@misc{Metzler2020, author = {Metzler, Ralf}, title = {Superstatistics and non-Gaussian diffusion}, series = {The European physical journal special topics}, volume = {229}, journal = {The European physical journal special topics}, number = {5}, publisher = {Springer}, address = {Heidelberg}, issn = {1951-6355}, doi = {10.1140/epjst/e2020-900210-x}, pages = {711 -- 728}, year = {2020}, abstract = {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.}, language = {en} } @article{ŚlęzakMetzlerMagdziarz2018, author = {Ślęzak, Jakub and Metzler, Ralf and Magdziarz, Marcin}, title = {Superstatistical generalised Langevin equation}, series = {New Journal of Physics}, volume = {20}, journal = {New Journal of Physics}, number = {023026}, publisher = {Deutsche Physikalische Gesellschaft / Institute of Physics}, address = {Bad Honnef und London}, issn = {1367-2630}, doi = {10.1088/1367-2630/aaa3d4}, pages = {1 -- 25}, year = {2018}, abstract = {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.}, language = {en} } @article{KrapfMetzler2019, author = {Krapf, Diego and Metzler, Ralf}, title = {Strange interfacial molecular dynamics}, series = {Physics today}, volume = {72}, journal = {Physics today}, number = {9}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0031-9228}, doi = {10.1063/PT.3.4294}, pages = {48 -- 54}, year = {2019}, language = {en} } @article{DahlenburgChechkinSchumeretal.2021, author = {Dahlenburg, Marcus and Chechkin, Aleksei and Schumer, Rina and Metzler, Ralf}, title = {Stochastic resetting by a random amplitude}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {103}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {5}, publisher = {American Physical Society}, address = {Woodbury, NY}, issn = {2470-0045}, doi = {10.1103/PhysRevE.103.052123}, pages = {22}, year = {2021}, abstract = {Stochastic resetting, a diffusive process whose amplitude is reset to the origin at random times, is a vividly studied strategy to optimize encounter dynamics, e.g., in chemical reactions. Here we generalize the resetting step by introducing a random resetting amplitude such that the diffusing particle may be only partially reset towards the trajectory origin or even overshoot the origin in a resetting step. We introduce different scenarios for the random-amplitude stochastic resetting process and discuss the resulting dynamics. Direct applications are geophysical layering (stratigraphy) and population dynamics or financial markets, as well as generic search processes.}, language = {en} } @article{XuZhouMetzleretal.2022, author = {Xu, Pengbo and Zhou, Tian and Metzler, Ralf and Deng, Weihua}, title = {Stochastic harmonic trapping of a L{\´e}vy walk}, series = {New journal of physics : the open-access journal for physics / Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics}, volume = {24}, journal = {New journal of physics : the open-access journal for physics / Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics}, number = {3}, publisher = {Deutsche Physikalische Gesellschaft}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/ac5282}, pages = {1 -- 28}, year = {2022}, abstract = {We introduce and study a L{\´e}vy 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.}, language = {en} } @article{MardoukhiChechkinMetzler2020, author = {Mardoukhi, Yousof and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Spurious ergodicity breaking in normal and fractional Ornstein-Uhlenbeck process}, series = {New Journal of Physics}, volume = {22}, journal = {New Journal of Physics}, publisher = {IOP}, address = {London}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab950b}, pages = {18}, year = {2020}, abstract = {The Ornstein-Uhlenbeck process is a stationary and ergodic Gaussian process, that is fully determined by its covariance function and mean. We show here that the generic definitions of the ensemble- and time-averaged mean squared displacements fail to capture these properties consistently, leading to a spurious ergodicity breaking. We propose to remedy this failure by redefining the mean squared displacements such that they reflect unambiguously the statistical properties of any stochastic process. In particular we study the effect of the initial condition in the Ornstein-Uhlenbeck process and its fractional extension. For the fractional Ornstein-Uhlenbeck process representing typical experimental situations in crowded environments such as living biological cells, we show that the stationarity of the process delicately depends on the initial condition.}, language = {en} } @article{KrapfLukatMarinarietal.2019, author = {Krapf, Diego and Lukat, Nils and Marinari, Enzo and Metzler, Ralf and Oshanin, Gleb and Selhuber-Unkel, Christine and Squarcini, Alessio and Stadler, Lorenz and Weiss, Matthias and Xu, Xinran}, title = {Spectral Content of a Single Non-Brownian Trajectory}, series = {Physical review : X, Expanding access}, volume = {9}, journal = {Physical review : X, Expanding access}, number = {1}, publisher = {American Physical Society}, address = {College Park}, issn = {2160-3308}, doi = {10.1103/PhysRevX.9.011019}, pages = {13}, year = {2019}, abstract = {Time-dependent processes are often analyzed using the power spectral density (PSD) calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble average. Frequently, the available experimental datasets are too small for such ensemble averages, and hence, it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from