Why is a hard sphere gas correlated? In stat mech we calculated the radial distribution function (a.k.a. pair correlation function) for a classical gas by using perturbation theory for the BBGKY hierarchy. (I could post more details of the calculation if you want, but it is a rather long winded but standard perturbation theory type calculation.) The result we got was
$$ g_2 (r) = \mathrm{e}^{-u(r)/T} \left[ 1 + n_0 \int\mathrm{d}^3r'\ f(r')f(|\vec{r}-\vec{r}'|)\right],$$
where $u(r)$ is the interaction potential, $T$ is the temprature, $n_0$ is the density and $f(r)=\mathrm{e}^{-u(r)/T}-1$ is the Mayer function. $g_2$ roughly measures the probability of finding two particles seperated by a distance $r$. $n_0$ is the small parameter of the perturbation theory.
If you then apply this result to hard sphere (infinite repulsive potential of diameter $a$), you get this:

Now it makes perfect sense that $g_2$ is zero for $r<a$. Also the asymptote to one at large $r$ is part of the definition of $g_2$, meaning that particles are uncorrelated at large distances. The problem is the peak at $r\sim a$ which implies that you are more likely to find particles clustered together, despite the complete absence of any attractive forces! Why is that?
Our lecturer seems to think it is because when two particles collide they stop, then bounce, hence spending more time in the vicinity of each other than for an ideal gas. But this seems dubious because perfect hard sphere collisions are instantaneous. I can imagine three possibilities:

*

*This argument could be formalized as a limit of soft sphere scattering and is the correct explanation of the correlation,

*there is some other (presumably entropic) explanation,

*the correlation doesn't exist - the perturbation theory gives a qualitatively wrong picture (seems unlikely in this case).

So what is it?
 A: I think gatsu's right: it's because of an entropic attraction resulting from the fact that two spheres whose centers are less than $2a$ apart leave more room for other spheres.  To see why this happens, it may help to draw a picture:

Here, the blue spheres all have radius $\frac12a$.  The gray dashed circle around each sphere has radius $a$, and shows the region from which the centers of other spheres are excluded.
Since the spheres A and B are more than $2a$ apart (measured from their centers), the total volume from which they together exclude other spheres simply equals the sum of their respective exclusion volumes.  Spheres B and C, however, are closer than $2a$ from each other, and so their exclusion zones partially overlap, leaving more room for sphere A and any other spheres in the gas.
Looking only at pairwise interactions, this explanation predicts that the quasi-attractive interaction between the particles should appear only at distances less than $2a$.  This matches your graph, which looks flat at $r/a > 2$.  Of course, a more through analysis would also consider clusters of three or more nearby particles, which ought to produce some effects at longer distances (at least at sufficiently high gas densities), but presumably your perturbation analysis also neglected these higher-order effects.
A: If two particles are close to each other, there is more space for the rest of the particles to move. This gives rise to an effective entropic attraction between the particles because when looking at two particles for different separations while "tracing out" over the degrees of freedom of the rest of the system, the entropy of the rest is higher when the two tagged particles you are looking at are close to each other.
In fact at high density, you should also observe oscillations in the g(r) and not a single peack. The width of the bumps in these oscillations is related to the particle size.
A: The entropy answers are all correct I think. Here is another little bit of the physics.
When two molecules are close, they shield each other from collisions with incoming molecules from the rest of the gas. In consequence, the net result of all the collisions is that each molecule of any given pair receives more nudges towards than away from the other. Similar statements apply for larger clusters. So we can predict that hard spheres will tend to cluster, even before we thought about entropy. But it is my hunch that this argument from collisions and the argument from entropy are two ways of talking about the same thing.
(Incidentally this clustering also gives some insight into why gases condense a little more readily than you might guess.)
