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 - GEN A1 - Ślęzak, Jakub A1 - Burnecki, Krzysztof A1 - Metzler, Ralf T1 - Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems T2 - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe N2 - Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as ‘superstatistics’ or ‘diffusing diffusivity’. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models.Westart from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 765 KW - diffusion KW - Langevin equation KW - Brownian yet non-Gaussian diffusion KW - diffusing diffusivity KW - superstatistics KW - autoregressive models KW - time series analysis KW - codifference Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-437923 SN - 1866-8372 IS - 765 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 - 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 - 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 - JOUR A1 - Ślęzak, Jakub A1 - Burnecki, Krzysztof A1 - Metzler, Ralf T1 - Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems JF - New Journal of Physics N2 - Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as ‘superstatistics’ or ‘diffusing diffusivity’. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models.Westart from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion. KW - diffusion KW - Langevin equation KW - Brownian yet non-Gaussian diffusion KW - diffusing diffusivity KW - superstatistics KW - autoregressive models KW - time series analysis KW - codifference Y1 - 2019 U6 - https://doi.org/10.1088/1367-2630/ab3366 SN - 1367-2630 VL - 21 PB - Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics CY - Bad Honnef und London ER -