Anomalous diffusion: temporal non-Markovianity and weak ergodicity breaking

2009 ◽  
Vol 2009 (08) ◽  
pp. P08025 ◽  
Author(s):  
Bartłomiej Dybiec
Keyword(s):  
Anomalous Diffusion ◽  
Weak Ergodicity ◽  
Weak Ergodicity Breaking ◽  
Ergodicity Breaking

scholarly journals Weak ergodicity breaking and ageing in anomalous diffusion

2015 ◽  
Vol 36 ◽  
pp. 1560007 ◽  
Author(s):  
Ralf Metzler
Keyword(s):  
Anomalous Diffusion ◽  
Large Scale ◽  
Diffusion Processes ◽  
Tracer Particle ◽  
Single Particle Tracking ◽  
Weak Ergodicity ◽  
Ergodic Processes ◽  
Mean Squared Displacement ◽  
Weak Ergodicity Breaking ◽  
Ergodicity Breaking

Modern single particle tracking techniques and many large scale simulations produce time series r(t) of the position of a tracer particle. Standardly these are evaluated in terms of the time averaged mean squared displacement. For ergodic processes such as Brownian motion, one can interpret the results of such an analysis in terms of the known theories for the corresponding ensemble averaged mean squared displacement, if only the measurement time is sufficiently long. In anomalous diffusion processes, that are widely observed over many orders of magnitude, the equivalence between (long) time and ensemble averages may be broken (weak ergodicity breaking). In such cases the time averages may no longer be interpreted in terms of ensemble theories. Here we collect some recent results on weakly non-ergodic systems with respect to the time averaged mean squared displacement and the inherent irreproducibility of individual measurements. We also address the phenomenon of ageing, the dependence of physical observables on the time span between initial preparation of the system and the start of the measurement.

scholarly journals Modeling Anomalous Diffusion by a Subordinated Integrated Brownian Motion

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Long Shi ◽  
Aiguo Xiao
Keyword(s):  
Brownian Motion ◽  
Continuous Time ◽  
Waiting Times ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Weak Ergodicity ◽  
Normal Diffusion ◽  
Weak Ergodicity Breaking ◽  
The Mean ◽  
Ergodicity Breaking

We consider a particular type of continuous time random walk where the jump lengths between subsequent waiting times are correlated. In a continuum limit, the process can be defined by an integrated Brownian motion subordinated by an inverse α-stable subordinator. We compute the mean square displacement of the proposed process and show that the process exhibits subdiffusion when 0<α<1/3, normal diffusion when α=1/3, and superdiffusion when 1/3<α<1. The time-averaged mean square displacement is also employed to show weak ergodicity breaking occurring in the proposed process. An extension to the fractional case is also considered.

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