TY - JOUR A1 - Awad, Emad A1 - Metzler, Ralf T1 - Crossover dynamics from superdiffusion to subdiffusion BT - models and solutions JF - Fractional calculus and applied analysis : an international journal for theory and applications N2 - The Cattaneo or telegrapher's equation describes the crossover from initial ballistic to normal diffusion. Here we study and survey time-fractional generalisations of this equation that are shown to produce the crossover of the mean squared displacement from superdiffusion to subdiffusion. Conditional solutions are derived in terms of Fox H-functions and the dth-order moments as well as the diffusive flux of the different models are derived. Moreover, the concept of the distribution-like is proposed as an alternative to the probability density function. KW - Cattaneo equation KW - telegrapher's equation KW - crossover dynamics KW - fractional dynamic equations KW - anomalous diffusion KW - superdiffusion and KW - subdiffusion KW - Fox H-functions Y1 - 2020 U6 - https://doi.org/10.1515/fca-2020-0003 SN - 1311-0454 SN - 1314-2224 VL - 23 IS - 1 SP - 55 EP - 102 PB - De Gruyter CY - Berlin ; Boston ER - TY - JOUR A1 - Bauer, Maximilian A1 - Godec, Aljaž A1 - Metzler, Ralf T1 - Diffusion of finite-size particles in two-dimensional channels with random wall configurations JF - Physical chemistry, chemical physics : PCCP ; a journal of European chemical societies N2 - Diffusion of chemicals or tracer molecules through complex systems containing irregularly shaped channels is important in many applications. Most theoretical studies based on the famed Fick–Jacobs equation focus on the idealised case of infinitely small particles and reflecting boundaries. In this study we use numerical simulations to consider the transport of finite-size particles through asymmetrical two-dimensional channels. Additionally, we examine transient binding of the molecules to the channel walls by applying sticky boundary conditions. We consider an ensemble of particles diffusing in independent channels, which are characterised by common structural parameters. We compare our results for the long-time effective diffusion coefficient with a recent theoretical formula obtained by Dagdug and Pineda [J. Chem. Phys., 2012, 137, 024107]. KW - anomalous diffusion KW - fractional dynamics KW - transport KW - nonergodicity KW - coefficient Y1 - 2014 U6 - https://doi.org/10.1039/C3CP55160A SN - 1463-9084 SN - 1463-9076 VL - 16 IS - 13 SP - 6118 EP - 6128 PB - RSC Publications CY - Cambridge ER - TY - JOUR A1 - Bodrova, Anna S. A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Ultraslow scaled Brownian motion JF - New journal of physics : the open-access journal for physics N2 - 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. KW - anomalous diffusion KW - stochastic processes KW - ageing Y1 - 2015 U6 - https://doi.org/10.1088/1367-2630/17/6/063038 SN - 1367-2630 VL - 17 PB - IOP Publ. Ltd. CY - Bristol 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 - TY - JOUR A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity JF - Soft matter N2 - 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. KW - anomalous diffusion KW - intracellular-transport KW - adenoassociated virus KW - infection pathway KW - escherichia-coli KW - endosomal escape KW - living cells KW - trafficking KW - cytoplasm KW - models Y1 - 2014 U6 - https://doi.org/10.1039/c3sm52846d SN - 2046-2069 VL - 2014 IS - 10 SP - 1591 EP - 1601 PB - Royal Society of Chemistry ER - TY - JOUR A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Ageing and confinement in non-ergodic heterogeneous diffusion processes JF - Journal of physics : A, Mathematical and theoretical N2 - 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. KW - stochastic processes KW - anomalous diffusion KW - ageing KW - weak ergodicity breaking Y1 - 2014 U6 - https://doi.org/10.1088/1751-8113/47/48/485002 SN - 1751-8113 SN - 1751-8121 VL - 47 IS - 48 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Cherstvy, Andrey G. A1 - Safdari, Hadiseh A1 - Metzler, Ralf T1 - Anomalous diffusion, nonergodicity, and ageing for exponentially and logarithmically time-dependent diffusivity BT - striking differences for massive versus massless particles JF - Journal of physics. D, Applied physics N2 - We investigate a diffusion process with a time-dependent diffusion coefficient, both exponentially increasing and decreasing in time, D(t)=D-0(e +/- 2 alpha t). For this (hypothetical) nonstationary