TY  - JOUR
A1  - Vilk, Ohad
A1  - Aghion, Erez
A1  - Nathan, Ran
A1  - Toledo, Sivan
A1  - Metzler, Ralf
A1  - Assaf, Michael
T1  - Classification of anomalous diffusion in animal movement data using power spectral analysis
JF  - Journal of physics : A, Mathematical and theoretical
N2  - The field of movement ecology has seen a rapid increase in high-resolution data in recent years, leading to the development of numerous statistical and numerical methods to analyse relocation trajectories. Data are often collected at the level of the individual and for long periods that may encompass a range of behaviours. 

Here, we use the power spectral density (PSD) to characterise the random movement patterns of a black-winged kite (Elanus caeruleus) and a white stork (Ciconia ciconia). The tracks are first segmented and clustered into different behaviours (movement modes), and for each mode we measure the PSD and the ageing properties of the process. 

For the foraging kite we find 1/f noise, previously reported in ecological systems mainly in the context of population dynamics, but not for movement data. We further suggest plausible models for each of the behavioural modes by comparing both the measured PSD exponents and the distribution of the single-trajectory PSD to known theoretical results and simulations.
KW  - diffusion
KW  - anomalous diffusion
KW  - power spectral analysis
KW  - ecological
KW  - movement data
Y1  - 2022
U6  - https://doi.org/10.1088/1751-8121/ac7e8f
SN  - 1751-8113
SN  - 1751-8121
VL  - 55
IS  - 33
PB  - IOP Publishing
CY  - Bristol
ER  - 
TY  - THES
A1  - Dörries, Timo Julian
T1  - Anomalous transport and non-Gaussian dynamics in mobile-immobile models
N2  - The mobile-immobile model (MIM) has been established in geoscience in the context of contaminant transport in groundwater. Here the tracer particles effectively immobilise, e.g., due to diffusion into dead-end pores or sorption. The main idea of the MIM is to split the total particle density into a mobile and an immobile density. Individual tracers switch between the mobile and immobile state following a two-state telegraph process, i.e., the residence times in each state are distributed exponentially. In geoscience the focus lies on the breakthrough curve (BTC), which is the concentration at a fixed location over time.  We apply the MIM to biological experiments with a special focus on anomalous scaling regimes of the  mean squared displacement (MSD) and non-Gaussian displacement distributions. As an exemplary system, we have analysed the motion of tau proteins, that diffuse freely inside axons of neurons. Their free diffusion thereby corresponds to the mobile state of the MIM. Tau proteins stochastically bind to microtubules, which effectively immobilises the tau proteins until they unbind and continue diffusing. Long immobilisation durations compared to the mobile durations give rise to distinct non-Gaussian Laplace shaped distributions. It is  accompanied by a plateau in the MSD for initially mobile tracer particles at relevant intermediate timescales. An equilibrium fraction of initially mobile tracers gives rise to non-Gaussian displacements at intermediate timescales, while the MSD remains linear at all times. In another setting  bio molecules diffuse in a biosensor and transiently bind to specific receptors, where advection becomes relevant in the mobile state. The plateau in the MSD observed for the advection-free setting and long immobilisation durations persists also for the case with advection. We find a new clear regime of anomalous diffusion with non-Gaussian distributions and a cubic scaling of the MSD. This regime emerges for initially mobile and for initially immobile tracers. For an equilibrium fraction of initially mobile tracers we observe an intermittent ballistic scaling of the MSD.  The long-time effective diffusion coefficient is enhanced by advection, which we physically explain with the variance of mobile durations.  Finally, we generalize the MIM to incorporate arbitrary immobilisation time distributions and focus on a Mittag-Leffler immobilisation time distribution with power-law tail ~ t^(-1-mu)  with 0<mu<1 and diverging mean immobilisation durations.  A fit of our model to the BTC of experimental data from tracer particles in aquifers matches the BTC including the power-law tail.  We use the fit parameters for plotting the displacement distributions and the MSD. We find Gaussian normal diffusion at short times and long-time power-law decay of mobile mass accompanied by anomalous diffusion at long times.  The long-time diffusion is subdiffusive in the advection-free setting, while it is either subdiffusive for 0<mu<1/2 or superdiffusive for 1/2<mu<1 when advection is present. In the long-time limit we show equivalence of our model to a bi-fractional diffusion equation.
