Fourier Transform of correlation functions

In summary, the correlation integral is similar to the convolution integral, only with a plus sign. The correlation function is diagonal in momentum space, which makes calculations easier. The static susceptibility sum rule is related to the Fourier transform of the correlation function at zero wavevector.
  • #1
elduderino
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0
Why are they useful, what do they denote (physically or otherwise)...
 
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  • #2
Maybe this will help? http://en.wikipedia.org/wiki/Convolution_theorem" [Broken].
 
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  • #3
Yes, the correlation integral acts very much like a convolution integral, only with a plus sign... but I am not able to understand why while calculating correlation functions, or even green's functions, authors tend to caclulate their Fourier transforms as well.

thank you
 
  • #4
The main reason is that in a translationally invariant system, Green's functions are diagonal in momentum space. This simplifies all calculations and turns matrix equations (e.g. Dyson equation) into algebraic equations that can be easily solved. The diagonality of correlation functions is one way of saying that "momentum is conserved", although the physical meaning of momentum varies from one model to another.
 
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  • #5
Usually the FT of a correlation function is what's experimentally measurable. Using neutron scattering or photoemission one measures the momentum of the emitted particle, and this directly relates to a moment correlation function or Green's function.
 
  • #6
static susceptibility sum rule for magnetic systems:

[tex]\chi = \frac{\partial M}{\partial H} = k_B T \int_V d^d\mathbf{r} G(\mathbf{r}) = k_B T \hat{G}(\mathbf{k} = \mathbf{0}) [/tex]

i.e., The magnetic susceptibility is related to the Fourier transform of the correlation function at zero wavevector.
 
  • #7
As an example:
Normally one detects light with a photo detector. A photo detection is converted into electronic signal after some kind of filtering (or whatever is the frequency response of the system). Physically for a square-law detector the Fourier transform (FT) of a correlation provides the frequency spectrum of the signal. That is the simplest of the physical explanation. For example, if two waves interfere in which one of the wave is Doppler shifted the correlation function (or the auto-correlation function to be accurate from the detected signal) will be a sin function while its Fourier will be peaked at a frequency corresponding to the Doppler shift. One reason why correlation function is measured versus Fourier is its large dynamic range not affordable in Fourier. One could measure correlation function all the way from nanoseconds to minutes. Try this time scale with Fourier and see the amount of data needed...
 
  • #8
When we want to study the Goldstone modes of the system, the Fourier transform is more desired because the Goldstone modes vanishes when k->0.
 
  • #9
thanks!
 

1. What is the Fourier Transform of a correlation function?

The Fourier Transform of a correlation function is a mathematical operation that transforms a function from the time domain to the frequency domain. It is used to analyze the frequency components present in a signal or data set.

2. Why is the Fourier Transform of correlation functions important in scientific research?

The Fourier Transform of correlation functions is important because it allows researchers to identify and analyze the underlying periodicity or frequency components in a signal. This information can be used to understand the behavior of systems and make predictions about future trends in data.

3. How is the Fourier Transform of correlation functions calculated?

The Fourier Transform of a correlation function is calculated by taking the product of the correlation function and a complex exponential function, and integrating over all time. This process can be done analytically or numerically using software or programming languages like MATLAB or Python.

4. What is the relationship between the Fourier Transform of correlation functions and power spectral density?

The power spectral density is the squared magnitude of the Fourier Transform of a correlation function. It represents the distribution of power or energy over different frequencies in a signal. Essentially, the Fourier Transform of a correlation function provides the frequency components, while the power spectral density gives the power associated with each frequency.

5. How is the Fourier Transform of correlation functions applied in different fields of science?

The Fourier Transform of correlation functions has broad applications in various fields of science, ranging from physics, mathematics, and engineering to biology and finance. It is used to analyze signals and data sets in fields such as signal processing, image processing, and time series analysis, among others.

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