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# Above under and beyond brownian motion talk

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This talk was Dedicated to Einstein's miracle year at his 26

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### Above under and beyond brownian motion talk

1. 1. Above and Under Brownian MotionBrownian Motion , Fractional Brownian Motion , Levy Flight, and beyond Seminar Talk at Beijing Normal University Xiong Wang 王雄 Centre for Chaos and Complex Networks City University of Hong Kong 1
2. 2. Outline Discrete Time Random walks Ordinary random walks Lévy flights Generalized central limit theorem Stable distribution Continuous time random walks Ordinary Diffusion Lévy Flights Fractional Brownian motion (subdiffusion) Ambivalent processes 2
3. 3. Part 1Discrete Time Random walks 3
4. 4. Ordinary random walks 4
5. 5. Central limit theorem 5
6. 6. Lévy ﬂights
7. 7. Lévy ﬂight scalessuperdiffusively
8. 8. Part 2Generalized central limittheorem 9
9. 9. Generalized central limittheorem A generalization due to Gnedenko and Kolmogorov states that the sum of a number of random variables with power-law tail distributions decreasing as 1 / | x | α + 1 where 0 < α < 2 (and therefore having infinite variance) will tend to a stable distribution f(x;α,0,c,0) as the number of variables grows. 10
10. 10. Stable distribution In probability theory, a random variable is said to be stable (or to have a stable distribution) if it has the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The stable distribution family is also sometimes referred to as the Lévy alpha- stable distribution. 11
11. 11.  Such distributions form a four-parameter family of continuous probability distributions parametrized by location and scale parameters μ and c, respectively, and two shape parameters β and α, roughly corresponding to measures of asymmetry and concentration, respectively (see the figures). C:chaosTalklevyStableDensityFunction.cdf
12. 12. Characteristic function ofStable distribution A random variable X is called stable if its characteristic function is given by 13
13. 13. Symmetric α-stable distributionswith unit scale factor 14
14. 14. Skewed centered stabledistributions with different β 15
15. 15. Unified normal and power law For α = 2 the distribution reduces to a Gaussian distribution with variance σ2 = 2c2 and mean μ; the skewness parameter β has no effect The asymptotic behavior is described, for α < 2 16
16. 16. Log-log plot of skewed centered stable distribution PDFs showing thepower law behavior for large x. Again the slope of the linear portionsis equal to -(α+1)
17. 17. Part 1Continuous time random walks 18
18. 18. spatial displacement ∆x and atemporal increment ∆t
19. 19. Ordinary Diffusion
20. 20. Lévy Flights
21. 21. Fractional Brownian motion(subdiffusion)
22. 22. 1d Fractional Brownian motion
23. 23. 2d Fractional Brownian motion
24. 24. Ambivalent processes
25. 25. Concluding RemarksThe ratio of the exponents α/β resembles theinterplay between sub- and superdiffusion.For β < 2α the ambivalent CTRW is effectivelysuperdiffusive,for β > 2α effectively subdiffusive.For β = 2α the process exhibits the samescaling as ordinary Brownian motion, despitethe crucial difference of inﬁnite moments and anon-Gaussian shape of the pdf W(x, t). 28
26. 26. Xiong Wang 王雄Centre for Chaos and Complex NetworksCity University of Hong KongEmail: wangxiong8686@gmail.com 29