N2  - In den Geowissenschaften wurde das 	"mobile-immobile model" (MIM) entwickelt, um den Transport von Verunreinigungen in Grundwässern zu beschreiben. Diese Verunreinigungen können in Sackgassenporen diffundieren oder an Oberflächen adsorbieren. Dabei bewegen sich die Verunreinigungen effektiv nicht. Der Grundgedanke des MIMs besteht darin, die Dichte der Verunreinigungen in eine Dichte aus mobilen Partikeln und eine Dichte aus immobilen Partikeln aufzuteilen. Jedes einzelne Teilchen der  Verunreinigung wechselt dabei zwischen dem mobilen Zustand, wo es diffundiert und sich durch Advektion bewegt und dem immobilen Zustand, wo es sich nicht bewegt. Je nach Zustand wird die Verunreinigung der mobilen oder der immobilen Dichte zugerechnet. Die Aufenthaltsdauern im mobilen und immobilen Zustand folgen jeweils einer Exponentialverteilung. Dies bedeutet, dass der Wechsel zwischen mobilen und immobilen Zustand durch einen Telegrafenprozess beschrieben werden kann. In den Geowissenschaften liegt der Fokus auf der "breakthrough curve" (BTC), was die Konzentrationskurve an einem festen Ort definiert. Der Grundgedanke dieser Arbeit besteht nun darin, das MIM auf biologische Systeme zu übertragen, wo beispielsweise Proteine zwischen einem diffusiven und einem immobilen Zustand wechseln. Dabei fokussieren wir uns auf Messgrößen, die üblicherweise in Experimenten mit einzelnen Molekülen oder Proteinen gemessen werden. Typische Messgrößen stellen die mittlere quadratische Verschiebung (MSD) und die Verschiebungsdichte dar, wobei wir besonderen Fokus auf nicht-lineare MSDs und nicht-Gaußsche Verteilungen legen. Als tragendes Beispiel für MIM in biologischen Systemen identifizieren wir Tau Proteine, welche in Nervenzellen diffundieren und für eine zufällige Dauer an Mikrotubuli binden, wobei sie effektiv unbeweglich werden. In der eindimensionalen Beschreibung des Systems sind die mittleren Bindungsdauern wesentlich länger als die durchschnittlichen Diffusionsdauern. Dadurch entsteht eine exponentielle Ortswahrscheinlichkeitsverteilung begleitet von einem Plateau im MSD auf mittleren Zeitskalen, wenn die Proteine zu Beginn alle mobil sind. Für eine Gleichgewichtsverteilung an mobilen Proteinen zu Beginn ist das MSD linear, wobei die Verteilung einer skalierten Laplaceverteilung entspricht. In anderen Systemen, wie beispielsweise biologischen Molekülen in Biosensoren, spielt Advektion eine wichtige Rolle. Das Plateau, welches wir im MSD für lange mittlere Immobilisierungsdauern finden, besteht auch mit Advektion. Zusätzlich finden wir ein neues Regime mit einem anomalen Skalierungsverhalten des MSDs. Dabei wächst das MSD kubisch unabhängig von der mittleren Immobilisierungsdauer, sofern die Advektionsgeschwindigkeit hinreichend groß ist. Für eine Gleichgewichtsverteilung an mobilen Molekülen finden wir ein ballistisches Regime im MSD. Das Langzeitverhalten vom MSD ist linear, wobei der effektive Diffusionskoeffizient durch Advektion verstärkt wird. Dieser Effekt ist in der Literatur bekannt. Wir bieten hier aber eine physikalische Erklärung über eine Kopplung der Advektion mit der Varianz der gesamten Aufenthaltsdauer im mobilen Zustand.  Abschließend erweitern wir das MIM, indem wir über exponentiell verteilte Immobilisierungsdauern hinausgehen und beliebige Verteilungen zulassen.  Das entsprechende Modell nennen wir extended MIM (EMIM).  Besonderen Fokus legen wir dabei auf eine Mittag-Leffler Verteilung, die sich  durch ein Potenzgesetz ~t^(-1-mu) mit 0<mu<1 für lange Immobilisierungsdauern auszeichnet. Solche Verteilungen werden in Experimenten gemessen. Das Potenzgesetz hat zur Folge, dass der Mittelwert divergiert.  Wir fitten EMIM zu einer BTC aus einem Experiment, in dem die Konzentration von fluoreszenter Farbe nach Durchlauf eines Grundwasserleiters gemessen wird.  Das gemessene Potenzgesetz der BTC kann durch MIM erklärt werden. Wir verwenden die erhaltenen Fitparameter, um das MSD und die Dichten darzustellen. Dabei finden wir ein lineares MSD mit Gaußscher Verteilung zu Beginn und eine nicht-Gaußsche Verteilung im Langzeitverhalten. Im Allgemeinen ist das MSD von EMIM mit der Mittag-Leffler Verteilung subdiffusiv im advektionsfreien Fall. Mit vorhandener Advektion finden wir Subdiffusion für 0<mu<1/2 und Superdiffusion für 1/2<mu<1. Im Langzeitlimes konvergiert EMIM zu einer bifraktionelen Diffusiongleichung.
KW  - anomalous diffusion
KW  - tau proteins
KW  - contaminant transport
KW  - mobile-immobile model (MIM)
KW  - anomale Diffusion
KW  - Schadstofftransport
KW  - mobile-immobile model (MIM)
KW  - Tau-Protein
Y1  - 2024
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-634959
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  - 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  - Xu, Pengbo
A1  - Deng, Weihua
A1  - Sandev, Trifce
T1  - Levy walk with parameter dependent velocity
BT  - hermite polynomial approach and numerical simulation
JF  - Journal of physics : A, Mathematical and theoretical
N2  - To analyze stochastic processes, one often uses integral transform (Fourier and Laplace) methods. However, for the time-space coupled cases, e.g. the Levy walk, sometimes the integral transform method may fail. Here we provide a Hermite polynomial expansion approach, being complementary to the integral transform method, to the Levy walk. Two approaches are compared for some already known results. We also consider the generalized Levy walk with parameter dependent velocity. Namely, we consider the Levy walk with velocity which depends on the walking length or on the duration of each step. Some interesting features of the generalized Levy walk are observed, including the special shapes of the probability density function, the first passage time distributions, and various diffusive behaviors of the mean squared displacement.