diffusion process we compute-both analytically and from extensive stochastic simulations-the behavior of the ensemble- and time-averaged mean-squared displacements (MSDs) of the particles, both in the over- and underdamped limits. Simple asymptotic relations derived for the short- and long-time behaviors are shown to be in excellent agreement with the results of simulations. The diffusive characteristics in the presence of ageing are also considered, with dramatic differences of the over- versus underdamped regime. Our results for D(t)=D-0(e +/- 2 alpha t) extend and generalize the class of diffusive systems obeying scaled Brownian motion featuring a power-law-like variation of the diffusivity with time, D(t) similar to t(alpha-1). We also examine the logarithmically increasing diffusivity, D(t)=D(0)log[t/tau(0)], as another fundamental functional dependence (in addition to the power-law and exponential) and as an example of diffusivity slowly varying in time. One of the main conclusions is that the behavior of the massive particles is predominantly ergodic, while weak ergodicity breaking is repeatedly found for the time-dependent diffusion of the massless particles at short times. The latter manifests itself in the nonequivalence of the (both nonaged and aged) MSD and the mean time-averaged MSD. The current findings are potentially applicable to a class of physical systems out of thermal equilibrium where a rapid increase or decrease of the particles' diffusivity is inherently realized. One biological system potentially featuring all three types of time-dependent diffusion (power-law-like, exponential, and logarithmic) is water diffusion in the brain tissues, as we thoroughly discuss in the end. KW - anomalous diffusion KW - scaled Brownian motion KW - stochastic processes KW - nonstationary diffusivity KW - water diffusion in the brain KW - nonergodicity Y1 - 2021 U6 - https://doi.org/10.1088/1361-6463/abdff0 SN - 0022-3727 SN - 1361-6463 VL - 54 IS - 19 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Dieterich, Peter A1 - Klages, Rainer A1 - Chechkin, Aleksei V. T1 - Fluctuation relations for anomalous dynamics generated by time-fractional Fokker-Planck equations JF - New journal of physics : the open-access journal for physics N2 - 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. KW - fluctuation relations KW - anomalous diffusion KW - stochastic processes KW - stochastic thermodynamics KW - Fokker-Planck equations Y1 - 2015 U6 - https://doi.org/10.1088/1367-2630/17/7/075004 SN - 1367-2630 VL - 17 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Ghosh, Surya K. A1 - Cherstvy, Andrey G. A1 - Grebenkov, Denis S. A1 - Metzler, Ralf T1 - Anomalous, non-Gaussian tracer diffusion in crowded two-dimensional environments JF - NEW JOURNAL OF PHYSICS N2 - A topic of intense current investigation pursues the question of how the highly crowded environment of biological cells affects the dynamic properties of passively diffusing particles. Motivated by recent experiments we report results of extensive simulations of the motion of a finite sized tracer particle in a heterogeneously crowded environment made up of quenched distributions of monodisperse crowders of varying sizes in finite circular two-dimensional domains. For given spatial distributions of monodisperse crowders we demonstrate how anomalous diffusion with strongly non-Gaussian features arises in this model system. We investigate both biologically relevant situations of particles released either at the surface of an inner domain or at the outer boundary, exhibiting distinctly different features of the observed anomalous diffusion for heterogeneous distributions of crowders. Specifically we reveal an asymmetric spreading of tracers even at moderate crowding. In addition to the mean squared displacement (MSD) and local diffusion exponent we investigate the magnitude and the amplitude scatter of the time averaged MSD of individual tracer trajectories, the non-Gaussianity parameter, and the van Hove correlation function. We also quantify how the average tracer diffusivity varies with the position in the domain with a heterogeneous radial distribution of crowders and examine the behaviour of the survival probability and the dynamics of the tracer survival probability. Inter alia, the systems we investigate are related to the passive transport of lipid molecules and proteins in two-dimensional crowded membranes or the motion in colloidal