Dokument-ID Dokumenttyp Verfasser/Autoren Herausgeber Haupttitel Abstract Auflage Verlagsort Verlag Erscheinungsjahr Seitenzahl Schriftenreihe Titel Schriftenreihe Bandzahl ISBN Quelle der Hochschulschrift Konferenzname Quelle:Titel Quelle:Jahrgang Quelle:Heftnummer Quelle:Erste Seite Quelle:Letzte Seite URN DOI Abteilungen
OPUS4-37424 Wissenschaftlicher Artikel Palyulin, Vladimir V.; Chechkin, Aleksei V.; Metzler, Ralf Space-fractional Fokker-Planck equation and optimization of random search processes in the presence of an external bias Based on the space-fractional Fokker-Planck equation with a delta-sink term, we study the efficiency of random search processes based on Levy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the preference of the searcher based on prior experience. While Levy flights turn out to be efficient search processes when the target is upstream relative to the starting point, in the downstream scenario, regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Levy flights, due to which Levy flights typically overshoot a point or small interval. Studying the solution of the fractional Fokker-Planck equation, we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favours Levy flights over the regular Brownian search. Contrary to the common belief that Levy flights with a Levy index alpha = 1 (i.e. Cauchy flights) are optimal for sparse targets, we find that the optimal value for alpha may range in the entire interval (1, 2) and explicitly include Brownian motion as the most efficient search strategy overall. Bristol IOP Publ. Ltd. 2014 32 Journal of statistical mechanics: theory and experiment 10.1088/1742-5468/2014/11/P11031 Institut für Physik und Astronomie
OPUS4-34805 Wissenschaftlicher Artikel Cherstvy, Andrey G.; Chechkin, Aleksei V.; Metzler, Ralf Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes 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. Bristol IOP Publ. Ltd. 2013 13 New journal of physics : the open-access journal for physics 15 15 10.1088/1367-2630/15/8/083039 Institut für Physik und Astronomie
OPUS4-35532 Wissenschaftlicher Artikel Chechkin, Aleksei V.; Lenz, F.; Klages, Rainer Normal and anomalous fluctuation relations for gaussian stochastic dynamics 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. Bristol IOP Publ. Ltd. 2012 12 Journal of statistical mechanics: theory and experiment 4 10.1088/1742-5468/2012/11/L11001 Institut für Physik und Astronomie
OPUS4-38043 Wissenschaftlicher Artikel Palyulin, Vladimir V.; Chechkin, Aleksei V.; Metzler, Ralf Levy flights do not always optimize random blind search for sparse targets It is generally believed that random search processes based on scale-free, Levy stable jump length distributions (Levy flights) optimize the search for sparse targets. Here we show that this popular search advantage is less universal than commonly assumed. We study the efficiency of a minimalist search model based on Levy flights in the absence and presence of an external drift (underwater current, atmospheric wind, a preference of the walker owing to prior experience, or a general bias in an abstract search space) based on two different optimization criteria with respect to minimal search time and search reliability (cumulative arrival probability). Although Levy flights turn out to be efficient search processes when the target is far from the starting point, or when relative to the starting point the target is upstream, we show that for close targets and for downstream target positioning regular Brownian motion turns out to be the advantageous search strategy. Contrary to claims that Levy flights with a critical exponent alpha = 1 are optimal for the search of sparse targets in different settings, based on our optimization parameters the optimal a may range in the entire interval (1, 2) and especially include Brownian motion as the overall most efficient search strategy. Washington National Acad. of Sciences 2014 6 Proceedings of the National Academy of Sciences of the United States of America 111 8 2931 2936 10.1073/pnas.1320424111 Institut für Physik und Astronomie
OPUS4-35890 Wissenschaftlicher Artikel Burnecki, Krzysztof; Wylomanska, Agnieszka; Beletskii, Aleksei; Gonchar, Vsevolod; Chechkin, Aleksei V. Recognition of stable distribution with levy index alpha close to 2 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. College Park American Physical Society 2012 8 Physical review : E, Statistical, nonlinear and soft matter physics 85 5 10.1103/PhysRevE.85.056711 Institut für Physik und Astronomie
OPUS4-37307 Wissenschaftlicher Artikel Cherstvy, Andrey G.; Chechkin, Aleksei V.; Metzler, Ralf Ageing and confinement in non-ergodic heterogeneous diffusion processes We study the effects of ageing-the time delay between initiation of the physical process at t = 0 and start of observation at some time t(a) > 0-and spatial confinement on the properties of heterogeneous diffusion processes (HDPs) with deterministic power-law space-dependent diffusivities, D(x) = D-0 vertical bar x vertical bar(alpha). From analysis of the ensemble and time averaged mean squared displacements and the ergodicity breaking parameter quantifying the inherent degree of irreproducibility of individual realizations of the HDP we obtain striking similarities to ageing subdiffusive continuous time random walks with scale-free waiting time distributions. We also explore how both processes can be distinguished. For confined HDPs we study the long-time saturation of the ensemble and time averaged particle displacements as well as the magnitude of the inherent scatter of time averaged displacements and contrast the outcomes to the results known for other anomalous diffusion processes under confinement. Bristol IOP Publ. Ltd. 2014 18 Journal of physics : A, Mathematical and theoretical 47 48 10.1088/1751-8113/47/48/485002 Institut für Physik und Astronomie