KW  - Hermite polynomial expansion
KW  - Levy walk
KW  - anomalous diffusion
KW  - parameter
KW  - dependent velocity
Y1  - 2020
U6  - https://doi.org/10.1088/1751-8121/ab7420
SN  - 1751-8113
SN  - 1751-8121
VL  - 53
IS  - 11
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
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  - 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  - Wang, Wei
A1  - Cherstvy, Andrey G.
A1  - Chechkin, Aleksei V.
A1  - Thapa, Samudrajit
A1  - Seno, Flavio
A1  - Liu, Xianbin
A1  - Metzler, Ralf
T1  - Fractional Brownian motion with random diffusivity
BT  - emerging residual nonergodicity below the correlation time
JF  - Journal of physics : A, Mathematical and theoretical
N2  - Numerous examples for a priori unexpected non-Gaussian behaviour for normal and anomalous diffusion have recently been reported in single-particle tracking experiments. Here, we address the case of non-Gaussian anomalous diffusion in terms of a random-diffusivity mechanism in the presence of power-law correlated fractional Gaussian noise. We study the ergodic properties of this model via examining the ensemble- and time-averaged mean-squared displacements as well as the ergodicity breaking parameter EB quantifying the trajectory-to-trajectory fluctuations of the latter. For long measurement times, interesting crossover behaviour is found as function of the correlation time tau characterising the diffusivity dynamics. We unveil that at short lag times the EB parameter reaches a universal plateau. The corresponding residual value of EB is shown to depend only on tau and the trajectory length. The EB parameter at long lag times, however, follows the same power-law scaling as for fractional Brownian motion. We also determine a corresponding plateau at short lag times for the discrete representation of fractional Brownian motion, absent in the continuous-time formulation. These analytical predictions are in excellent agreement with results of computer simulations of the underlying stochastic processes. Our findings can help distinguishing and categorising certain nonergodic and non-Gaussian features of particle displacements, as observed in recent single-particle tracking experiments.
KW  - stochastic processes
KW  - anomalous diffusion
KW  - fractional Brownian motion
KW  - diffusing diffusivity
KW  - weak ergodicity breaking
Y1  - 2020
U6  - https://doi.org/10.1088/1751-8121/aba467
SN  - 1751-8113
SN  - 1751-8121
VL  - 53
IS  - 47
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  - GEN
A1  - Xu, Pengbo
A1  - Zhou, Tian
A1  - Metzler, Ralf
A1  - Deng, Weihua
T1  - Stochastic harmonic trapping of a Lévy walk: transport and first-passage dynamics under soft resetting strategies
T2  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - We introduce and study a Lévy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1262 
KW  - diffusion
KW  - anomalous diffusion
KW  - stochastic resetting
KW  - Levy walks
Y1  - 2022
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-560402
SN  - 1866-8372
SP  - 1
EP  - 28
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - JOUR
A1  - Xu, Pengbo
A1  - Zhou, Tian
A1  - Metzler, Ralf
A1  - Deng, Weihua
T1  - Stochastic harmonic trapping of a Lévy walk
BT  - transport and first-passage dynamics under soft resetting strategies
JF  - New journal of physics : the open-access journal for physics / Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics
N2  - We introduce and study a Lévy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic.
KW  - diffusion
KW  - anomalous diffusion
KW  - stochastic resetting
KW  - Levy walks
Y1  - 2022
U6  - https://doi.org/10.1088/1367-2630/ac5282
SN  - 1367-2630
VL  - 24
IS  - 3
SP  - 1
EP  - 28
PB  - Deutsche Physikalische Gesellschaft
CY  - Bad Honnef
ER  - 
TY  - JOUR
A1  - Sandev, Trifce
A1  - Metzler, Ralf
A1  - Chechkin, Aleksei V.
T1  - From continuous time random walks to the generalized diffusion equation
JF  - Fractional calculus and applied analysis : an international journal for theory and applications
N2  - We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.
KW  - continuous time random walk (CTRW)
KW  - generalized diffusion equation
KW  - Mittag-Leffler functions
KW  - anomalous diffusion
Y1  - 2018
U6  - https://doi.org/10.1515/fca-2018-0002
SN  - 1311-0454
SN  - 1314-2224
VL  - 21
IS  - 1
SP  - 10
EP  - 28
PB  - De Gruyter
CY  - Berlin
ER  - 
TY  - JOUR
A1  - Vitali, Silvia
A1  - Sposini, Vittoria
A1  - Sliusarenko, Oleksii
A1  - Paradisi, Paolo
A1  - Castellani, Gastone
A1  - Pagnini, Gianni
T1  - Langevin equation in complex media and anomalous diffusion
JF  - Interface : journal of the Royal Society
N2  - The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modelling approaches for the description of anomalous diffusion in biological systems, such as the very complex and heterogeneous cell environment. Nevertheless, many questions are still open, such as the joint manifestation of statistical features in agreement with different models that can also be somewhat alternative to each other, e.g. continuous time random walk and fractional Brownian motion. To overcome these limitations, we propose a stochastic diffusion model with additive noise and linear friction force (linear Langevin equation), thus involving the explicit modelling of velocity dynamics. The complexity of the medium is parametrized via a population of intensity parameters (relaxation time and diffusivity of velocity), thus introducing an additional randomness, in addition to white noise, in the particle's dynamics. We prove that, for proper distributions of these parameters, we can get both Gaussian anomalous diffusion, fractional diffusion and its generalizations.