solutions or emulsions in effectively two-dimensional geometries, as well as inside supercrowded, surface adhered cells. KW - anomalous diffusion KW - crowded fluids KW - stochastic processes Y1 - 2016 U6 - https://doi.org/10.1088/1367-2630/18/1/013027 SN - 1367-2630 VL - 18 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Ghosh, Surya K. A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf ED - Metzler, Ralf T1 - Non-universal tracer diffusion in crowded media of non-inert obstacles JF - Physical Chemistry Chemical Physics N2 - We study the diffusion of a tracer particle, which moves in continuum space between a lattice of excluded volume, immobile non-inert obstacles. In particular, we analyse how the strength of the tracer–obstacle interactions and the volume occupancy of the crowders alter the diffusive motion of the tracer. From the details of partitioning of the tracer diffusion modes between trapping states when bound to obstacles and bulk diffusion, we examine the degree of localisation of the tracer in the lattice of crowders. We study the properties of the tracer diffusion in terms of the ensemble and time averaged mean squared displacements, the trapping time distributions, the amplitude variation of the time averaged mean squared displacements, and the non-Gaussianity parameter of the diffusing tracer. We conclude that tracer–obstacle adsorption and binding triggers a transient anomalous diffusion. From a very narrow spread of recorded individual time averaged trajectories we exclude continuous type random walk processes as the underlying physical model of the tracer diffusion in our system. For moderate tracer–crowder attraction the motion is found to be fully ergodic, while at stronger attraction strength a transient disparity between ensemble and time averaged mean squared displacements occurs. We also put our results into perspective with findings from experimental single-particle tracking and simulations of the diffusion of tagged tracers in dense crowded suspensions. Our results have implications for the diffusion, transport, and spreading of chemical components in highly crowded environments inside living cells and other structured liquids. KW - fluorescence correlation spectroscopy KW - single-particle tracking KW - anomalous diffusion KW - living cells KW - physiological consequences KW - langevin equation KW - infection pathway KW - excluded volume KW - brownian-motion KW - random-walks Y1 - 2014 SN - 1463-9076 VL - 3 IS - 17 SP - 1847 EP - 1858 PB - The Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Godec, Aljaz A1 - Bauer, Maximilian A1 - Metzler, Ralf T1 - Collective dynamics effect transient subdiffusion of inert tracers in flexible gel networks JF - New journal of physics : the open-access journal for physics N2 - Based on extensive Brownian dynamics simulations we study the thermal motion of a tracer bead in a cross-linked, flexible gel in the limit when the tracer particle size is comparable to or even larger than the equilibrium mesh size of the gel. The analysis of long individual trajectories of the tracer demonstrates the existence of pronounced transient anomalous diffusion. From the time averaged mean squared displacement and the time averaged van Hove correlation functions we elucidate the many-body origin of the non-Brownian tracer bead dynamics. Our results shed new light onto the ongoing debate over the physical origin of steric tracer interactions with structured environments. KW - anomalous diffusion KW - gel network KW - van Hove correlation Y1 - 2014 U6 - https://doi.org/10.1088/1367-2630/16/9/092002 SN - 1367-2630 VL - 16 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Godec, Aljaz A1 - Chechkin, Aleksei V. A1 - Barkai, Eli A1 - Kantz, Holger A1 - Metzler, Ralf T1 - Localisation and universal fluctuations in ultraslow diffusion processes JF - Journal of physics : A, Mathematical and theoretical N2 - 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. KW - Sinai diffusion KW - anomalous diffusion KW - quenched energy landscape Y1 - 2014 U6 - https://doi.org/10.1088/1751-8113/47/49/492002 SN - 1751-8113 SN - 1751-8121 VL - 47 IS - 49 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Guggenberger, Tobias A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions JF - Journal of physics : A, Mathematical and theoretical N2 - We study the stochastic motion of particles driven by long-range correlated fractional Gaussian noise (FGN) in a superharmonic