Above under and beyond brownian motion talk
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Above under and beyond brownian motion talk




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




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Above under and beyond brownian motion talk Above under and beyond brownian motion talk Presentation Transcript

  • 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
  • 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
  • Part 1Discrete Time Random walks 3 View slide
  • Ordinary random walks 4 View slide
  • Central limit theorem 5
  • Lévy flights
  • Lévy flight scalessuperdiffusively
  • Part 2Generalized central limittheorem 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
  • 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
  •  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
  • Characteristic function ofStable distribution A random variable X is called stable if its characteristic function is given by 13
  • Symmetric α-stable distributionswith unit scale factor 14
  • Skewed centered stabledistributions with different β 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
  • 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)
  • Part 1Continuous time random walks 18
  • spatial displacement ∆x and atemporal increment ∆t
  • Ordinary Diffusion
  • Lévy Flights
  • Fractional Brownian motion(subdiffusion)
  • 1d Fractional Brownian motion
  • 2d Fractional Brownian motion
  • Ambivalent processes
  • 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
  • Xiong Wang 王雄Centre for Chaos and Complex NetworksCity University of Hong KongEmail: wangxiong8686@gmail.com 29