OPUS4-37299 Wissenschaftlicher Artikel Godec, Aljaz; Chechkin, Aleksei V.; Barkai, Eli; Kantz, Holger; Metzler, Ralf Localisation and universal fluctuations in ultraslow diffusion processes We study ultraslow diffusion processes with logarithmic mean squared displacement (MSD) < x(2)(t)> similar or equal to log(gamma)t. Comparison of annealed (renewal) continuous time random walks (CTRWs) with logarithmic waiting time distribution psi(tau) similar or equal to 1/(tau log(1+gamma)tau) and Sinai diffusion in quenched random landscapes reveals striking similarities, despite the great differences in their physical nature. In particular, they exhibit a weakly non-ergodic disparity of the time-averaged and ensemble-averaged MSDs. Remarkably, for the CTRW we observe that the fluctuations of time averages become universal, with an exponential suppression of mobile trajectories. We discuss the fundamental connection between the Golosov localization effect and non-ergodicity in the sense of the disparity between ensemble-averaged MSD and time-averaged MSD. Bristol IOP Publ. Ltd. 2014 10 Journal of physics : A, Mathematical and theoretical 47 49 10.1088/1751-8113/47/49/492002 Institut für Physik und Astronomie
OPUS4-35401 Wissenschaftlicher Artikel Sliusarenko, O. Yu.; Surkov, D. A.; Gonchar, V. Yu.; Chechkin, Aleksei V. Stationary states in bistable system driven by Levy noise 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. Heidelberg Springer 2013 6 European physical journal special topics 216 1 133 138 10.1140/epjst/e2013-01736-0 Institut für Physik und Astronomie
OPUS4-35613 Wissenschaftlicher Artikel Chechkin, Aleksei V.; Zaid, Irwin M.; Lomholt, Michael A.; Sokolov, Igor M.; Metzler, Ralf Bulk-mediated diffusion on a planar surface full solution 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. College Park American Physical Society 2012 11 Physical review : E, Statistical, nonlinear and soft matter physics 86 4 10.1103/PhysRevE.86.041101 Institut für Physik und Astronomie
OPUS4-35363 Wissenschaftlicher Artikel Chechkin, Aleksei V.; Zaid, I. M.; Lomholt, M. A.; Sokolov, I. M.; Metzler, Ralf Bulk-mediated surface diffusion on a cylinder in the fast exchange limit 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. Les Ulis EDP Sciences 2013 13 Mathematical modelling of natural phenomena 8 2 114 126 10.1051/mmnp/20138208 Institut für Chemie
OPUS4-40053 Wissenschaftlicher Artikel Cherstvy, Andrey G.; Vinod, Deepak; Aghion, Erez; Chechkin, Aleksei V.; Metzler, Ralf Time averaging, ageing and delay analysis of financial time series We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black-Scholes-Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics. London IOP 2017 11 New journal of physics 19 1 11 10.1088/1367-2630/aa7199 Institut für Physik und Astronomie
OPUS4-40054 misc Cherstvy, Andrey G.; Vinod, Deepak; Aghion, Erez; Chechkin, Aleksei V.; Metzler, Ralf Time averaging, ageing and delay analysis of financial time series We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black-Scholes-Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics. 2017 11 urn:nbn:de:kobv:517-opus4-400541 Institut für Physik und Astronomie
OPUS4-38517 Wissenschaftlicher Artikel Sandev, Trifce; Chechkin, Aleksei V.; Korabel, Nickolay; Kantz, Holger; Sokolov, Igor M.; Metzler, Ralf Distributed-order diffusion equations and multifractality: Models and solutions We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided. College Park American Physical Society 2015 19 Physical review : E, Statistical, nonlinear and soft matter physics 92 4 10.1103/PhysRevE.92.042117 Institut für Physik und Astronomie
OPUS4-38154 Wissenschaftlicher Artikel Nezhadhaghighi, M. Ghasemi; Chechkin, Aleksei V.; Metzler, Ralf Numerical approach to unbiased and driven generalized elastic model From scaling arguments and numerical simulations, we investigate the properties of the generalized elastic model (GEM) that is used to describe various physical systems such as polymers, membranes, single-file systems, or rough interfaces. We compare analytical and numerical results for the subdiffusion exponent beta characterizing the growth of the mean squared displacement <(delta h)(2)> of the field h described by the GEM dynamic equation. We study the scaling properties of the qth order moments with time, finding that the interface fluctuations show no intermittent behavior. We also investigate the ergodic properties of the process h in terms of the ergodicity breaking parameter and the distribution of the time averaged mean squared displacement. Finally, we study numerically the driven GEM with a constant, localized perturbation and extract the characteristics of the average drift for a tagged probe. Melville American Institute of Physics 2014 9 The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr 140 2 10.1063/1.4858425 Institut für Physik und Astronomie