KW  - anomalous diffusion
KW  - heterogeneous media
KW  - biological transport
KW  - Gaussian processes
KW  - space-time fractional diffusion equation
KW  - fractional Brownian motion
Y1  - 2018
U6  - https://doi.org/10.1098/rsif.2018.0282
SN  - 1742-5689
SN  - 1742-5662
VL  - 15
IS  - 145
PB  - Royal Society
CY  - London
ER  - 
TY  - JOUR
A1  - Sandev, Trifce
A1  - Tomovski, Zivorad
A1  - Dubbeldam, Johan L. A.
A1  - Chechkin, Aleksei V.
T1  - Generalized diffusion-wave equation with memory kernel
JF  - Journal of physics : A, Mathematical and theoretical
N2  - We study generalized diffusion-wave equation in which the second order time derivative is replaced by an integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular cases. We consider different memory kernels of the integro-differential operator, derive corresponding fundamental solutions, specify the conditions of their non-negativity and calculate the mean squared displacement for all cases. In particular, we introduce and study generalized diffusion-wave equations with a regularized Prabhakar derivative of single and distributed orders. The equations considered can be used for modeling the broad spectrum of anomalous diffusion processes and various transitions between different diffusion regimes.
KW  - diffusion-wave equation
KW  - Mittag-Leffler function
KW  - anomalous diffusion
Y1  - 2018
U6  - https://doi.org/10.1088/1751-8121/aaefa3
SN  - 1751-8113
SN  - 1751-8121
VL  - 52
IS  - 1
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Sliusarenko, Oleksii Yu
A1  - Vitali, Silvia
A1  - Sposini, Vittoria
A1  - Paradisi, Paolo
A1  - Chechkin, Aleksei V.
A1  - Castellani, Gastone
A1  - Pagnini, Gianni
T1  - Finite-energy Levy-type motion through heterogeneous ensemble of Brownian particles
JF  - Journal of physics : A, Mathematical and theoretical
N2  - Complex systems are known to display anomalous diffusion, whose signature is a space/time scaling x similar to t(delta) with delta not equal 1/2 in the probability density function (PDF). Anomalous diffusion can emerge jointly with both Gaussian, e.g. fractional Brownian motion, and power-law decaying distributions, e.g. Levy Flights or Levy Walks (LWs). Levy flights get anomalous scaling, but, being jumps of any size allowed even at short times, have infinite position variance, infinite energy and discontinuous paths. LWs, which are based on random trapping events, overcome these limitations: they resemble a Levy-type power-law distribution that is truncated in the large displacement range and have finite moments, finite energy and, even with discontinuous velocity, they are continuous. However, LWs do not take into account the role of strong heterogeneity in many complex systems, such as biological transport in the crowded cell environment. In this work we propose and discuss a model describing a heterogeneous ensemble of Brownian particles (HEBP). Velocity of each single particle obeys a standard underdamped Langevin equation for the velocity, with linear friction term and additive Gaussian noise. Each particle is characterized by its own relaxation time and velocity diffusivity. We show that, for proper distributions of relaxation time and velocity diffusivity, the HEBP resembles some LW statistical features, in particular power-law decaying PDF, long-range correlations and anomalous diffusion, at the same time keeping finite position moments and finite energy. The main differences between the HEBP model and two different LWs are investigated, finding that, even when both velocity and position PDFs are similar, they differ in four main aspects: (i) LWs are biscaling, while HEBP is monoscaling; (ii) a transition from anomalous (delta = 1/2) to normal (delta = 1/2) diffusion in the long-time regime is seen in the HEBP and not in LWs; (iii) the power-law index of the position PDF and the space/time diffusion scaling are independent in the HEBP, while they both depend on the scaling of the interevent time PDF in LWs; (iv) at variance with LWs, our HEBP model obeys a fluctuation-dissipation theorem.
KW  - anomalous diffusion
KW  - heterogeneous ensemble of Brownian particles
KW  - biological transport
KW  - Langevin equation
KW  - Gaussian processes
KW  - fractional diffusion
KW  - Levy walk
Y1  - 2019
U6  - https://doi.org/10.1088/1751-8121/aafe90
SN  - 1751-8113
SN  - 1751-8121
VL  - 52
IS  - 9
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - GEN
A1  - Thapa, Samudrajit
A1  - Wyłomańska, Agnieszka
A1  - Sikora, Grzegorz
A1  - Wagner, Caroline E.
A1  - Krapf, Diego
A1  - Kantz, Holger
A1  - Chechkin, Aleksei V.
A1  - Metzler, Ralf
T1  - Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories
T2  - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1118 
KW  - diffusion
KW  - anomalous diffusion
KW  - large-deviation statistic
KW  - time-averaged mean squared displacement
KW  - Chebyshev inequality
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-493494
SN  - 1866-8372
IS  - 1118
ER  - 
TY  - JOUR
A1  - Thapa, Samudrajit
A1  - Wyłomańska, Agnieszka
A1  - Sikora, Grzegorz
A1  - Wagner, Caroline E.
