ; IOP},
  address   = {Bad Honnef ; London},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/abd50e},
  pages     = {22},
  year      = {2021},
  abstract  = {Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations.},
  language  = {en}
}
@article{CherstvyChechkinMetzler2014,
  author    = {Cherstvy, Andrey G. and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity},
  series = {Soft matter},
  volume    = {2014},
  journal   = {Soft matter},
  number    = {10},
  publisher = {Royal Society of Chemistry},
  issn      = {2046-2069},
  doi       = {10.1039/c3sm52846d},
  pages     = {1591 -- 1601},
  year      = {2014},
  abstract  = {We study the thermal Markovian diffusion of tracer particles in a 2D medium with spatially varying diffusivity D(r), mimicking recently measured, heterogeneous maps of the apparent diffusion coefficient in biological cells. For this heterogeneous diffusion process (HDP) we analyse the mean squared displacement (MSD) of the tracer particles, the time averaged MSD, the spatial probability density function, and the first passage time dynamics from the cell boundary to the nucleus. Moreover we examine the non-ergodic properties of this process which are important for the correct physical interpretation of time averages of observables obtained from single particle tracking experiments. From extensive computer simulations of the 2D stochastic Langevin equation we present an in-depth study of this HDP. In particular, we find that the MSDs along the radial and azimuthal directions in a circular domain obey anomalous and Brownian scaling, respectively. We demonstrate that the time averaged MSD stays linear as a function of the lag time and the system thus reveals a weak ergodicity breaking. Our results will enable one to rationalise the diffusive motion of larger tracer particles such as viruses or submicron beads in biological cells.},
  language  = {en}
}
@misc{CherstvyChechkinMetzler2014,
  author    = {Cherstvy, Andrey G. and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity},
  number    = {168},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-74021},
  pages     = {1591 -- 1601},
  year      = {2014},
  abstract  = {We study the thermal Markovian diffusion of tracer particles in a 2D medium with spatially varying diffusivity D(r), mimicking recently measured, heterogeneous maps of the apparent diffusion coefficient in biological cells. For this heterogeneous diffusion process (HDP) we analyse the mean squared displacement (MSD) of the tracer particles, the time averaged MSD, the spatial probability density function, and the first passage time dynamics from the cell boundary to the nucleus. Moreover we examine the non-ergodic properties of this process which are important for the correct physical interpretation of time averages of observables obtained from single particle tracking experiments. From extensive computer simulations of the 2D stochastic Langevin equation we present an in-depth study of this HDP. In particular, we find that the MSDs along the radial and azimuthal directions in a circular domain obey anomalous and Brownian scaling, respectively. We demonstrate that the time averaged MSD stays linear as a function of the lag time and the system thus reveals a weak ergodicity breaking. Our results will enable one to rationalise the diffusive motion of larger tracer particles such as viruses or submicron beads in biological cells.},
  language  = {en}
}
@article{WangCherstvyChechkinetal.2020,
  author    = {Wang, Wei and Cherstvy, Andrey G. and Chechkin, Aleksei V. and Thapa, Samudrajit and Seno, Flavio and Liu, Xianbin and Metzler, Ralf},
  title     = {Fractional Brownian motion with random diffusivity},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {53},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {47},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8121/aba467},
  pages     = {34},
  year      = {2020},
  abstract  = {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.},
  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{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{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}
}
@misc{ThapaWyłomańskaSikoraetal.2021,
  author    = {Thapa, Samudrajit and Wyłomańska, Agnieszka and Sikora, Grzegorz and Wagner, Caroline E. and Krapf, Diego and Kantz, Holger and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {1118},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-49349},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-493494},
  pages     = {24},
  year      = {2021},
  abstract  = {Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations.},
  language  = {en}
}
@article{GuggenbergerChechkinMetzler2021,
  author    = {Guggenberger, Tobias and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {54},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {29},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8121/ac019b},
  pages     = {17},
  year      = {2021},
  abstract  = {We study the stochastic motion of particles driven by long-range correlated fractional Gaussian noise (FGN) in a superharmonic external potential of the form U(x) proportional to x(2n) (n is an element of N). When the noise is considered to be external, the resulting overdamped motion is described by the non-Markovian Langevin equation for fractional Brownian motion. For this case we show the existence of long time, stationary probability density functions (PDFs) the shape of which strongly deviates from the naively expected Boltzmann PDF in the confining potential U(x). We analyse in detail the temporal approach to stationarity as well as the shape of the non-Boltzmann stationary PDF. A typical characteristic is that subdiffusive, antipersistent (with negative autocorrelation) motion tends to effect an accumulation of probability close to the origin as compared to the corresponding Boltzmann distribution while the opposite trend occurs for superdiffusive (persistent) motion. For this latter case this leads to distinct bimodal shapes of the PDF. This property is compared to a similar phenomenon observed for Markovian Levy flights in superharmonic potentials. We also demonstrate that the motion encoded in the fractional Langevin equation driven by FGN always relaxes to the Boltzmann distribution, as in this case the fluctuation-dissipation theorem is fulfilled.},
  language  = {en}
}
@article{ChechkinZaidLomholtetal.2013,
  author    = {Chechkin, Aleksei V. and Zaid, I. M. and Lomholt, M. A. and Sokolov, Igor M. and Metzler, Ralf},
  title     = {Bulk-mediated surface diffusion on a cylinder in the fast exchange limit},
  series = {Mathematical modelling of natural phenomena},
  volume    = {8},
  journal   = {Mathematical modelling of natural phenomena},
  number    = {2},
  publisher = {EDP Sciences},
  address   = {Les Ulis},
  issn      = {0973-5348},
  doi       = {10.1051/mmnp/20138208},
  pages     = {114 -- 126},
  year      = {2013},
  abstract  = {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.},
  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{ChechkinZaidLomholtetal.2013,
  author    = {Chechkin, Aleksei V. and Zaid, Irwin M. and Lomholt, Michael A. and Sokolov, Igor M. and Metzler, Ralf},
  title     = {Bulk-mediated surface diffusion on a cylinder in the fast exchange limit},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  number    = {593},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-41548},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-415480},
  pages     = {114 -- 126},
  year      = {2013},
  abstract  = {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.},
  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{MolinaGarciaSandevSafdarietal.2018,
  author    = {Molina-Garcia, Daniel and Sandev, Trifce and Safdari, Hadiseh and Pagnini, Gianni and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Crossover from anomalous to normal diffusion},
  series = {New Journal of Physics},
  volume    = {20},
  journal   = {New Journal of Physics},
  publisher = {IOP Publishing Ltd},
  address   = {London und Bad Honnef},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/aae4b2},
  pages     = {28},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}
@misc{MolinaGarciaSandevSafdarietal.2019,
  author    = {Molina-Garcia, Daniel and Sandev, Trifce and Safdari, Hadiseh and Pagnini, Gianni and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Crossover from anomalous to normal diffusion},
  series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe},
  number    = {507},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-42259},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-422590},
  pages     = {28},
  year      = {2019},
  abstract  = {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.},
  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{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{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}
}