@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{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{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{SandevTomovskiDubbeldametal.2018,
  author    = {Sandev, Trifce and Tomovski, Zivorad and Dubbeldam, Johan L. A. and Chechkin, Aleksei V.},
  title     = {Generalized diffusion-wave equation with memory kernel},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {52},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {1},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8121/aaefa3},
  pages     = {22},
  year      = {2018},
  abstract  = {We study generalized diffusion-wave equation in which the second order time derivative is replaced by an integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular cases. We consider different memory kernels of the integro-differential operator, derive corresponding fundamental solutions, specify the conditions of their non-negativity and calculate the mean squared displacement for all cases. In particular, we introduce and study generalized diffusion-wave equations with a regularized Prabhakar derivative of single and distributed orders. The equations considered can be used for modeling the broad spectrum of anomalous diffusion processes and various transitions between different diffusion regimes.},
  language  = {en}
}