A1  - Krapf, Diego
A1  - Kantz, Holger
A1  - Chechkin, Aleksei V.
A1  - Metzler, Ralf
T1  - Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories
JF  - New Journal of Physics
N2  - Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations.
KW  - diffusion
KW  - anomalous diffusion
KW  - large-deviation statistic
KW  - time-averaged mean squared displacement
KW  - Chebyshev inequality
Y1  - 2020
U6  - https://doi.org/10.1088/1367-2630/abd50e
SN  - 1367-2630
VL  - 23
PB  - Dt. Physikalische Ges. ; IOP
CY  - Bad Honnef ; London
ER  - 
TY  - THES
A1  - Thapa, Samudrajit
T1  - Deciphering anomalous diffusion in complex systems using Bayesian inference and large deviation theory
N2  - The development of methods such as super-resolution microscopy (Nobel prize in Chemistry, 2014) and multi-scale computer modelling (Nobel prize in Chemistry, 2013) have provided scientists with powerful tools to study microscopic systems. Sub-micron particles or even fluorescently labelled single molecules can now be tracked for long times in a variety of systems such as living cells, biological membranes, colloidal solutions etc. at spatial and temporal resolutions previously inaccessible. Parallel to such single-particle tracking experiments, super-computing techniques enable simulations of large atomistic or coarse-grained systems such as biologically relevant membranes or proteins from picoseconds to seconds, generating large volume of data. These have led to an unprecedented rise in the number of reported cases of anomalous diffusion wherein the characteristic features of Brownian motion—namely linear growth of the mean squared displacement with time and the Gaussian form of the probability density function (PDF) to find a particle at a given position at some fixed time—are routinely violated. This presents a big challenge in identifying the underlying stochastic process and also estimating the corresponding parameters of the process to completely describe the observed behaviour. Finding the correct physical mechanism which leads to the observed dynamics is of paramount importance, for example, to understand the first-arrival time of transcription factors which govern gene regulation, or the survival probability of a pathogen in a biological cell post drug administration. Statistical Physics provides useful methods that can be applied to extract such vital information. This cumulative dissertation, based on five publications, focuses on the development, implementation and application of such tools with special emphasis on Bayesian inference and large deviation theory. Together with the implementation of Bayesian model comparison and parameter estimation methods for models of diffusion, complementary tools are developed based on different observables and large deviation theory to classify stochastic processes and gather pivotal information. Bayesian analysis of the data of micron-sized particles traced in mucin hydrogels at different pH conditions unveiled several interesting features and we gained insights into, for example, how in going from basic to acidic pH, the hydrogel becomes more heterogeneous and phase separation can set in, leading to observed non-ergodicity (non-equivalence of time and ensemble averages) and non-Gaussian PDF. With large deviation theory based analysis we could detect, for instance, non-Gaussianity in seeming Brownian diffusion of beads in aqueous solution, anisotropic motion of the beads in mucin at neutral pH conditions, and short-time correlations in climate data. Thus through the application of the developed methods to biological and meteorological datasets crucial information is garnered about the underlying stochastic processes and significant insights are obtained in understanding the physical nature of these systems.
N2  - Die Entwicklung von Methoden wie der superauflösenden Mikroskopie (Nobelpreis für Chemie, 2014) und der Multiskalen-Computermodellierung (Nobelpreis für Chemie, 2013) hat Wis- senschaftlern mächtige Werkzeuge zur Untersuchung mikroskopischer Systeme zur Verfügung gestellt. Submikrometer Partikel und sogar einzelne fluoreszent markierte Moleküle können heute über lange Beobachtungszeiten in einer Vielzahl von Systemen, wie z.B. lebenden Zellen, biologischen Membranen und kolloidalen Suspensionen, mit bisher unerreichter räumlicher und zeitlicher Auflösung verfolgt werden. Neben solchen Einzelpartikelverfolgungsexperi- menten ermöglichen Supercomputer die Simulation großer atomistischer oder coarse-grained Systeme, wie z..B. biologisch relevante Membranen oder Proteine, über wenige Picosekunden bis hin zu einigen Sekunden, wobei große Datenmengen produziert werden. Diese haben zu einem beispiellosen Anstieg in der Zahl berichteter Fälle von anomaler Diffusion geführt, bei welcher die charakteristischen Eigenschaften der Brownschen Diffusion—nämlich das lineare Wachstum der mittleren quadratischen Verschiebung mit der Zeit und die Gaußsche Form der Wahrscheinlichkeitsdichtefunktion ein Partikel an einem gegebenen Ort und zu gegebener Zeit zu finden—verletzt sind. Dies stellt eine große Herausforderung bei der Identifizierung des zugrundeliegenden stochastischen Prozesses und der Schätzung der zugehörigen Prozess- parameter dar, was zur vollständigen Beschreibung des beobachteten Verhaltens nötig ist. Das Auffinden des korrekten physikalischen Mechanismus, welcher zum beobachteten Verhal- ten führt, ist von überragender Bedeutung, z.B. beim Verständnis der, die Genregulation steuernden, first-arrival time von Transkriptionsfaktoren oder der Überlebensfunktion eines Pathogens in einer biologischen Zelle nach Medikamentengabe. Die statistische Physik stellt