Deriving relations for a hard sphere phase diagram

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.

• +1. There's still a lot of undefined symbols or variables, maybe you can tell us what they mean? Oct 25 '20 at 20:30
• I have added some extra definitions. hope they help @NikeDattani Oct 25 '20 at 21:56
• 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. Oct 25 '20 at 22:14