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 — gtspring2009:spieker_blog:correlationfcts [2010/02/02 07:55] (current) Line 1: Line 1: + ===== Correlation Function ===== + + **08/​28/​09** I went to the library yesterday to see if I could find a book that explains correlation functions a little deeper than we have in class but in a way that I can understand, but I didn't find one.  I am going to ask Dr. Wiesenfeld if he knows of any resources I could use to help me in the case of large velocity vector fields. ​ The one dimensional case that we have seen in class is of the Langevin Equation where the correlation function is given by: + + <​latex>​ + $C(\tau , t) = \langle v(t)v(t+\tau) \rangle$ + ​ + + I am guessing that in the case of of the velocity fields we are working with, the appropriate calculation (and I think it can be shown) would be: + + <​latex>​ + $C(\tau) = \frac{1}{T}\int_0^T \vec{v}(t)\cdot\vec{v}(t+\tau)dt$ + ​ + + I don't know if anything interesting would emerge from performing this calculation on the periodic orbits, but it might be worth a shot looking at the correlation function for both turbulent and decaying to laminar trajectories. ​ In the scope of the class I am taking, the correlation function of a seemingly random function can be transformed into a power spectrum of the function via a Fourier transform by the Wiener-Khintchine Theorem: ​ + + <​latex>​ + S(\Omega) = \frac{1}{2\pi}\int_{-\infty}^{\infty}C(\tau)e^{-i\Omega\tau}d\tau + ​ + + and otherwise latent patterns present themselves in frequency space. ​ I thought that maybe such patterns might emerge if we performed this calculation,​ but Daniel Borrero didn't seem to think that the calculation would be that simple, so I will learn more before I make any further conjectures. ​