nützliche Methoden bereit, die angewendet werden können, um solch wichtige Informationen zu erhalten. Der Schwerpunkt der vorliegenden, auf fünf Publikationen basierenden, kumulativen Dissertation liegt auf der Entwicklung, Implementierung und Anwendung solcher Methoden, mit einem besonderen Schwerpunkt auf der Bayesschen Inferenz und der Theorie der großen Abweichungen. Zusammen mit der Implementierung eines Bayesschen Modellvergleichs und Methoden zur Parameterschätzung für Diffusionsmodelle werden ergänzende Methoden en- twickelt, welche auf unterschiedliche Observablen und der Theorie der großen Abweichungen basieren, um stochastische Prozesse zu klassifizieren und wichtige Informationen zu erhalten. Die Bayessche Analyse der Bewegungsdaten von Mikrometerpartikeln, welche in Mucin- hydrogelen mit verschiedenen pH-Werten verfolgt wurden, enthüllte mehrere interessante Eigenschaften und wir haben z.B. Einsichten darüber gewonnen, wie der Übergang von basischen zu sauren pH-Werten die Heterogenität des Hydrogels erhöht und Phasentrennung einsetzen kann, was zur beobachteten Nicht-Ergodizität (Inäquivalenz von Zeit- und En- semblemittelwert) und nicht-Gaußscher Wahrscheinlichkeitsdichte führt. Mit einer auf der Theorie der großen Abweichungen basierenden Analyse konnten wir, z.B., nicht-Gaußsches Verhalten bei der scheinbaren Brownschen Diffusion von Partikeln in wässriger Lösung, anisotrope Bewegung von Partikeln in Mucin bei neutralem pH-Wert und Kurzzeitkorrela- tionen in Klimadaten detektieren. Folglich werden durch die Anwendung der entwickelten Methoden auf biologische und meteorologische Daten entscheidende Informationen über die zugrundeliegenden stochastischen Prozesse gesammelt und bedeutende Erkenntnisse für das Verständnis der Eigenschaften dieser Systeme erhalten.
KW  - anomalous diffusion
KW  - Bayesian inference
KW  - large deviation theory
KW  - statistical physics
Y1  - 2020
ER  - 
TY  - JOUR
A1  - Wang, Wei
A1  - Seno, Flavio
A1  - Sokolov, Igor M.
A1  - Chechkin, Aleksei V.
A1  - Metzler, Ralf
T1  - Unexpected crossovers in correlated random-diffusivity processes
JF  - New Journal of Physics
N2  - The passive and active motion of micron-sized tracer particles in crowded liquids and inside living biological cells is ubiquitously characterised by 'viscoelastic' anomalous diffusion, in which the increments of the motion feature long-ranged negative and positive correlations. While viscoelastic anomalous diffusion is typically modelled by a Gaussian process with correlated increments, so-called fractional Gaussian noise, an increasing number of systems are reported, in which viscoelastic anomalous diffusion is paired with non-Gaussian displacement distributions. Following recent advances in Brownian yet non-Gaussian diffusion we here introduce and discuss several possible versions of random-diffusivity models with long-ranged correlations. While all these models show a crossover from non-Gaussian to Gaussian distributions beyond some correlation time, their mean squared displacements exhibit strikingly different behaviours: depending on the model crossovers from anomalous to normal diffusion are observed, as well as a priori unexpected dependencies of the effective diffusion coefficient on the correlation exponent. Our observations of the non-universality of random-diffusivity viscoelastic anomalous diffusion are important for the analysis of experiments and a better understanding of the physical origins of 'viscoelastic yet non-Gaussian' diffusion.
KW  - diffusion
KW  - anomalous diffusion
KW  - non-Gaussianity
KW  - fractional Brownian motion
Y1  - 2020
U6  - https://doi.org/10.1088/1367-2630/aba390
SN  - 1367-2630
VL  - 22
PB  - Dt. Physikalische Ges.
CY  - Bad Honnef
ER  - 
TY  - GEN
A1  - Wang, Wei
A1  - Seno, Flavio
A1  - Sokolov, Igor M.
A1  - Chechkin, Aleksei V.
A1  - Metzler, Ralf
T1  - Unexpected crossovers in correlated random-diffusivity processes
N2  - The passive and active motion of micron-sized tracer particles in crowded liquids and inside living biological cells is ubiquitously characterised by 'viscoelastic' anomalous diffusion, in which the increments of the motion feature long-ranged negative and positive correlations. While viscoelastic anomalous diffusion is typically modelled by a Gaussian process with correlated increments, so-called fractional Gaussian noise, an increasing number of systems are reported, in which viscoelastic anomalous diffusion is paired with non-Gaussian displacement distributions. Following recent advances in Brownian yet non-Gaussian diffusion we here introduce and discuss several possible versions of random-diffusivity models with long-ranged correlations. While all these models show a crossover from non-Gaussian to Gaussian distributions beyond some correlation time, their mean squared displacements exhibit strikingly different behaviours: depending on the model crossovers from anomalous to normal diffusion are observed, as well as a priori unexpected dependencies of the effective diffusion coefficient on the correlation exponent. Our observations of the non-universality of random-diffusivity viscoelastic anomalous diffusion are important for the analysis of experiments and a better understanding of the physical origins of 'viscoelastic yet non-Gaussian' diffusion.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1006 
KW  - diffusion
KW  - anomalous diffusion
KW  - non-Gaussianity
KW  - fractional Brownian motion
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-480049
SN  - 1866-8372
IS  - 1006
ER  - 
TY  - JOUR
A1  - Sandev, Trifce
A1  - Iomin, Alexander
A1  - Kantz, Holger
A1  - Metzler, Ralf
A1  - Chechkin, Aleksei V.
