@article{MetzlerJeonCherstvyetal.2014,
  author    = {Metzler, Ralf and Jeon, Jae-Hyung and Cherstvy, Andrey G. and Barkai, Eli},
  title     = {Anomalous diffusion models and their properties},
  series = {physical chemistry, chemical physics : PCCP},
  volume    = {2014},
  journal   = {physical chemistry, chemical physics : PCCP},
  number    = {16},
  issn      = {1463-9076},
  doi       = {10.1039/c4cp03465a},
  pages     = {24128 -- 24164},
  year      = {2014},
  abstract  = {Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.},
  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{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}
}