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
OPUS4-39317 Wissenschaftlicher Artikel Bodrova, Anna; Chechkin, Aleksei V.; Cherstvy, Andrey G.; Metzler, Ralf Quantifying non-ergodic dynamics of force-free granular gases Brownian motion is ergodic in the Boltzmann-Khinchin sense that long time averages of physical observables such as the mean squared displacement provide the same information as the corresponding ensemble average, even at out-of-equilibrium conditions. This property is the fundamental prerequisite for single particle tracking and its analysis in simple liquids. We study analytically and by event-driven molecular dynamics simulations the dynamics of force-free cooling granular gases and reveal a violation of ergodicity in this Boltzmann-Khinchin sense as well as distinct ageing of the system. Such granular gases comprise materials such as dilute gases of stones, sand, various types of powders, or large molecules, and their mixtures are ubiquitous in Nature and technology, in particular in Space. We treat-depending on the physical-chemical properties of the inter-particle interaction upon their pair collisions-both a constant and a velocity-dependent (viscoelastic) restitution coefficient epsilon. Moreover we compare the granular gas dynamics with an effective single particle stochastic model based on an underdamped Langevin equation with time dependent diffusivity. We find that both models share the same behaviour of the ensemble mean squared displacement (MSD) and the velocity correlations in the limit of weak dissipation. Qualitatively, the reported non-ergodic behaviour is generic for granular gases with any realistic dependence of epsilon on the impact velocity of particles. Cambridge Royal Society of Chemistry 2015 8 Physical chemistry, chemical physics : a journal of European Chemical Societies 17 34 21791 21798 10.1039/c5cp02824h Institut für Physik und Astronomie
OPUS4-38596 Wissenschaftlicher Artikel Safdari, Hadiseh; Cherstvy, Andrey G.; Chechkin, Aleksei V.; Thiel, Felix; Sokolov, Igor M.; Metzler, Ralf Quantifying the non-ergodicity of scaled Brownian motion We examine the non-ergodic properties of scaled Brownian motion (SBM), a non-stationary stochastic process with a time dependent diffusivity of the form D(t) similar or equal to t(alpha-1). We compute the ergodicity breaking parameter EB in the entire range of scaling exponents a, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths T and short lag times Delta the EB parameter as function of the scaling exponent a has no divergence at alpha - 1/2 and present the asymptotes for EB in different limits. We generalize the analytical and simulations results for the time averaged and ergodic properties of SBM in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion. Bristol IOP Publ. Ltd. 2015 18 Journal of physics : A, Mathematical and theoretical 48 37 10.1088/1751-8113/48/37/375002 Institut für Physik und Astronomie
OPUS4-39007 Wissenschaftlicher Artikel Safdari, Hadiseh; Chechkin, Aleksei V.; Jafari, Gholamreza R.; Metzler, Ralf Aging scaled Brownian motion Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time. College Park American Physical Society 2015 9 Physical review : E, Statistical, nonlinear and soft matter physics 91 4 10.1103/PhysRevE.91.042107 Institut für Physik und Astronomie
OPUS4-9714 Wissenschaftlicher Artikel Bodrova, Anna S.; Chechkin, Aleksei V.; Cherstvy, Andrey G.; Safdari, Hadiseh; Sokolov, Igor M.; Metzler, Ralf Underdamped scaled Brownian motion It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases. London Nature Publishing Group 2016 Scientific reports 6 10.1038/srep30520 Institut für Physik und Astronomie
OPUS4-9715 misc Bodrova, Anna S.; Chechkin, Aleksei V.; Cherstvy, Andrey G.; Safdari, Hadiseh; Sokolov, Igor M.; Metzler, Ralf Underdamped scaled Brownian motion It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases. 2016 16 urn:nbn:de:kobv:517-opus4-97158 Institut für Physik und Astronomie
OPUS4-8518 Wissenschaftlicher Artikel Bodrova, Anna; Chechkin, Aleksei V.; Cherstvy, Andrey G.; Metzler, Ralf Quantifying non-ergodic dynamics of force-free granular gases Brownianmotion is ergodic in the Boltzmann-Khinchin sense that long time averages of physical observables such as the mean squared displacement provide the same information as the corresponding ensemble average, even at out-of-equilibrium conditions. This property is the fundamental prerequisite for single particle tracking and its analysis in simple liquids. We study analytically and by event-driven molecular dynamics simulations the dynamics of force-free cooling granular gases and reveal a violation of ergodicity in this Boltzmann- Khinchin sense as well as distinct ageing of the system. Such