TY - JOUR A1 - Cherstvy, Andrey G. A1 - Nagel, Oliver A1 - Beta, Carsten A1 - Metzler, Ralf T1 - Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells? Based on the statistical analysis of experimental single-cell tracking data of the two-dimensional motion of the Dictyostelium discoideum amoeboid cells, we quantify their diffusive behaviour based on a number of standard and complementary statistical indicators. We compute the ensemble- and time-averaged mean-squared displacements (MSDs) of the diffusing amoebae cells and observe a pronounced spread of short-time diffusion coefficients and anomalous MSD-scaling exponents for individual cells. The distribution functions of the cell displacements, the long-tailed distribution of instantaneous speeds, and the velocity autocorrelations are also computed. In particular, we observe a systematic superdiffusive short-time behaviour for the ensemble- and time-averaged MSDs of the amoeboid cells. Also, a clear anti-correlation of scaling exponents and generalised diffusivity values for different cells is detected. Most significantly, we demonstrate that the distribution function of the cell displacements has a strongly non-Gaussian shape andusing a rescaled spatio-temporal variablethe cell-displacement data collapse onto a universal master curve. The current analysis of single-cell motions can be implemented for quantifying diffusive behaviours in other living-matter systems, in particular, when effects of active transport, non-Gaussian displacements, and heterogeneity of the population are involved in the dynamics. Y1 - 2018 U6 - https://doi.org/10.1039/c8cp04254c SN - 1463-9076 SN - 1463-9084 VL - 20 IS - 35 SP - 23034 EP - 23054 PB - Royal Society of Chemistry CY - Cambridge ER - TY - GEN A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes N2 - We consider diffusion processes with a spatially varying diffusivity giving rise to anomalous diffusion. Such heterogeneous diffusion processes are analysed for the cases of exponential, power-law, and logarithmic dependencies of the diffusion coefficient on the particle position. Combining analytical approaches with stochastic simulations, we show that the functional form of the space-dependent diffusion coefficient and the initial conditions of the diffusing particles are vital for their statistical and ergodic properties. In all three cases a weak ergodicity breaking between the time and ensemble averaged mean squared displacements is observed. We also demonstrate a population splitting of the time averaged traces into fast and slow diffusers for the case of exponential variation of the diffusivity as well as a particle trapping in the case of the logarithmic diffusivity. Our analysis is complemented by the quantitative study of the space coverage, the diffusive spreading of the probability density, as well as the survival probability. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 236 KW - anomalous diffusion KW - disordered media KW - fractional dynamics KW - infection pathway KW - inhomogeneous-media KW - intracellular-transport KW - langevin equation KW - living cells KW - random-walks KW - single-particle tracking Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-94468 SP - 20220 EP - 20235 ER - TY - JOUR A1 - Metzler, Ralf A1 - Jeon, Jae-Hyung A1 - Cherstvy, Andrey G. A1 - Barkai, Eli T1 - Anomalous diffusion models and their properties BT - non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking JF - physical chemistry, chemical physics : PCCP N2 - Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion. KW - intermittent chaotic systems KW - Fokker-Planck equations KW - time random-walks KW - fluorescence photobleaching recovery KW - fluctuation-dissipation theorem KW - fractional dynamics approach KW - photon-counting statistics KW - weak ergodicity breaking KW - flight search patterns KW - levy flights Y1 - 2014 U6 - https://doi.org/10.1039/c4cp03465a SN - 1463-9076 SN - 1463-9084 VL - 2014 IS - 16 SP - 24128 EP - 24164 ER - TY - GEN A1 - Metzler, Ralf A1 - Jeon, Jae-Hyung A1 - Cherstvy, Andrey G. A1 - Barkai, Eli T1 - Anomalous diffusion models and their properties BT - non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking N2 - Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 174 KW - Fokker-Planck equations KW - flight search patterns KW - fluctuation-dissipation theorem KW - fluorescence photobleaching recovery KW - fractional dynamics approach KW - intermittent chaotic systems KW - levy flights KW - photon-counting statistics KW - time random-walks KW - weak ergodicity breaking Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-74448 SP - 24128 EP - 24164 ER - TY - GEN 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 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 168 KW - adenoassociated virus KW - anomalous diffusion KW - cytoplasm KW - endosomal escape KW - escherichia-coli KW - infection pathway KW - intracellular-transport KW - living cells KW - models KW - trafficking Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-74021 IS - 168 SP - 1591 EP - 1601 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 - GEN A1 - Ghosh, Surya K. A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Non-universal tracer diffusion in crowded media of non-inert obstacles 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 186 KW - escence 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 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-77128 SP - 1847 EP - 1858 PB - The Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Shin, Jaeoh A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf ED - Metzler, Ralf T1 - Kinetics of polymer looping with macromolecular crowding: effects of volume fraction and crowder size JF - Soft Matter N2 - The looping of polymers such as DNA is a fundamental process in the molecular biology of living cells, whose interior is characterised by a high degree of molecular crowding. We here investigate in detail