gtspring2009:spieker_blog:correlationfcts

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+ | ===== Correlation Function ===== | ||

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+ | **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 $ | ||

+ | </latex> | ||

+ | |||

+ | 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 $ | ||

+ | </latex> | ||

+ | |||

+ | 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 | ||

+ | </latex> | ||

+ | |||

+ | 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. | ||

gtspring2009/spieker_blog/correlationfcts.txt ยท Last modified: 2010/02/02 07:55 (external edit)