granular gases comprise materials such as dilute gases of stones, sand, various types of powders, or large molecules, and their mixtures are ubiquitous in Nature and technology, in particular in Space. We treat—depending on the physical-chemical properties of the inter-particle interaction upon their pair collisions—both a constant and a velocity-dependent (viscoelastic) restitution coefficient e. Moreover we compare the granular gas dynamics with an effective single particle stochastic model based on an underdamped Langevin equation with time dependent diffusivity. We find that both models share the same behaviour of the ensemble mean squared displacement (MSD) and the velocity correlations in the limit of weak dissipation. Qualitatively, the reported non-ergodic behaviour is generic for granular gases with any realistic dependence of e on the impact velocity of particles. 2015 8 Physical chemistry, chemical physics : PCCP ; a journal of European Chemical Societies 17 21791 21798 10.1039/C5CP02824H Institut für Chemie
OPUS4-8520 misc Bodrova, Anna; Chechkin, Aleksei V.; Cherstvy, Andrey G.; Metzler, Ralf Quantifying non-ergodic dynamics of force-free granular gases Brownianmotion is ergodic in the Boltzmann-Khinchin sense that long time averages of physical observables such as the mean squared displacement provide the same information as the corresponding ensemble average, even at out-of-equilibrium conditions. This property is the fundamental prerequisite for single particle tracking and its analysis in simple liquids. We study analytically and by event-driven molecular dynamics simulations the dynamics of force-free cooling granular gases and reveal a violation of ergodicity in this Boltzmann-Khinchin sense as well as distinct ageing of the system. Such granular gases comprise materials such as dilute gases of stones, sand, various types of powders, or large molecules, and their mixtures are ubiquitous in Nature and technology, in particular in Space. We treat—depending on the physical-chemical properties of the inter-particle interaction upon their pair collisions—both a constant and a velocity-dependent (viscoelastic) restitution coefficient e. Moreover we compare the granular gas dynamics with an effective single particle stochastic model based on an underdamped Langevin equation with time dependent diffusivity. We find that both models share the same behaviour of the ensemble mean squared displacement (MSD) and the velocity correlations in the limit of weak dissipation. Qualitatively, the reported non-ergodic behaviour is generic for granular gases with any realistic dependence of e on the impact velocity of particles. 2015 urn:nbn:de:kobv:517-opus4-85200 Institut für Chemie
OPUS4-40817 misc Burnecki, Krzysztof; Wylomanska, Agnieszka; Chechkin, Aleksei V. Discriminating between light- and heavy-tailed distributions with limit theorem In this paper we propose an algorithm to distinguish between light- and heavy-tailed probability laws underlying random datasets. The idea of the algorithm, which is visual and easy to implement, is to check whether the underlying law belongs to the domain of attraction of the Gaussian or non-Gaussian stable distribution by examining its rate of convergence. The method allows to discriminate between stable and various non-stable distributions. The test allows to differentiate between distributions, which appear the same according to standard Kolmogorov-Smirnov test. In particular, it helps to distinguish between stable and Student's t probability laws as well as between the stable and tempered stable, the cases which are considered in the literature as very cumbersome. Finally, we illustrate the procedure on plasma data to identify cases with so-called L-H transition. 2015 23 Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe 495 urn:nbn:de:kobv:517-opus4-408172 Mathematisch-Naturwissenschaftliche Fakultät
OPUS4-7629 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 ?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. Cambridge The Royal Society of Chemistry 2014 7 Physical chemistry, chemical physics : PCCP 30 16 15811 15817 10.1039/C4CP02019G Institut für Physik und Astronomie
OPUS4-7630 misc 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 ?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. 2014 6 15811 15817 urn:nbn:de:kobv:517-opus4-76302 Institut für Physik und Astronomie
OPUS4-34573 Wissenschaftlicher Artikel Schulz, Johannes H. P.; Chechkin, Aleksei V.; Metzler, Ralf Correlated continuous time random walks - combining scale-invariance with long-range memory for spatial and temporal dynamics 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. Bristol IOP Publ. Ltd. 2013 22 Journal of physics : A, Mathematical and theoretical 46 47 10.1088/1751-8113/46/47/475001 Institut für Physik und Astronomie
