What does the behavior of the pair correlation function look like in the vicinity of the critical point? What does the g(r) look like near the critical point?
I know what the pair correlation function (radial distribution function) should look like for a solid, which has regular packing and therefore the g(r) will not decay with time. And I know what it looks like for, say, a dense Lennard-Jones fluid. But I don't know what it will look like near the critical point, where the correlation length diverges.
I found here a figure that shows the g(r) for a Lenanrd-Jones fluid in the critical region. It appears that the initial peak is lower than the initial peak of g(r) near the triple point and that it then attenuates to zero rather quickly, without any of the normal lesser peaks. How does this show that the correlation length diverges?
 A: The pair correlation function $g(r)$ is defined via the density-density correlation function
$$
S_{nn}(r) =\langle n(0) n(r)\rangle .
$$
Typically, we define $g(r)=S_{nn}(r)/\langle n\rangle^2$.
The density $\delta n = n-n_{cr}$ is also the order parameter for the critical point on the liquid-gas phase transition. This means that the correlation function of $\delta n$ exhibits critical behavior. In particular, it has a long range tail
$$
G(r) \sim \frac{1}{r^{d-2+\eta}},
$$ 
where $\eta$ is the correlation function critical exponent. For $d=3$ Ising universality this exponent is pretty small, $\eta\simeq 0.04$.  
This implies that near the critical point the pair correlation function $g(r)-1$ has a $1/r$ tail. This may be a little hard to see on the plot you link to, because they zoom in on fairly short distances.
A: The radial (pair) distribution function is not the correct correlation function for this problem.  At the critical point it is fluctuations in average properties that appear with all possible wavelengths.  You need a correlation function that captures this - the density correlation function: see Mathematics of Complexity and Dynamical Systems
