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In Torquato's book "Random Heterogeneous Materials", he has written: $$\frac{p}{\rho kT} = 1+2^{d-1}\eta g_2 (D^{+})\tag{1}$$ where $g_2(D^+)$ is the contact value from the right-side of the radial distribution function, and $\eta$ is a dimensionless reduced density, where $d$ is the dimension, $\eta = \rho v(D)$, where $v(D)$ is the volume of a cylinder with radius $D$.

After a couple lines, he states that for hard spheres, via the Ornstein-Zernike equation, we can rewrite the above equation in terms of the direct correlation function $c(r)$ as $$\frac{p}{\rho kT} = 1+2^{d-1}\eta [c(D^+)-c(D^-)]\tag{2}.$$

How does he reach this conclusion?

Ornstein-Zernike states: $$h(r_{12}) = c(r_{12}) + \rho \int d\mathbf{r}_3 c(r_{13})h(r_{32})\tag{3}$$ which after a Fourier transform becomes, where $r_i$ is the position of the particle $i$, and $r_{ij}$ is the displacement vector $r_i-r_j$, $$\hat{C} (\mathbf{k}) = \frac{\hat{H}(\mathbf{k})}{1+\rho \hat{H}(\mathbf{k})}\tag{4}$$ where, $\hat{C} = \mathcal{F}[c], \hat{H} = \mathcal{F} [h]$, where $\mathcal{F}$ is the Fourier transform. However, I don't see how to simplify this to the second equation he has. I would appreciate any advice you have.

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  • $\begingroup$ +1. There's still a lot of undefined symbols or variables, maybe you can tell us what they mean? $\endgroup$ Oct 25 '20 at 20:30
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    $\begingroup$ I have added some extra definitions. hope they help @NikeDattani $\endgroup$
    – megamence
    Oct 25 '20 at 21:56
  • $\begingroup$ That's much better. I think $r$ and $r_{ij}$ are not defined, and while $D^{\pm}$ might be deducible, it's not totally clear. $\endgroup$ Oct 25 '20 at 22:14

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