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.