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  -