Ultraslow diffusion and weak ergodicity breaking in right triangular billiards

2017 ◽  
Vol 95 (3) ◽  
Author(s):  
Junxiang Huang ◽  
Hong Zhao
Keyword(s):  
Weak Ergodicity ◽  
Weak Ergodicity Breaking ◽  
Ergodicity Breaking

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.

scholarly journals Time-average based on scaling law in anomalous diffusions

2015 ◽  
Vol 29 (13) ◽  
pp. 1550059 ◽  
Author(s):  
Hyun-Joo Kim
Keyword(s):  
Single Molecule ◽  
Scaling Law ◽  
Lag Time ◽  
Markovian Model ◽  
Weak Ergodicity ◽  
Single Molecule Tracking ◽  
Memory Enhancement ◽  
Weak Ergodicity Breaking ◽  
Time Average ◽  
Ergodicity Breaking

To solve the obscureness in measurement brought about from the weak ergodicity breaking appeared in anomalous diffusions, we have suggested the time-averaged mean squared displacement (MSD) [Formula: see text] with an integral interval depending linearly on the lag time τ. For the continuous time random walk describing a subdiffusive behavior, we have found that [Formula: see text] like that of the ensemble-averaged MSD, which makes it be possible to measure the proper exponent values through time-average in experiments like a single molecule tracking. Also, we have found that it has originated from the scaling nature of the MSD at an aging time in anomalous diffusion and confirmed them through numerical results of the other microscopic non-Markovian model showing subdiffusions and superdiffusions with the origin of memory enhancement.

scholarly journals Weak Ergodicity Breaking of Receptor Motion in Living Cells Stemming from Random Diffusivity

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
Carlo Manzo ◽  
Juan A. Torreno-Pina ◽  
Pietro Massignan ◽  
Gerald J. Lapeyre ◽  
Maciej Lewenstein ◽  
...  
Keyword(s):  
Living Cells ◽  
Weak Ergodicity ◽  
Weak Ergodicity Breaking ◽  
Ergodicity Breaking

scholarly journals Strong ergodicity breaking in aging of mean-field spin glasses

2020 ◽  
Vol 117 (30) ◽  
pp. 17522-17527
Author(s):  
Massimo Bernaschi ◽  
Alain Billoire ◽  
Andrea Maiorano ◽  
Giorgio Parisi ◽  
Federico Ricci-Tersenghi
Keyword(s):  
Spin Glass ◽  
Spin Glasses ◽  
Mean Field ◽  
Relaxation Processes ◽  
Large Set ◽  
Weak Ergodicity ◽  
Strong Ergodicity ◽  
Weak Ergodicity Breaking ◽  
Ergodicity Breaking ◽  
Mean Field Spin Glasses

Out-of-equilibrium relaxation processes show aging if they become slower as time passes. Aging processes are ubiquitous and play a fundamental role in the physics of glasses and spin glasses and in other applications (e.g., in algorithms minimizing complex cost/loss functions). The theory of aging in the out-of-equilibrium dynamics of mean-field spin glass models has achieved a fundamental role, thanks to the asymptotic analytic solution found by Cugliandolo and Kurchan. However, this solution is based on assumptions (e.g., the weak ergodicity breaking hypothesis) which have never been put under a strong test until now. In the present work, we present the results of an extraordinary large set of numerical simulations of the prototypical mean-field spin glass models, namely the Sherrington–Kirkpatrick and the Viana–Bray models. Thanks to a very intensive use of graphics processing units (GPUs), we have been able to run the latter model for more than264spin updates and thus safely extrapolate the numerical data both in the thermodynamical limit and in the large times limit. The measurements of the two-times correlation functions in isothermal aging after a quench from a random initial configuration to a temperatureT<Tcprovides clear evidence that, at large times, such correlations do not decay to zero as expected by assuming weak ergodicity breaking. We conclude that strong ergodicity breaking takes place in mean-field spin glasses aging dynamics which, asymptotically, takes place in a confined configurational space. Theoretical models for the aging dynamics need to be revised accordingly.

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