@misc{JeonChechkinMetzler2014, author = {Jeon, Jae-Hyung and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-76302}, pages = {15811 -- 15817}, year = {2014}, abstract = {Anomalous diffusion is frequently described by scaled Brownian motion (SBM){,} a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is ?x2(t)? [similar{,} equals] 2K(t)t with K(t) [similar{,} equals] t[small alpha]-1 for 0 < [small alpha] < 2. SBM may provide a seemingly adequate description in the case of unbounded diffusion{,} for which its probability density function coincides with that of fractional Brownian motion. Here we show that free SBM is weakly non-ergodic but does not exhibit a significant amplitude scatter of the time averaged mean squared displacement. More severely{,} we demonstrate that under confinement{,} the dynamics encoded by SBM is fundamentally different from both fractional Brownian motion and continuous time random walks. SBM is highly non-stationary and cannot provide a physical description for particles in a thermalised stationary system. Our findings have direct impact on the modelling of single particle tracking experiments{,} in particular{,} under confinement inside cellular compartments or when optical tweezers tracking methods are used.}, language = {en} } @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} } @article{BurneckiWylomanskaBeletskiietal.2012, author = {Burnecki, Krzysztof and Wylomanska, Agnieszka and Beletskii, Aleksei and Gonchar, Vsevolod and Chechkin, Aleksei V.}, title = {Recognition of stable distribution with levy index alpha close to 2}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {85}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {5}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.85.056711}, pages = {8}, year = {2012}, abstract = {We address the problem of recognizing alpha-stable Levy distribution with Levy index close to 2 from experimental data. We are interested in the case when the sample size of available data is not large, thus the power law asymptotics of the distribution is not clearly detectable, and the shape of the empirical probability density function is close to a Gaussian. We propose a testing procedure combining a simple visual test based on empirical fourth moment with the Anderson-Darling and Jarque-Bera statistical tests and we check the efficiency of the method on simulated data. Furthermore, we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator-type fusion device and demonstrate that the phenomenon of the L-H transition from low confinement, L mode, to a high confinement, H mode, which occurs in this device is accompanied by the transition from Levy to Gaussian fluctuation statistics.}, language = {en} } @article{ChechkinLenzKlages2012, author = {Chechkin, Aleksei V. and Lenz, F. and Klages, Rainer}, title = {Normal and anomalous fluctuation relations for gaussian stochastic dynamics}, series = {Journal of statistical mechanics: theory and experiment}, journal = {Journal of statistical mechanics: theory and experiment}, number = {4}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1742-5468}, doi = {10.1088/1742-5468/2012/11/L11001}, pages = {12}, year = {2012}, abstract = {We study transient work fluctuation relations (FRs) for Gaussian stochastic systems generating anomalous diffusion. For this purpose we use a Langevin approach by employing two different types of additive noise: (i) internal noise where the fluctuation dissipation relation of the second kind (FDR II) holds, and (ii) external noise without FDR II. For internal noise we demonstrate that the existence of FDR II implies the existence of the fluctuation dissipation relation of the first kind (FDR I), which in turn leads to conventional (normal) forms of transient work FRs. For systems driven by external noise we obtain violations of normal FRs, which we call anomalous FRs. We derive them in the long-time limit and demonstrate the existence of logarithmic factors in FRs for intermediate times. We also outline possible experimental verifications.}, language = {en} } @article{ChechkinZaidLomholtetal.2012, author = {Chechkin, Aleksei V. and Zaid, Irwin M. and Lomholt, Michael A. and Sokolov, Igor M. and Metzler, Ralf}, title = {Bulk-mediated diffusion on a planar surface full solution}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {86}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.86.041101}, pages = {11}, year = {2012}, abstract = {We consider the effective surface motion of a particle that intermittently unbinds from a planar surface and performs bulk excursions. Based on a random-walk approach, we derive the diffusion equations for surface and bulk diffusion including the surface-bulk coupling. From these exact dynamic equations, we analytically obtain the propagator of the effective surface motion. This approach allows us to deduce a superdiffusive, Cauchy-type behavior on the surface, together with exact cutoffs limiting the Cauchy form. Moreover, we study the long-time dynamics for the surface motion.}, language = {en} }