General Method for Calculating Excluded Volume

Question

In section 5.3 of Kardar's Statistical Physics of Particles, the van der Waals equation is given as:

$[P+\frac{u_0 \Omega}{2}(\frac{N}{V})^2][V-\frac{N\Omega}{2}]=Nk_BT$

The van der Waals parameters are identified as $a = \frac{u_0 \Omega}{2}$ and $b=\frac{\Omega}{2}$. Here, $\Omega$ is the volume excluded around each particle (to the centers of the other atoms). The parameter $b$ is interpreted as the effective excluded volume for low densities due to the fact that for low densities ($\Omega \ll V$) the contribution of coordinates to the partition function is:

$V(V-\Omega)(V-2\Omega)...(V-(N-1)\Omega) \approx (V-\frac{N\Omega}{2})^N$ (Equation 5.46)

which explains the factor of $\frac{1}{2}$ in $b$. However, the author goes on to say that:

Of course, the above result is only approximate since the effects of excluded volume involving more than two particles are not correctly taken into account. The relatively simple form of Eq. (5.46) is only exact for spatial dimensions d = 1, and at infinity.

I have some questions:

1- What does the author exactly mean when he says the expression is exact for $d=1$ and at "infinity"? Does he mean infinite number of particles?

2- How to incorporate the effects of excluded volume involving more than two particles? As an example I would be glad if someone explained it for 3 particles.

3- Is there a general method or algorithm to find the excluded volume involving any number of particles for any dimension $d$? If so, I would like to see some sources explaining these methods.

Answer 1

1. He means the spatial dimension. In $d=1$ if you consider a 3-particle hard-ball cluster (3 hard-ball particles each overlapping with the remaining 2) it can only happen by taking two overlapping particles and slapping the third one in the middle. The 3-particle overlap will be the same as the overlap between two initial particles so I believe it factors out. In $d \rightarrow \infty$ I would expect the 3-particle (and higher-order contributions) to vanish. If you keep the particle volume constant and increase the spatial dimension, the particle radius has to decrease. It becomes progressively less likely for 3-particles to meet each other at the same time. In both cases, this is only an idea, but it should be quite easy to show through a direct calculation.

2. and 3. The cluster expansion seems to be an answer (although I am not sure since I only briefly heard about it). It is defined for any intermolecular potential but it should be quite simple for hard-ball potential. I think it would boil to the inclusion-exclusion principle.