
9:52 PM (17 minutes ago)



fyi only
Kolmogorov studies the distribution law for velocity differences far from the boundary of a domain for hydrodynamic equation for a large number of particles. He takes a point p0 in the domain and considers the distribution of velocity differences for paths of particles x^i, i=1,2,…,N.
y^i=v(x^i,t)v(x0,t)
where ‘local homogeneity’ simply means that the reference point choice x0 does not affect the distribution F_N of these y^i. He defines local isotropy where the distribution is invariant to rotations and reflections. His brilliance is to find these nice conditions when the Reynold’s number R=LU/nu where nu is the viscosity is high because nu > 0. Then he does a direct calculation using an ingenious change of coordinates for the energy dispersion which is basically E(v^2) and concludes that the distribution F_N will depend only on this energy dispersion and nu. With these he finally argues that under conditions implied by his situation F_N does not depend at all on nu.
Kolmogorov’s law is that energy in a certain wave number (k) range is uniquely determined by dissipation energy e0 and viscosity nu
E(k) = e0^{2/3} k^{5/3} f(k*eta)
Which is the famous Kolmogorov’s scaling law.
The Mathrev attachment explains in more detail.
This entire analysis is very useful as inspiration for what happens to volatility on a graph especially because scaling laws are known to hold for financial data (implied by Hurst>0.5 vaguely). On a graph when we have a large number of ordinary differential equations or delay differential equations there is also the possibility of applying the ergodic theorems directly.
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Preview attachment Screenshot from 20150604 21:39:13.png
Screenshot from 20150604 21:39:13.png
Preview attachment mathrev_kolmogorov.pdf
mathrev_kolmogorov.pdf
Preview attachment kolmogorovscalingpaper1951.pdf
kolmogorovscalingpaper1951.pdf
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(from R.)
Enjoy.
🙂
Enjoy.