S(f, T), the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable parametrized by frequency f and observation time T, for a broad family of anomalous diffusions-fractional Brownian motion with Hurst index H-and derive exactly its probability density function. We show that S(f, T) is proportional-up to a random numerical factor whose universal distribution we determine-to the ensemble-averaged PSD. For subdiffusion (H < 1/2), we find that S(f, T) similar to A/f(2H+1) with random amplitude A. In sharp contrast, for superdiffusion (H > 1/2) S(f, T) similar to BT2H-1/f(2) with random amplitude B. Remarkably, for H > 1/2 the PSD exhibits the same frequency dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably, for H > 1/2 the PSD is ageing and is dependent on T. Our predictions for both sub-and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels and by extensive simulations.}, language = {en} } @article{SposiniMetzlerOshanin2019, author = {Sposini, Vittoria and Metzler, Ralf and Oshanin, Gleb}, title = {Single-trajectory spectral analysis of scaled Brownian motion}, series = {New Journal of Physics}, volume = {21}, journal = {New Journal of Physics}, publisher = {Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics}, address = {Bad Honnef und London}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab2f52}, pages = {16}, year = {2019}, abstract = {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.}, language = {en} } @article{GrebenkovMetzlerOshanin2022, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Search efficiency in the Adam-Delbruck reduction-of-dimensionality scenario versus direct diffusive search}, series = {New journal of physics : the open-access journal for physics}, volume = {24}, journal = {New journal of physics : the open-access journal for physics}, number = {8}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/ac8824}, pages = {32}, year = {2022}, abstract = {The time instant-the first-passage time (FPT)-when a diffusive particle (e.g., a ligand such as oxygen or a signalling protein) for the first time reaches an immobile target located on the surface of a bounded three-dimensional domain (e.g., a hemoglobin molecule or the cellular nucleus) is a decisive characteristic time-scale in diverse biophysical and biochemical processes, as well as in intermediate stages of various inter- and intra-cellular signal transduction pathways. Adam and Delbruck put forth the reduction-of-dimensionality concept, according to which a ligand first binds non-specifically to any point of the surface on which the target is placed and then diffuses along this surface until it locates the target. In this work, we analyse the efficiency of such a scenario and confront it with the efficiency of a direct search process, in which the target is approached directly from the bulk and not aided by surface diffusion. We consider two situations: (i) a single ligand is launched from a fixed or a random position and searches for the target, and (ii) the case of 'amplified' signals when N ligands start either from the same point or from random positions, and the search terminates when the fastest of them arrives to the target. For such settings, we go beyond the conventional analyses, which compare only the mean values of the corresponding FPTs. Instead, we calculate the full probability density function of FPTs for both scenarios and study its integral characteristic-the 'survival' probability of a target up to time t. On this basis, we examine how the efficiencies of both scenarios are controlled by a variety of parameters and single out realistic conditions in which the reduction-of-dimensionality scenario outperforms the direct search.}, language = {en} } @article{CherstvyVinodAghionetal.2021, author = {Cherstvy, Andrey G. and Vinod, Deepak and Aghion, Erez and Sokolov, Igor M. and Metzler, Ralf}, title = {Scaled geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {103}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {6}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.103.062127}, pages = {11}, year = {2021}, abstract = {Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic stochastic process of geometric Brownian motion (GBM). The innate property of such models and of real stock-market prices is the roughly exponential growth of prices with time [on average, in crisis-free times]. We here explore the ensemble- and time averages of a multiplicative-noise stochastic process with power-law-like time-dependent volatility, sigma(t) similar to t(alpha), named scaled GBM (SGBM). For SGBM, the mean-squared displacement (MSD) computed for an ensemble of statistically equivalent trajectories can grow faster than exponentially in time, while the time-averaged MSD (TAMSD)-based on a sliding-window averaging along a single trajectory-is always linear at short lag times Delta. The proportionality factor between these the two averages of the time series is Delta/T at short lag times, where T is the trajectory length, similarly to GBM. This discrepancy of the scaling relations and pronounced nonequivalence of the MSD and TAMSD at Delta/T << 1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with s (t)-modulation, the main focus of our analysis. The analytical predictions for the MSD and mean TAMSD for SGBM are in quantitative agreement with the results of stochastic computer simulations.}, language = {en} }