@article{SchulzChechkinMetzler2013,
  author    = {Schulz, Johannes H. P. and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Correlated continuous time random walks - combining scale-invariance with long-range memory for spatial and temporal dynamics},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {46},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {47},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8113/46/47/475001},
  pages     = {22},
  year      = {2013},
  abstract  = {Standard continuous time random walk (CTRW) models are renewal processes in the sense that at each jump a new, independent pair of jump length and waiting time are chosen. Globally, anomalous diffusion emerges through scale-free forms of the jump length and/or waiting time distributions by virtue of the generalized central limit theorem. Here we present a modified version of recently proposed correlated CTRW processes, where we incorporate a power-law correlated noise on the level of both jump length and waiting time dynamics. We obtain a very general stochastic model, that encompasses key features of several paradigmatic models of anomalous diffusion: discontinuous, scale-free displacements as in Levy flights, scale-free waiting times as in subdiffusive CTRWs, and the long-range temporal correlations of fractional Brownian motion (FBM). We derive the exact solutions for the single-time probability density functions and extract the scaling behaviours. Interestingly, we find that different combinations of the model parameters lead to indistinguishable shapes of the emerging probability density functions and identical scaling laws. Our model will be useful for describing recent experimental single particle tracking data that feature a combination of CTRW and FBM properties.},
  language  = {en}
}
@article{CherstvyChechkinMetzler2013,
  author    = {Cherstvy, Andrey G. and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {15},
  journal   = {New journal of physics : the open-access journal for physics},
  number    = {15},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/15/8/083039},
  pages     = {13},
  year      = {2013},
  abstract  = {We demonstrate the non-ergodicity of a simple Markovian stochastic process with space-dependent diffusion coefficient D(x). For power-law forms D(x) similar or equal to vertical bar x vertical bar(alpha), this process yields anomalous diffusion of the form < x(2)(t)> similar or equal to t(2/(2-alpha)). Interestingly, in both the sub- and superdiffusive regimes we observe weak ergodicity breaking: the scaling of the time-averaged mean-squared displacement <(delta(2)(Delta))over bar> remains linear in the lag time Delta and thus differs from the corresponding ensemble average < x(2)(t)>. We analyse the non-ergodic behaviour of this process in terms of the time-averaged mean- squared displacement (delta(2)) over bar and its random features, i.e. the statistical distribution of (delta(2)) over bar and the ergodicity breaking parameters. The heterogeneous diffusion model represents an alternative approach to non- ergodic, anomalous diffusion that might be particularly relevant for diffusion in heterogeneous media.},
  language  = {en}
}
@article{SliusarenkoSurkovGoncharetal.2013,
  author    = {Sliusarenko, O. Yu. and Surkov, D. A. and Gonchar, V. Yu. and Chechkin, Aleksei V.},
  title     = {Stationary states in bistable system driven by Levy noise},
  series = {European physical journal special topics},
  volume    = {216},
  journal   = {European physical journal special topics},
  number    = {1},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {1951-6355},
  doi       = {10.1140/epjst/e2013-01736-0},
  pages     = {133 -- 138},
  year      = {2013},
  abstract  = {We study the properties of the probability density function (PDF) of a bistable system driven by heavy tailed white symmetric L,vy noise. The shape of the stationary PDF is found analytically for the particular case of the L,vy index alpha = 1 (Cauchy noise). For an arbitrary L,vy index we employ numerical methods based on the solution of the stochastic Langevin equation and space fractional kinetic equation. In contrast to the bistable system driven by Gaussian noise, in the L,vy case, the positions of maxima of the stationary PDF do not coincide with the positions of minima of the bistable potential. We provide a detailed study of the distance between the maxima and the minima as a function of the depth of the potential and the L,vy noise parameters.},
  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}
}
@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}
}