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?