@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{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}
}