@misc{VilkAghionAvgaretal.2022, author = {Vilk, Ohad and Aghion, Erez and Avgar, Tal and Beta, Carsten and Nagel, Oliver and Sabri, Adal and Sarfati, Raphael and Schwartz, Daniel K. and Weiß, Matthias and Krapf, Diego and Nathan, Ran and Metzler, Ralf and Assaf, Michael}, title = {Unravelling the origins of anomalous diffusion}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1303}, issn = {1866-8372}, doi = {10.25932/publishup-57764}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-577643}, pages = {16}, year = {2022}, abstract = {Anomalous diffusion or, more generally, anomalous transport, with nonlinear dependence of the mean-squared displacement on the measurement time, is ubiquitous in nature. It has been observed in processes ranging from microscopic movement of molecules to macroscopic, large-scale paths of migrating birds. Using data from multiple empirical systems, spanning 12 orders of magnitude in length and 8 orders of magnitude in time, we employ a method to detect the individual underlying origins of anomalous diffusion and transport in the data. This method decomposes anomalous transport into three primary effects: long-range correlations ("Joseph effect"), fat-tailed probability density of increments ("Noah effect"), and nonstationarity ("Moses effect"). We show that such a decomposition of real-life data allows us to infer nontrivial behavioral predictions and to resolve open questions in the fields of single-particle tracking in living cells and movement ecology.}, language = {en} } @article{VilkAghionAvgaretal.2022, author = {Vilk, Ohad and Aghion, Erez and Avgar, Tal and Beta, Carsten and Nagel, Oliver and Sabri, Adal and Sarfati, Raphael and Schwartz, Daniel K. and Weiß, Matthias and Krapf, Diego and Nathan, Ran and Metzler, Ralf and Assaf, Michael}, title = {Unravelling the origins of anomalous diffusion}, series = {Physical review research / American Physical Society}, volume = {4}, journal = {Physical review research / American Physical Society}, number = {3}, publisher = {American Physical Society}, address = {College Park, MD}, issn = {2643-1564}, doi = {10.1103/PhysRevResearch.4.033055}, pages = {16}, year = {2022}, abstract = {Anomalous diffusion or, more generally, anomalous transport, with nonlinear dependence of the mean-squared displacement on the measurement time, is ubiquitous in nature. It has been observed in processes ranging from microscopic movement of molecules to macroscopic, large-scale paths of migrating birds. Using data from multiple empirical systems, spanning 12 orders of magnitude in length and 8 orders of magnitude in time, we employ a method to detect the individual underlying origins of anomalous diffusion and transport in the data. This method decomposes anomalous transport into three primary effects: long-range correlations ("Joseph effect"), fat-tailed probability density of increments ("Noah effect"), and nonstationarity ("Moses effect"). We show that such a decomposition of real-life data allows us to infer nontrivial behavioral predictions and to resolve open questions in the fields of single-particle tracking in living cells and movement ecology.}, language = {en} } @article{PetreskaPejovSandevetal.2022, author = {Petreska, Irina and Pejov, Ljupco and Sandev, Trifce and Kocarev, Ljupčo and Metzler, Ralf}, title = {Tuning of the dielectric relaxation and complex susceptibility in a system of polar molecules: a generalised model based on rotational diffusion with resetting}, series = {Fractal and fractional}, volume = {6}, journal = {Fractal and fractional}, number = {2}, publisher = {MDPI AG, Fractal Fract Editorial Office}, address = {Basel}, issn = {2504-3110}, doi = {10.3390/fractalfract6020088}, pages = {23}, year = {2022}, abstract = {The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.}, language = {en} } @article{SposiniKrapfMarinarietal.2022, author = {Sposini, Vittoria and Krapf, Diego and Marinari, Enzo and Sunyer, Raimon and Ritort, Felix and Taheri, Fereydoon and Selhuber-Unkel, Christine and Benelli, Rebecca and Weiss, Matthias and Metzler, Ralf and Oshanin, Gleb}, title = {Towards a robust criterion of anomalous diffusion}, series = {Communications Physics}, volume = {5}, journal = {Communications Physics}, publisher = {Springer Nature}, address = {London}, issn = {2399-3650}, doi = {10.1038/s42005-022-01079-8}, pages = {10}, year = {2022}, abstract = {Anomalous-diffusion, the departure of the spreading dynamics of diffusing particles from the traditional law of Brownian-motion, is a signature feature of a large number of complex soft-matter and biological systems. Anomalous-diffusion emerges due to a variety of physical mechanisms, e.g., trapping interactions or the viscoelasticity of the environment. However, sometimes systems dynamics are erroneously claimed to be anomalous, despite the fact that the true motion is Brownian—or vice versa. This ambiguity in establishing whether the dynamics as normal or anomalous can have far-reaching consequences, e.g., in predictions for reaction- or relaxation-laws. Demonstrating that a system exhibits normal- or anomalous-diffusion is highly desirable for a vast host of applications. Here, we present a criterion for anomalous-diffusion based on the method of power-spectral analysis of single trajectories. The robustness of this criterion is studied for trajectories of fractional-Brownian-motion, a ubiquitous stochastic process for the description of anomalous-diffusion, in the presence of two types of measurement errors. In particular, we find that our criterion is very robust for subdiffusion. Various tests on surrogate data in absence or presence of additional positional noise demonstrate the efficacy of this method in practical contexts. Finally, we provide a proof-of-concept based on diverse experiments exhibiting both normal and anomalous-diffusion.