@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}
}
@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{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}
}
@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}
}
@article{XuDengSandev2020,
  author    = {Xu, Pengbo and Deng, Weihua and Sandev, Trifce},
  title     = {Levy walk with parameter dependent velocity},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {53},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {11},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8121/ab7420},
  pages     = {26},
  year      = {2020},
  abstract  = {To analyze stochastic processes, one often uses integral transform (Fourier and Laplace) methods. However, for the time-space coupled cases, e.g. the Levy walk, sometimes the integral transform method may fail. Here we provide a Hermite polynomial expansion approach, being complementary to the integral transform method, to the Levy walk. Two approaches are compared for some already known results. We also consider the generalized Levy walk with parameter dependent velocity. Namely, we consider the Levy walk with velocity which depends on the walking length or on the duration of each step. Some interesting features of the generalized Levy walk are observed, including the special shapes of the probability density function, the first passage time distributions, and various diffusive behaviors of the mean squared displacement.},
  language  = {en}
}