the looping dynamics of flexible polymer chains in the presence of different degrees of crowding. From the analysis of the looping–unlooping rates and the looping probabilities of the chain ends we show that the presence of small crowders typically slows down the chain dynamics but larger crowders may in fact facilitate the looping. We rationalise these non-trivial and often counterintuitive effects of the crowder size on the looping kinetics in terms of an effective solution viscosity and standard excluded volume. It is shown that for small crowders the effect of an increased viscosity dominates, while for big crowders we argue that confinement effects (caging) prevail. The tradeoff between both trends can thus result in the impediment or facilitation of polymer looping, depending on the crowder size. We also examine how the crowding volume fraction, chain length, and the attraction strength of the contact groups of the polymer chain affect the looping kinetics and hairpin formation dynamics. Our results are relevant for DNA looping in the absence and presence of protein mediation, DNA hairpin formation, RNA folding, and the folding of polypeptide chains under biologically relevant high-crowding conditions. KW - gene-regulation kinetics KW - physiological consequences KW - spatial-organization KW - anomalous diffusion KW - folding kinetics KW - living cells KW - dna coiling KW - in-vitro KW - dynamics KW - mixtures Y1 - 2014 SN - 1744-683X SP - 472 EP - 488 PB - The Royal Society of Chemistry CY - Cambridge ER - TY - GEN A1 - Shin, Jaeoh A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Kinetics of polymer looping with macromolecular crowding: effects of volume fraction and crowder size N2 - The looping of polymers such as DNA is a fundamental process in the molecular biology of living cells, whose interior is characterised by a high degree of molecular crowding. We here investigate in detail the looping dynamics of flexible polymer chains in the presence of different degrees of crowding. From the analysis of the looping–unlooping rates and the looping probabilities of the chain ends we show that the presence of small crowders typically slows down the chain dynamics but larger crowders may in fact facilitate the looping. We rationalise these non-trivial and often counterintuitive effects of the crowder size on the looping kinetics in terms of an effective solution viscosity and standard excluded volume. It is shown that for small crowders the effect of an increased viscosity dominates, while for big crowders we argue that confinement effects (caging) prevail. The tradeoff between both trends can thus result in the impediment or facilitation of polymer looping, depending on the crowder size. We also examine how the crowding volume fraction, chain length, and the attraction strength of the contact groups of the polymer chain affect the looping kinetics and hairpin formation dynamics. Our results are relevant for DNA looping in the absence and presence of protein mediation, DNA hairpin formation, RNA folding, and the folding of polypeptide chains under biologically relevant high-crowding conditions. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 185 KW - gene-regulation kinetics KW - physiological consequences KW - spatial-organization KW - anomalous diffusion KW - folding kinetics KW - living cells KW - dna coiling KW - in-vitro KW - dynamics KW - mixtures Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-76961 SP - 472 EP - 488 PB - The Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes N2 - We consider diffusion processes with a spatially varying diffusivity giving rise to anomalous diffusion. Such heterogeneous diffusion processes are analysed for the cases of exponential, power-law, and logarithmic dependencies of the diffusion coefficient on the particle position. Combining analytical approaches with stochastic simulations, we show that the functional form of the space-dependent diffusion coefficient and the initial conditions of the diffusing particles are vital for their statistical and ergodic properties. In all three cases a weak ergodicity breaking between the time and ensemble averaged mean squared displacements is observed. We also demonstrate a population splitting of the time averaged traces into fast and slow diffusers for the case of exponential variation of the diffusivity as well as a particle trapping in the case of the logarithmic diffusivity. Our analysis is complemented by the quantitative study of the space coverage, the diffusive spreading of the probability density, as well as the survival probability. Y1 - 2008 UR - http://pubs.rsc.org/en/content/articlehtml/2013/cp/c3cp53056f U6 - https://doi.org/10.1039/C3CP53056F ER - TY - JOUR A1 - Bauer, Maximilian A1 - Godec, Aljaz A1 - Metzler, Ralf T1 - Diffusion of finite-size particles in two-dimensional channels with random wall configurations JF - Physical chemistry, chemical physics : 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 Y1 - 2014 U6 - https://doi.org/10.1039/c3cp55160a SN - 1463-9076 SN - 1463-9084 VL - 16 IS - 13 SP - 6118 EP - 6128 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Jeon, Jae-Hyung A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - 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. Y1 - 2014 U6 - https://doi.org/10.1039/c4cp02019g SN - 1463-9076 SN - 1463-9084 VL - 16 IS - 30 SP - 15811 EP - 15817 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Metzler, Ralf A1 - Jeon, Jae-Hyung A1 - Cherstvy, Andrey G. A1 - Barkai, Eli T1 - Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion. Y1 - 2014 U6 - https://doi.org/10.1039/c4cp03465a SN - 1463-9076 SN - 1463-9084 VL - 16 IS - 44 SP - 24128 EP - 24164 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - de Carvalho, Sidney J. A1 - Metzler, Ralf A1 - Cherstvy, Andrey G. T1 - Inverted critical adsorption of polyelectrolytes in confinement JF - Soft matter N2 - What are the fundamental laws for the