OPUS4-7860 Wissenschaftlicher Artikel Metzler, Ralf; Cherstvy, Andrey G.; Chechkin, Aleksei V.; Bodrova, Anna S. 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 . 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. Bad Honnef, London Dt. Physikalische Ges., IOP 2015 New journal of physics : the open-access journal for physics 17 063038 10.1088/1367-2630/17/6/063038 Institut für Physik und Astronomie
OPUS4-7861 misc Metzler, Ralf; Cherstvy, Andrey G.; Chechkin, Aleksei V.; Bodrova, Anna S. 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 . 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. 2015 New journal of physics : the open-access journal for physics urn:nbn:de:kobv:517-opus4-78618 Institut für Physik und Astronomie
OPUS4-7400 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. Royal Society of Chemistry 2014 11 Soft matter 2014 10 1591 1601 10.1039/c3sm52846d Mathematisch-Naturwissenschaftliche Fakultät
OPUS4-7402 misc 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. 2014 11 168 1591 1601 urn:nbn:de:kobv:517-opus4-74021 Institut für Chemie
OPUS4-40974 misc Sposini, Vittoria; Chechkin, Aleksei V.; Flavio, Seno; Pagnini, Gianni; Metzler, Ralf Random diffusivity from stochastic equations Brownian yet non-Gaussian dynamics was observed. These are processes characterised by a linear growth in time of the mean squared displacement, yet the probability density function of the particle displacement is distinctly non-Gaussian, and often of exponential(Laplace) shape. This apparently ubiquitous behaviour observed in very different physical systems has been interpreted as resulting from diffusion in inhomogeneous environments and mathematically represented through a variable, stochastic diffusion coefficient. Indeed different models describing a fluctuating diffusivity have been studied. Here we present a new view of the stochastic basis describing time dependent random diffusivities within a broad spectrum of distributions. Concretely, our study is based on the very generic class of the generalised Gamma distribution. Two models for the particle spreading in such random diffusivity settings are studied. The first belongs to the class of generalised grey Brownian motion while the second follows from the idea of diffusing diffusivities. The two processes exhibit significant characteristics which reproduce experimental results from different biological and physical systems. We promote these two physical models for the description of stochastic particle motion in complex environments. 2018 33 New Journal of Physics urn:nbn:de:kobv:517-opus4-409743 Institut für Physik und Astronomie
OPUS4-40973 Wissenschaftlicher Artikel Sposini, Vittoria; Chechkin, Aleksei V.; Seno, Flavio; Pagnini, Gianni; Metzler, Ralf Random diffusivity from stochastic equations A considerable number of systems have recently been reported in which Brownian yet non-Gaussian dynamics was observed. These are processes characterised by a linear growth in time of the mean squared displacement, yet the probability density function of the particle displacement is distinctly non-Gaussian, and often of exponential(Laplace) shape. This apparently ubiquitous behaviour observed in very different physical systems has been interpreted as resulting from diffusion in inhomogeneous environments and mathematically represented through a variable, stochastic diffusion coefficient. Indeed different models describing a fluctuating diffusivity have been studied. Here we present a new view of the stochastic basis describing time dependent random diffusivities within a broad spectrum of distributions. Concretely, our study is based on the very generic class of the generalised Gamma distribution. Two models for the particle spreading in such random diffusivity settings are studied. The first belongs to the class of generalised grey Brownian motion while the second follows from the idea of diffusing diffusivities. The two processes exhibit significant characteristics which reproduce experimental results from different biological and physical systems. We promote these two physical models for the description of stochastic particle motion in complex environments. Bad Honnef und London Deutsche Physikalische Gesellschaft / Institute of Physics 2018 32 New Journal of Physics 1 33 10.1088/1367-2630/aab696 Institut für Physik und Astronomie
OPUS4-41548 misc Chechkin, Aleksei V.; Zaid, Irwin M.; Lomholt, Michael A.; Sokolov, Igor 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. 2013 13 Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe 593 114 126 urn:nbn:de:kobv:517-opus4-415480 10.25932/publishup-41548 Mathematisch-Naturwissenschaftliche Fakultät