T1  - Comb Model with Slow and Ultraslow Diffusion
JF  - Mathematical modelling of natural phenomena
N2  - We consider a generalized diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyze the probability distribution functions and we derive the mean squared displacement in x and y directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both x and y directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.
KW  - comb-like model
KW  - anomalous diffusion
KW  - mean squared displacement
KW  - probability density function
Y1  - 2016
U6  - https://doi.org/10.1051/mmnp/201611302
SN  - 0973-5348
SN  - 1760-6101
VL  - 11
SP  - 18
EP  - 33
PB  - EDP Sciences
CY  - Les Ulis
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  - GEN
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
T2  - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe
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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 755 
KW  - anomalous diffusion
KW  - fractional Brownian motion
KW  - reflecting boundary conditions
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436665
SN  - 1866-8372
IS  - 755
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  - GEN
A1  - Sposini, Vittoria
A1  - Metzler, Ralf
A1  - Oshanin, Gleb
T1  - Single-trajectory spectral analysis of scaled Brownian motion
T2  - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe
N2  - Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 753 
KW  - diffusion
KW  - anomalous diffusion
KW  - power spectral analysis
KW  - single trajectory analysis
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436522
SN  - 1866-8372
IS  - 753
ER  - 
TY  - JOUR
A1  - Sposini, Vittoria
A1  - Metzler, Ralf
A1  - Oshanin, Gleb
T1  - Single-trajectory spectral analysis of scaled Brownian motion
JF  - New Journal of Physics
N2  - Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.
KW  - diffusion
KW  - anomalous diffusion
KW  - power spectral analysis
KW  - single trajectory analysis
Y1  - 2019
U6  - https://doi.org/10.1088/1367-2630/ab2f52
SN  - 1367-2630
VL  - 21
PB  - Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics
CY  - Bad Honnef und London
ER  - 
TY  - GEN
A1  - Ślęzak, Jakub
A1  - Metzler, Ralf
A1  - Magdziarz, Marcin
T1  - Codifference can detect ergodicity breaking and non-Gaussianity
T2  - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe
N2  - We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a nonlinear analogue of the mean squared displacement.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 748 
KW  - diffusion
KW  - anomalous diffusion
KW  - stochastic time series
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436178
IS  - 748
ER  - 
TY  - JOUR
A1  - Ślęzak, Jakub
A1  - Metzler, Ralf
A1  - Magdziarz, Marcin
T1  - Codifference can detect ergodicity breaking and non-Gaussianity
JF  - New Journal of Physics
N2  - We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a nonlinear analogue of the mean squared displacement.
KW  - diffusion
KW  - stochastic time series
KW  - anomalous diffusion
Y1  - 2019
U6  - https://doi.org/10.1088/1367-2630/ab13f3
SN  - 1367-2630
VL  - 21
PB  - Deutsche Physikalische Gesellschaft
CY  - Bad Honnef
ER  - 
TY  - GEN
A1  - Chechkin, Aleksei V.
A1  - Zaid, Irwin M.
A1  - Lomholt, Michael A.
A1  - Sokolov, Igor M.
A1  - Metzler, Ralf
T1  - Bulk-mediated surface diffusion on a cylinder in the fast exchange limit
T2  - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe
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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 593 
KW  - Bulk-mediated diffusion;
KW  - anomalous diffusion
KW  - Levy flights
KW  - stochastic processes
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-415480
SN  - 1866-8372
IS  - 593
SP  - 114
EP  - 126
ER  - 
TY  - GEN
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
T2  - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe
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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 507 
KW  - anomalous diffusion
KW  - truncated power-law correlated noise
KW  - lipid bilayer membrane dynamics
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-422590
SN  - 1866-8372
IS  - 507
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  - Ślęzak, Jakub
A1  - Metzler, Ralf
A1  - Magdziarz, Marcin
T1  - Superstatistical generalised Langevin equation
BT  - non-Gaussian viscoelastic anomalous diffusion
JF  - New Journal of Physics
N2  - Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.
KW  - anomalous diffusion
KW  - generalised langevin equation
KW  - superstatistics
KW  - non-Gaussian diffusion
Y1  - 2018
U6  - https://doi.org/10.1088/1367-2630/aaa3d4
SN  - 1367-2630
VL  - 20
IS  - 023026
SP  - 1
EP  - 25
PB  - Deutsche Physikalische Gesellschaft / Institute of Physics
CY  - Bad Honnef und London
ER  - 
TY  - GEN
A1  - Ślęzak, Jakub
A1  - Metzler, Ralf
A1  - Magdziarz, Marcin
T1  - Superstatistical generalised Langevin equation
BT  - non-Gaussian viscoelastic anomalous diffusion
N2  - Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 413 
KW  - anomalous diffusion
KW  - generalised langevin equation
KW  - superstatistics
KW  - non-Gaussian diffusion
Y1  - 2018
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-409315
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  - 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  - 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  - Sandev, Trifce
A1  - Chechkin, Aleksei V.