adsorption of charged polymers onto oppositely charged surfaces, for convex, planar, and concave geometries? This question is at the heart of surface coating applications, various complex formation phenomena, as well as in the context of cellular and viral biophysics. It has been a long-standing challenge in theoretical polymer physics; for realistic systems the quantitative understanding is however often achievable only by computer simulations. In this study, we present the findings of such extensive Monte-Carlo in silico experiments for polymer–surface adsorption in confined domains. We study the inverted critical adsorption of finite-length polyelectrolytes in three fundamental geometries: planar slit, cylindrical pore, and spherical cavity. The scaling relations extracted from simulations for the critical surface charge density sc—defining the adsorption–desorption transition—are in excellent agreement with our analytical calculations based on the ground-state analysis of the Edwards equation. In particular, we confirm the magnitude and scaling of sc for the concave interfaces versus the Debye screening length 1/k and the extent of confinement a for these three interfaces for small ka values. For large ka the critical adsorption condition approaches the known planar limit. The transition between the two regimes takes place when the radius of surface curvature or half of the slit thickness a is of the order of 1/k. We also rationalize how sc(k) dependence gets modified for semi-flexible versus flexible chains under external confinement. We examine the implications of the chain length for critical adsorption—the effect often hard to tackle theoretically—putting an emphasis on polymers inside attractive spherical cavities. The applications of our findings to some biological systems are discussed, for instance the adsorption of nucleic acids onto the inner surfaces of cylindrical and spherical viral capsids. Y1 - 2015 U6 - https://doi.org/10.1039/C5SM00635J SN - 1744-6848 SN - 1744-683X IS - 11 SP - 4430 EP - 4443 PB - Royal Society of Chemistry CY - London ER - TY - GEN A1 - de Carvalho, Sidney J. A1 - Metzler, Ralf A1 - Cherstvy, Andrey G. T1 - Inverted critical adsorption of polyelectrolytes in confinement N2 - What are the fundamental laws for the adsorption of charged polymers onto oppositely charged surfaces, for convex, planar, and concave geometries? This question is at the heart of surface coating applications, various complex formation phenomena, as well as in the context of cellular and viral biophysics. It has been a long-standing challenge in theoretical polymer physics; for realistic systems the quantitative understanding is however often achievable only by computer simulations. In this study, we present the findings of such extensive Monte-Carlo in silico experiments for polymer–surface adsorption in confined domains. We study the inverted critical adsorption of finite-length polyelectrolytes in three fundamental geometries: planar slit, cylindrical pore, and spherical cavity. The scaling relations extracted from simulations for the critical surface charge density sc—defining the adsorption–desorption transition—are in excellent agreement with our analytical calculations based on the ground-state analysis of the Edwards equation. In particular, we confirm the magnitude and scaling of sc for the concave interfaces versus the Debye screening length 1/k and the extent of confinement a for these three interfaces for small ka values. For large ka the critical adsorption condition approaches the known planar limit. The transition between the two regimes takes place when the radius of surface curvature or half of the slit thickness a is of the order of 1/k. We also rationalize how sc(k) dependence gets modified for semi-flexible versus flexible chains under external confinement. We examine the implications of the chain length for critical adsorption—the effect often hard to tackle theoretically—putting an emphasis on polymers inside attractive spherical cavities. The applications of our findings to some biological systems are discussed, for instance the adsorption of nucleic acids onto the inner surfaces of cylindrical and spherical viral capsids. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 214 Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-89562 SP - 4430 EP - 4443 ER - TY - JOUR A1 - Mardoukhi, Yousof A1 - Jeon, Jae-Hyung A1 - Metzler, Ralf T1 - Geometry controlled anomalous diffusion in random fractal geometries BT - looking beyond the infinite cluster JF - Physical chemistry, chemical physics : PCCP ; a journal of European Chemical Societies N2 - We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law BT� h with h o 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided. Y1 - 2015 U6 - https://doi.org/10.1039/c5cp03548a SN - 1439-7641 IS - 17 SP - 30134 EP - 30147 PB - Wiley-VCH Verl. CY - Weinheim ER - TY - GEN A1 - Mardoukhi, Yousof A1 - Jeon, Jae-Hyung A1 - Metzler, Ralf T1 - Geometry controlled anomalous diffusion in random fractal geometries BT - looking beyond the infinite cluster N2 - We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law BT� h with h o 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 207 Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-85247 ER - TY - JOUR A1 - Bodrova, Anna A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Quantifying non-ergodic dynamics of force-free granular gases JF - Physical chemistry, chemical physics : PCCP ; a journal of European Chemical Societies N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1039/C5CP02824H SN - 1463-9084 IS - 17 SP - 21791 EP - 21798 ER - TY - GEN A1 - Bodrova, Anna A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Quantifying non-ergodic dynamics of force-free granular gases N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 206 Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-85200 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 -