@article{EliazarMetzlerReuveni2019, author = {Eliazar, Iddo and Metzler, Ralf and Reuveni, Shlomi}, title = {Poisson-process limit laws yield Gumbel max-min and min-max}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {100}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.100.022129}, pages = {12}, year = {2019}, abstract = {"A chain is only as strong as its weakest link" says the proverb. But what about a collection of statistically identical chains: How long till all chains fail? The answer to this question is given by the max-min of a matrix whose (i,j)entry is the failure time of link j of chain i: take the minimum of each row, and then the maximum of the rows' minima. The corresponding min-max is obtained by taking the maximum of each column, and then the minimum of the columns' maxima. The min-max applies to the storage of critical data. Indeed, consider multiple backup copies of a set of critical data items, and consider the (i,j) matrix entry to be the time at which item j on copy i is lost; then, the min-max is the time at which the first critical data item is lost. In this paper we address random matrices whose entries are independent and identically distributed random variables. We establish Poisson-process limit laws for the row's minima and for the columns' maxima. Then, we further establish Gumbel limit laws for the max-min and for the min-max. The limit laws hold whenever the entries' distribution has a density, and yield highly applicable approximation tools and design tools for the max-min and min-max of large random matrices. A brief of the results presented herein is given in: Gumbel central limit theorem for max-min and min-max}, language = {en} } @article{EliazarMetzlerReuveni2019, author = {Eliazar, Iddo and Metzler, Ralf and Reuveni, Shlomi}, title = {Gumbel central limit theorem for max-min and min-max}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {100}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.100.020104}, pages = {6}, year = {2019}, abstract = {The max-min and min-max of matrices arise prevalently in science and engineering. However, in many real-world situations the computation of the max-min and min-max is challenging as matrices are large and full information about their entries is lacking. Here we take a statistical-physics approach and establish limit laws—akin to the central limit theorem—for the max-min and min-max of large random matrices. The limit laws intertwine random-matrix theory and extreme-value theory, couple the matrix dimensions geometrically, and assert that Gumbel statistics emerge irrespective of the matrix entries' distribution. Due to their generality and universality, as well as their practicality, these results are expected to have a host of applications in the physical sciences and beyond.}, 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{TeomyMetzler2019, author = {Teomy, Eial and Metzler, Ralf}, title = {Transport in exclusion processes with one-step memory: density dependence and optimal acceleration}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {52}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {38}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ab37e4}, pages = {19}, year = {2019}, abstract = {We study a lattice gas of persistent walkers, in which each site is occupied by at most one particle and the direction each particle attempts to move to depends on its last step. We analyse the mean squared displacement (MSD) of the particles as a function of the particle density and their persistence (the tendency to continue moving in the same direction). For positive persistence the MSD behaves as expected: it increases with the persistence and decreases with the density. However, for strong anti-persistence we find two different regimes, in which the dependence of the MSD on the density is non-monotonic. For very strong anti-persistence there is an optimal density at which the MSD reaches a maximum. In an intermediate regime, the MSD as a function of the density exhibits both a minimum and a maximum, a phenomenon which has not been observed before. We derive a mean-field theory which qualitatively explains this behaviour.}, language = {en} } @article{TeomyMetzler2019, author = {Teomy, Eial and Metzler, Ralf}, title = {Correlations and transport in exclusion processes with general finite memory}, series = {Journal of statistical mechanics: theory and experiment}, volume = {2019}, journal = {Journal of statistical mechanics: theory and experiment}, number = {10}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1742-5468}, doi = {10.1088/1742-5468/ab47fb}, pages = {31}, year = {2019}, language = {en} } @article{Metzler2019, author = {Metzler, Ralf}, title = {Brownian motion and beyond: first-passage, power spectrum, non-Gaussianity, and anomalous diffusion}, series = {Journal of statistical mechanics: theory and experiment}, volume = {2019}, journal = {Journal of statistical mechanics: theory and experiment}, number = {11}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1742-5468}, doi = {10.1088/1742-5468/ab4988}, pages = {18}, year = {2019}, abstract = {Brownian motion is a ubiquitous physical phenomenon across the sciences. After its discovery by Brown and intensive study since the first half of the 20th century, many different aspects of Brownian motion and stochastic processes in general have been addressed in Statistical Physics. In particular, there now exists a very large range of applications of stochastic processes in various disciplines. Here we provide a summary of some of the recent developments in the field of stochastic processes, highlighting both the experimental findings and theoretical frameworks.}, 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} } @misc{SposiniMetzlerOshanin2019, author = {Sposini, Vittoria and Metzler, Ralf and Oshanin, Gleb}, title = {Single-trajectory spectral analysis of scaled Brownian motion}, series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe}, number = {753}, issn = {1866-8372}, doi = {10.25932/publishup-43652}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436522}, 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{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} } @misc{GrebenkovMetzlerOshanin2019, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control}, series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe}, number = {527}, issn = {1866-8372}, doi = {10.25932/publishup-42298}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-422989}, pages = {12}, year = {2019}, abstract = {Textbook concepts of diffusion-versus kinetic-control are well-defined for reaction-kinetics involving macroscopic concentrations of diffusive reactants that are adequately described by rate-constants—the inverse of the mean-first-passage-time to the reaction-event. In contradiction, an open important question is whether the mean-first-passage-time alone is a sufficient measure for biochemical reactions that involve nanomolar reactant concentrations. Here, using a simple yet generic, exactly solvable model we study the effect of diffusion and chemical reaction-limitations on the full reaction-time distribution. We show that it has a complex structure with four distinct regimes delineated by three characteristic time scales spanning a window of several decades. Consequently, the reaction-times are defocused: no unique time-scale characterises the reaction-process, diffusion- and kinetic-control can no longer be disentangled, and it is imperative to know the full reaction-time distribution. We introduce the concepts of geometry- and reaction-control, and also quantify each regime by calculating the corresponding reaction depth.}, language = {en} }