TY - JOUR A1 - Sliusarenko, O. Yu. A1 - Surkov, D. A. A1 - Gonchar, V. Yu. A1 - Chechkin, Aleksei V. T1 - Stationary states in bistable system driven by Levy noise JF - European physical journal special topics N2 - 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. Y1 - 2013 U6 - https://doi.org/10.1140/epjst/e2013-01736-0 SN - 1951-6355 VL - 216 IS - 1 SP - 133 EP - 138 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Schulz, Johannes H. P. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Correlated continuous time random walks - combining scale-invariance with long-range memory for spatial and temporal dynamics JF - Journal of physics : A, Mathematical and theoretical N2 - 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. Y1 - 2013 U6 - https://doi.org/10.1088/1751-8113/46/47/475001 SN - 1751-8113 SN - 1751-8121 VL - 46 IS - 47 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes JF - New journal of physics : the open-access journal for physics N2 - 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. Y1 - 2013 U6 - https://doi.org/10.1088/1367-2630/15/8/083039 SN - 1367-2630 VL - 15 IS - 15 PB - IOP Publ. Ltd. CY - Bristol ER - TY - GEN A1 - Chechkin, Aleksei V. A1 - Zaid, Irwin M. A1 - Lomholt, Michael A. A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Bulk-mediated surface diffusion on a cylinder in the fast exchange limit T2 - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 593 KW - Bulk-mediated diffusion; KW - anomalous diffusion KW - Levy flights KW - stochastic processes Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-415480 SN - 1866-8372 IS - 593 SP - 114 EP - 126 ER - TY - JOUR A1 - Chechkin, Aleksei V. A1 - Zaid, I. M. A1 - Lomholt, M. A. A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Bulk-mediated surface diffusion on a cylinder in the fast exchange limit JF - Mathematical modelling of natural phenomena N2 - 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. KW - Bulk-mediated diffusion KW - anomalous diffusion KW - Levy flights KW - stochastic processes Y1 - 2013 U6 - https://doi.org/10.1051/mmnp/20138208 SN - 0973-5348 VL - 8 IS - 2 SP - 114 EP - 126 PB - EDP Sciences CY - Les Ulis ER -