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This is what I understand so far (But I might already be wrong):

The pair distribution function (PDF) $g^{(2)}(\textbf{x},\textbf{x}')$ is the probability of finding a particle at x if there is another particle located at x'. For homogeneous densities, the PDF does not depend on the exact location sites but on the distance between the two particles: $g^{(2)}(\textbf{x}-\textbf{x}')$.

Now, analogously to the PDF, we talked about the first order coherence function (also: one particle density matrix) $G^{(1)}(\textbf{x},\textbf{x}')= \sqrt{n(\textbf{x})n(\textbf{x}')}g^{(1)}(\textbf{x},\textbf{x}')$.

This is where I'm getting completely lost. What does the first order function $g^{(1)}(\textbf{x},\textbf{x}')$ represent? Is it the one-particle analog to $g^{(2)}(\textbf{x},\textbf{x}')$? If so, what does it mean? Surely, talking about the "probability of finding a particle at one site if there's another particle at another site" does not make sense if we're dealing with just a single particle.

Also: Is "Coherence function" just another way of saying "correlation function"? Or is a "coherence function" a function that somehow relates to 'the' "correlation function"? what is the physical meaning of the correlation function/ coherence function?

Any help is highly appreciated, thanks!

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Think of experiments where a particle interferes with itself, like in the double slit. The coherence function tells you if one part of the wavefunction is capable of interfering with another part. For example, if the part exiting the left slit can interfere with the one on the right. If something (like a detector, or bumping into stuff) adds random phase to one part of the wavefunction, coherence goes down and you get a washed-out interference pattern.

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