external potential of the form U(x) proportional to x(2n) (n is an element of N). When the noise is considered to be external, the resulting overdamped motion is described by the non-Markovian Langevin equation for fractional Brownian motion. For this case we show the existence of long time, stationary probability density functions (PDFs) the shape of which strongly deviates from the naively expected Boltzmann PDF in the confining potential U(x). We analyse in detail the temporal approach to stationarity as well as the shape of the non-Boltzmann stationary PDF. A typical characteristic is that subdiffusive, antipersistent (with negative autocorrelation) motion tends to effect an accumulation of probability close to the origin as compared to the corresponding Boltzmann distribution while the opposite trend occurs for superdiffusive (persistent) motion. For this latter case this leads to distinct bimodal shapes of the PDF. This property is compared to a similar phenomenon observed for Markovian Levy flights in superharmonic potentials. We also demonstrate that the motion encoded in the fractional Langevin equation driven by FGN always relaxes to the Boltzmann distribution, as in this case the fluctuation-dissipation theorem is fulfilled. KW - anomalous diffusion KW - Boltzmann distribution KW - non-Gaussian distribution Y1 - 2021 U6 - https://doi.org/10.1088/1751-8121/ac019b SN - 1751-8113 SN - 1751-8121 VL - 54 IS - 29 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Guggenberger, Tobias A1 - Pagnini, Gianni A1 - Vojta, Thomas A1 - Metzler, Ralf T1 - Fractional Brownian motion in a finite interval BT - correlations effect depletion or accretion zones of particles near boundaries JF - New Journal of Physics N2 - Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) 〈X² (t)〉 ⋍ tᵅ with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers. KW - anomalous diffusion KW - fractional Brownian motion KW - reflecting boundary conditions Y1 - 2019 U6 - https://doi.org/10.1088/1367-2630/ab075f SN - 1367-2630 VL - 21 PB - Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics CY - Bad Honnef und London ER - TY - JOUR A1 - Krüsemann, Henning A1 - Godec, Aljaz A1 - Metzler, Ralf T1 - Ageing first passage time density in continuous time random walks and quenched energy landscapes JF - Journal of physics : A, Mathematical and theoretical N2 - We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function Psi (t) similar or equal to t(-1-alpha) (0 <= alpha <= 2). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time t(a), the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs with 0 < alpha < 1 describe systems such as charge carriers in amorphous semiconducters, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range 1 < alpha < 2, for sufficiently long ta. We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the ageing CTRW results. With small bias or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape. KW - first passage KW - random walks KW - anomalous diffusion Y1 - 2015 U6 - https://doi.org/10.1088/1751-8113/48/28/285001 SN - 1751-8113 SN - 1751-8121 VL - 48 IS - 28 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Metzler, Ralf T1 - Superstatistics and non-Gaussian diffusion JF - The European physical journal special topics N2 - Brownian motion and viscoelastic anomalous diffusion in homogeneous environments are intrinsically Gaussian processes. In a growing number of systems, however, non-Gaussian displacement distributions of these processes are being reported. The physical cause of the non-Gaussianity is typically seen in different forms of disorder. These include, for instance, imperfect "ensembles" of tracer particles, the presence of local variations of the tracer mobility in heteroegenous environments, or cases in which the speed or persistence of moving nematodes or cells are distributed. From a theoretical point of view stochastic descriptions based on distributed ("superstatistical") transport coefficients as well as time-dependent generalisations based on stochastic transport parameters with built-in finite correlation time are invoked. After a brief review of the history of Brownian motion and the famed Gaussian displacement distribution, we here provide a brief introduction to the phenomenon of non-Gaussianity and the stochastic modelling in terms of superstatistical and diffusing-diffusivity approaches. KW - Brownian diffusion KW - anomalous diffusion KW - dynamics KW - kinetic-theory