# Meaning of and relationship between pair distribution function/ coherence functions/ correlation function

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!