OPUS4-38319 Wissenschaftlicher Artikel Cherstvy, Andrey G.; Chechkin, Aleksei V.; Metzler, Ralf Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity We study the thermal Markovian diffusion of tracer particles in a 2D medium with spatially varying diffusivity D(r), mimicking recently measured, heterogeneous maps of the apparent diffusion coefficient in biological cells. For this heterogeneous diffusion process (HDP) we analyse the mean squared displacement (MSD) of the tracer particles, the time averaged MSD, the spatial probability density function, and the first passage time dynamics from the cell boundary to the nucleus. Moreover we examine the non-ergodic properties of this process which are important for the correct physical interpretation of time averages of observables obtained from single particle tracking experiments. From extensive computer simulations of the 2D stochastic Langevin equation we present an in-depth study of this HDP. In particular, we find that the MSDs along the radial and azimuthal directions in a circular domain obey anomalous and Brownian scaling, respectively. We demonstrate that the time averaged MSD stays linear as a function of the lag time and the system thus reveals a weak ergodicity breaking. Our results will enable one to rationalise the diffusive motion of larger tracer particles such as viruses or submicron beads in biological cells. Cambridge Royal Society of Chemistry 2014 11 Soft matter 10 10 1591 1601 10.1039/c3sm52846d Institut für Physik und Astronomie
OPUS4-38257 Wissenschaftlicher Artikel Jeon, Jae-Hyung; Chechkin, Aleksei V.; Metzler, Ralf Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion 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 < x(2)(t) similar or equal to 2K(t)t with K(t) similar or equal to t(alpha-1) for 0 < 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. Cambridge Royal Society of Chemistry 2014 7 Physical chemistry, chemical physics : a journal of European Chemical Societies 16 30 15811 15817 10.1039/c4cp02019g Institut für Chemie
OPUS4-38715 Wissenschaftlicher Artikel Sandev, Trifce; Chechkin, Aleksei V.; Kantz, Holger; Metzler, Ralf Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel 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. Berlin De Gruyter 2015 33 Fractional calculus and applied analysis : an international journal for theory and applications 18 4 1006 1038 10.1515/fca-2015-0059 Institut für Physik und Astronomie
OPUS4-38753 Wissenschaftlicher Artikel Dieterich, Peter; Klages, Rainer; Chechkin, Aleksei V. Fluctuation relations for anomalous dynamics generated by time-fractional Fokker-Planck equations Anomalous dynamics characterized by non-Gaussian probability distributions (PDFs) and/or temporal long-range correlations can cause subtle modifications of conventional fluctuation relations (FRs). As prototypes we study three variants of a generic time-fractional Fokker-Planck equation with constant force. Type A generates superdiffusion, type B subdiffusion and type C both super-and subdiffusion depending on parameter variation. Furthermore type C obeys a fluctuation-dissipation relation whereas A and B do not. We calculate analytically the position PDFs for all three cases and explore numerically their strongly non-Gaussian shapes. While for type C we obtain the conventional transient work FR, type A and type B both yield deviations by featuring a coefficient that depends on time and by a nonlinear dependence on the work. We discuss possible applications of these types of dynamics and FRs to experiments. Bristol IOP Publ. Ltd. 2015 14 New journal of physics : the open-access journal for physics 17 10.1088/1367-2630/17/7/075004 Institut für Physik und Astronomie
OPUS4-38374 Wissenschaftlicher Artikel Pavlyukevich, Ilya; Li, Yongge; Xu, Yong; Chechkin, Aleksei V. Directed transport induced by spatially modulated Levy flights In this paper we study the dynamics of a particle in a ratchet potential subject to multiplicative alpha-stable Levy noise, alpha is an element of(0, 2), in the limit of a noise amplitude epsilon -> 0. We compare the dynamics for Ito and Marcus multiplicative noises and obtain the explicit asymptotics of the escape time in the wells and transition probabilities between the wells. A detailed analysis of the noise-induced current is performed for the Seebeck ratchet with a weak multiplicative noise for alpha is an element of(0, 2]. Bristol IOP Publ. Ltd. 2015 21 Journal of physics : A, Mathematical and theoretical 48 49 10.1088/1751-8113/48/49/495004 Institut für Physik und Astronomie
OPUS4-38814 Wissenschaftlicher Artikel Bodrova, Anna S.; Chechkin, Aleksei V.; Cherstvy, Andrey G.; Metzler, Ralf Ultraslow scaled Brownian motion We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form D(t) similar or equal to 1/t. For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations. Bristol IOP Publ. Ltd. 2015 16 New journal of physics : the open-access journal for physics 17 10.1088/1367-2630/17/6/063038 Institut für Physik und Astronomie