KW - models KW - motion KW - nanoparticles KW - nonergodicity KW - statistics KW - subdiffusion Y1 - 2020 U6 - https://doi.org/10.1140/epjst/e2020-900210-x SN - 1951-6355 SN - 1951-6401 VL - 229 IS - 5 SP - 711 EP - 728 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Metzler, Ralf A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Bodrova, Anna S. T1 - Ultraslow scaled Brownian motion JF - New journal of physics : the open-access journal for physics N2 - 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. KW - anomalous diffusion KW - stochastic processes KW - ageing Y1 - 2015 U6 - https://doi.org/10.1088/1367-2630/17/6/063038 SN - 1367-2630 VL - 17 IS - 063038 PB - Dt. Physikalische Ges., IOP CY - Bad Honnef, London ER - TY - JOUR A1 - Molina-Garcia, Daniel A1 - Sandev, Trifce A1 - Safdari, Hadiseh A1 - Pagnini, Gianni A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Crossover from anomalous to normal diffusion BT - truncated power-law noise correlations and applications to dynamics in lipid bilayers JF - New Journal of Physics N2 - Abstract The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive–diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive–diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion–diffusion and subdiffusion–subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling. KW - anomalous diffusion KW - truncated power-law correlated noise KW - lipid bilayer membrane dynamics Y1 - 2018 U6 - https://doi.org/10.1088/1367-2630/aae4b2 SN - 1367-2630 VL - 20 PB - IOP Publishing Ltd CY - London und Bad Honnef ER - TY - JOUR A1 - Mutothya, Nicholas Mwilu A1 - Xu, Yong A1 - Li, Yongge A1 - Metzler, Ralf T1 - Characterising stochastic motion in heterogeneous media driven by coloured non-Gaussian noise JF - Journal of physics : A, Mathematical and theoretical N2 - We study the stochastic motion of a test particle in a heterogeneous medium in terms of a position dependent diffusion coefficient mimicking measured deterministic diffusivity gradients in biological cells or the inherent heterogeneity of geophysical systems. Compared to previous studies we here investigate the effect of the interplay of anomalous diffusion effected by position dependent diffusion coefficients and coloured non-Gaussian noise. The latter is chosen to be distributed according to Tsallis' q-distribution, representing a popular example for a non-extensive statistic. We obtain the ensemble and time averaged mean squared displacements for this generalised process and establish its non-ergodic properties as well as analyse the non-Gaussian nature of the associated displacement distribution. We consider both non-stratified and stratified environments. KW - diffusion KW - anomalous diffusion KW - non-extensive statistics KW - coloured KW - noise KW - heterogeneous diffusion process Y1 - 2021 U6 - https://doi.org/10.1088/1751-8121/abfba6 SN - 1751-8113 SN - 1751-8121 VL - 54 IS - 29 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Palyulin, Vladimir V. A1 - Ala-Nissila, Tapio A1 - Metzler, Ralf ED - Metzler, Ralf T1 - Polymer translocation: the first two decades and the recent diversification JF - Soft matter N2 - Probably no other field of statistical physics at the borderline of soft matter and biological physics has caused such a flurry of papers as polymer translocation since the 1994 landmark paper by Bezrukov, Vodyanoy, and Parsegian and the study of Kasianowicz in 1996. Experiments, simulations, and theoretical approaches are still contributing novel insights to date, while no universal consensus on the statistical understanding of polymer translocation has been reached. We here collect the published results, in particular, the famous–infamous debate on the scaling exponents governing the translocation process. We put these results into perspective and discuss where the field is going. In particular, we argue that the phenomenon of polymer translocation is non-universal and highly sensitive to the exact specifications of the models and experiments used towards its analysis. KW - solid-state nanopores KW - single-stranded-dna KW - posttranslational protein translocation KW - anomalous diffusion KW - monte-carlo KW - structured polynucleotides KW - dynamics simulation KW - equation approach KW - osmotic-pressure KW - membrane channel Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-76266 SN - 1744-683X VL - 45 IS - 10 SP - 9016 EP - 9037 PB - the Royal Society of Chemistry CY - Cambridge ER -