}, language = {en} } @misc{SposiniKrapfMarinarietal.2022, author = {Sposini, Vittoria and Krapf, Diego and Marinari, Enzo and Sunyer, Raimon and Ritort, Felix and Taheri, Fereydoon and Selhuber-Unkel, Christine and Benelli, Rebecca and Weiss, Matthias and Metzler, Ralf and Oshanin, Gleb}, title = {Towards a robust criterion of anomalous diffusion}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1313}, issn = {1866-8372}, doi = {10.25932/publishup-58596}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-585967}, pages = {10}, year = {2022}, abstract = {Anomalous-diffusion, the departure of the spreading dynamics of diffusing particles from the traditional law of Brownian-motion, is a signature feature of a large number of complex soft-matter and biological systems. Anomalous-diffusion emerges due to a variety of physical mechanisms, e.g., trapping interactions or the viscoelasticity of the environment. However, sometimes systems dynamics are erroneously claimed to be anomalous, despite the fact that the true motion is Brownian—or vice versa. This ambiguity in establishing whether the dynamics as normal or anomalous can have far-reaching consequences, e.g., in predictions for reaction- or relaxation-laws. Demonstrating that a system exhibits normal- or anomalous-diffusion is highly desirable for a vast host of applications. Here, we present a criterion for anomalous-diffusion based on the method of power-spectral analysis of single trajectories. The robustness of this criterion is studied for trajectories of fractional-Brownian-motion, a ubiquitous stochastic process for the description of anomalous-diffusion, in the presence of two types of measurement errors. In particular, we find that our criterion is very robust for subdiffusion. Various tests on surrogate data in absence or presence of additional positional noise demonstrate the efficacy of this method in practical contexts. Finally, we provide a proof-of-concept based on diverse experiments exhibiting both normal and anomalous-diffusion.}, language = {en} } @article{VinodCherstvyMetzleretal.2022, author = {Vinod, Deepak and Cherstvy, Andrey G. and Metzler, Ralf and Sokolov, Igor M.}, title = {Time-averaging and nonergodicity of reset geometric Brownian motion with drift}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {106}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {3}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.106.034137}, pages = {36}, year = {2022}, abstract = {How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Delta) not equal TAMSD(Delta) and Variance(Delta) not equal TAMSD(Delta) at short lag times Delta and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Delta/T << 1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.}, language = {en} } @misc{XuZhouMetzleretal.2022, author = {Xu, Pengbo and Zhou, Tian and Metzler, Ralf and Deng, Weihua}, title = {Stochastic harmonic trapping of a L{\´e}vy walk: transport and first-passage dynamics under soft resetting strategies}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {1866-8372}, doi = {10.25932/publishup-56040}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-560402}, 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{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{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{WangCherstvyMetzleretal.2022, author = {Wang, Wei and Cherstvy, Andrey G. and Metzler, Ralf and Sokolov, Igor M.}, title = {Restoring ergodicity of stochastically reset anomalous-diffusion processes}, series = {Physical Review Research}, volume = {4}, journal = {Physical Review Research}, edition = {1}, publisher = {American Physical Society}, address = {College Park, Maryland, United States}, issn = {2643-1564}, doi = {10.1103/PhysRevResearch.4.013161}, pages = {013161-1 -- 013161-13}, year = {2022}, abstract = {How do different reset protocols affect ergodicity of a diffusion process in single-particle-tracking experiments? We here address the problem of resetting of an arbitrary stochastic anomalous-diffusion process (ADP) from the general mathematical points of view and assess ergodicity of such reset ADPs for an arbitrary resetting protocol. The process of stochastic resetting describes the events of the instantaneous restart of a particle's motion via randomly distributed returns to a preset initial position (or a set of those). The waiting times of such resetting events obey the Poissonian, Gamma, or more generic distributions with specified conditions regarding the existence of moments. Within these general approaches, we derive general analytical results and support them by computer simulations for the behavior of the reset mean-squared displacement (MSD), the new reset increment-MSD (iMSD), and the mean reset time-averaged MSD (TAMSD). For parental nonreset ADPs with the MSD(t)∝ tμ we find a generic behavior and a switch of the short-time growth of the reset iMSD and mean reset TAMSDs from ∝ _μ for subdiffusive to ∝ _1 for superdiffusive reset ADPs. The critical condition for a reset ADP that recovers its ergodicity is found to be more general than that for the nonequilibrium stationary state, where obviously the iMSD and the mean TAMSD are equal. The consideration of the new statistical quantifier, the iMSD—as compared to the standard MSD—restores the ergodicity of an arbitrary reset ADP in all situations when the μth moment of the waiting-time distribution of resetting events is finite. Potential applications of these new resetting results are, inter alia, in the area of biophysical and soft-matter systems.}, language = {en} }