TY - GEN
A1 - Barkai, Eli
A1 - Garini, Yuval
A1 - Metzler, Ralf
T1 - Electrostatic effects in living cells Reply
T2 - PHYSICS TODAY
Y1 - 2013
SN - 0031-9228
(print)
SN - 1945-0699
(online)
VL - 66
IS - 7
SP - 11
EP - 11
PB - AMER INST PHYSICS
CY - MELVILLE
ER -
TY - JOUR
A1 - Chechkin, Aleksei V.
A1 - Zaid, I. M.
A1 - Lomholt, M. A.
A1 - Sokolov, I. 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 - http://dx.doi.org/10.1051/mmnp/20138208
SN - 0973-5348 (print)
VL - 8
IS - 2
SP - 114
EP - 126
PB - EDP Sciences
CY - Les Ulis
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 - http://dx.doi.org/10.1039/c4cp03465a
SN - 1463-9076 (print)
SN - 1463-9084 (online)
VL - 16
IS - 44
SP - 24128
EP - 24164
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 - http://dx.doi.org/10.1039/c4cp02019g
SN - 1463-9076 (print)
SN - 1463-9084 (online)
VL - 16
IS - 30
SP - 15811
EP - 15817
PB - Royal Society of Chemistry
CY - Cambridge
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 - http://dx.doi.org/10.1039/c3cp55160a
SN - 1463-9076 (print)
SN - 1463-9084 (online)
VL - 16
IS - 13
SP - 6118
EP - 6128
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 - Postprints 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 - 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 - http://dx.doi.org/10.1039/C5CP02824H
SN - 1463-9084 (online)
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 - Postprints 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 - 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 - http://dx.doi.org/10.1039/c5cp03548a
SN - 1439-7641 (online)
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 - Postprints 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 -