A1  - Kantz, Holger
A1  - Metzler, Ralf
T1  - Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel
JF  - Fractional calculus and applied analysis : an international journal for theory and applications
N2  - 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.
KW  - continuous time random walk (CTRW)
KW  - Fokker-Planck-Smoluchowski equation
KW  - Mittag-Leffler functions
KW  - anomalous diffusion
KW  - multi-scaling
Y1  - 2015
U6  - https://doi.org/10.1515/fca-2015-0059
SN  - 1311-0454
SN  - 1314-2224
VL  - 18
IS  - 4
SP  - 1006
EP  - 1038
PB  - De Gruyter
CY  - Berlin
ER  - 
TY  - JOUR
A1  - Safdari, Hadiseh
A1  - Cherstvy, Andrey G.
A1  - Chechkin, Aleksei V.
A1  - Thiel, Felix
A1  - Sokolov, Igor M.
A1  - Metzler, Ralf
T1  - Quantifying the non-ergodicity of scaled Brownian motion
JF  - Journal of physics : A, Mathematical and theoretical
N2  - 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.
KW  - scaled Brownian motion
KW  - anomalous diffusion
KW  - ageing
Y1  - 2015
U6  - https://doi.org/10.1088/1751-8113/48/37/375002
SN  - 1751-8113
SN  - 1751-8121
VL  - 48
IS  - 37
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Shin, Jaeoh
A1  - Cherstvy, Andrey G.
A1  - Metzler, Ralf
T1  - Self-subdiffusion in solutions of star-shaped crowders: non-monotonic effects of inter-particle interactions
JF  - New journal of physics : the open-access journal for physics
N2  - We examine by extensive computer simulations the self-diffusion of anisotropic star-like particles in crowded two-dimensional solutions. We investigate the implications of the area coverage fraction phi of the crowders and the crowder-crowder adhesion properties on the regime of transient anomalous diffusion. We systematically compute the mean squared displacement (MSD) of the particles, their time averaged MSD, and the effective diffusion coefficient. The diffusion is ergodic in the limit of long traces, such that the mean time averaged MSD converges towards the ensemble averaged MSD, and features a small residual amplitude spread of the time averaged MSD from individual trajectories. At intermediate time scales, we quantify the anomalous diffusion in the system. Also, we show that the translational-but not rotational-diffusivity of the particles Dis a nonmonotonic function of the attraction strength between them. Both diffusion coefficients decrease as the power law D(phi) similar to (1 - phi/phi*)(2 ... 2.4) with the area fraction phi occupied by the crowders and the critical value phi*. Our results might be applicable to rationalising the experimental observations of non-Brownian diffusion for a number of standard macromolecular crowders used in vitro to mimic the cytoplasmic conditions of living cells.
KW  - anomalous diffusion
KW  - crowded fluids
KW  - stochastic processes
Y1  - 2015
U6  - https://doi.org/10.1088/1367-2630/17/11/113028
SN  - 1367-2630
VL  - 17
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Palyulin, Vladimir V.
A1  - Metzler, Ralf
T1  - Speeding up the first-passage for subdiffusion by introducing a finite potential barrier
JF  - Journal of physics : A, Mathematical and theoretical
N2  - We show that for a subdiffusive continuous time random walk with scale-free waiting time distribution the first-passage dynamics on a finite interval can be optimized by introduction of a piecewise linear potential barrier. Analytical results for the survival probability and first-passage density based on the fractional Fokker-Planck equation are shown to agree well with Monte Carlo simulations results. As an application we discuss an improved design for efficient translocation of gradient copolymers compared to homopolymer translocation in a quasi-equilibrium approximation.
KW  - first passage
KW  - anomalous diffusion
KW  - potential landscape
KW  - polymer translocation
Y1  - 2014
U6  - https://doi.org/10.1088/1751-8113/47/3/032002
SN  - 1751-8113
SN  - 1751-8121
VL  - 47
IS  - 3
PB  - IOP Publ. Ltd.
CY  - Bristol
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  - 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  - 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  - Sandev, Trifce
A1  - Metzler, Ralf
A1  - Tomovski, Zivorad
T1  - Velocity and displacement correlation functions for fractional generalized Langevin equations
JF  - Fractional calculus and applied analysis : an international journal for theory and applications
N2  - We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle.
KW  - fractional generalized Langevin equation
KW  - frictional memory kernel
KW  - variances
KW  - mean square displacement
KW  - anomalous diffusion
Y1  - 2012
U6  - https://doi.org/10.2478/s13540-012-0031-2
SN  - 1311-0454
VL  - 15
IS  - 3
SP  - 426
EP  - 450
PB  - Versita
CY  - Warsaw
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  - 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  - GEN
A1  - Metzler, Ralf
A1  - Cherstvy, Andrey G.
A1  - Chechkin, Aleksei V.
A1  - Bodrova, Anna S.
T1  - Ultraslow scaled Brownian motion
T2  - 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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 188 
KW  - anomalous diffusion
KW  - stochastic processes
KW  - ageing
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-78618
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  - 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  - 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  -