# 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?

• Throw us a bone here. Which correlation function are you talking about? Just saying that it's denoted $g(r)$ isn't giving much of a hint. And what work have you done toward determining what $g(r)$ looks like at its critical point, and what problem are you encountering when trying to do that? This question needs work from you, or it will just get closed. – Red Act Aug 29 '14 at 22:29
• @RedAct Act I'm talking about the pair-correlation function, i.e. radial distribution function. I know what it 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 Lennard-Jones fluid. But I don't know what it will look like near the critical point, where the correlation length diverges. – user2561523 Aug 29 '14 at 23:05
• Add that and more to the question. – BMS Aug 30 '14 at 1:38
• The closest thing I can find to what you're asking for is Figure 2 of courses.physics.illinois.edu/phys466/sp2013/projects/1998/team5 , which shows a set of radial distribution functions based on a simulation of hard disks on a sphere for a variety of different densities during a fluid-solid phase transition. That's not a good enough find that I would make an answer out of it, but hopefully it's helpful? – Red Act Aug 30 '14 at 4:16
• @RedAct Thanks. I found a book online that shows the g(r) near the critical point, but I don't understand its implications for the correlation length. I've since edited my question. – user2561523 Aug 30 '14 at 10:03

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.