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Quantifying the non-ergodicity of scaled Brownian motion

  • We examine the non-ergodic properties of scaled Brownian motion (SBM), a non-stationary stochastic process with a time dependent diffusivity of the form D(t) similar or equal to t(alpha-1). We compute the ergodicity breaking parameter EB in the entire range of scaling exponents a, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths T and short lag times Delta the EB parameter as function of the scaling exponent a has no divergence at alpha - 1/2 and present the asymptotes for EB in different limits. We generalize the analytical and simulations results for the time averaged and ergodic properties of SBM in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractionalWe examine the non-ergodic properties of scaled Brownian motion (SBM), a non-stationary stochastic process with a time dependent diffusivity of the form D(t) similar or equal to t(alpha-1). We compute the ergodicity breaking parameter EB in the entire range of scaling exponents a, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths T and short lag times Delta the EB parameter as function of the scaling exponent a has no divergence at alpha - 1/2 and present the asymptotes for EB in different limits. We generalize the analytical and simulations results for the time averaged and ergodic properties of SBM in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion.show moreshow less

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Metadaten
Author:Hadiseh Safdari, Andrey G. Cherstvy, Aleksei V. ChechkinORCiDGND, Felix Thiel, Igor M. Sokolov, Ralf MetzlerORCiDGND
DOI:https://doi.org/10.1088/1751-8113/48/37/375002
ISSN:1751-8113 (print)
ISSN:1751-8121 (online)
Parent Title (English):Journal of physics : A, Mathematical and theoretical
Publisher:IOP Publ. Ltd.
Place of publication:Bristol
Document Type:Article
Language:English
Year of first Publication:2015
Year of Completion:2015
Release Date:2017/03/27
Tag:ageing; anomalous diffusion; scaled Brownian motion
Volume:48
Issue:37
Pagenumber:18
Funder:Academy of Finland (Suomen Akatemia, Finland Distinguished Professorship); Deutsche Forschungsgemeinschaft; IMU Berlin Einstein Foundation
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
Peer Review:Referiert