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  - 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  -