# How do I find $g(r)$ from the Ornstein-Zernike equation for a hard-sphere fluid in Percus-Yevick Approximation?

I have been reading this paper from Thiele [J. Chem. Phys. 39, 474 (1963)], who has obtained the direct correlation function $$c(r)$$ for a hard-sphere system using the Percus-Yevick approximation.

My question is, how do I find $$g(r)$$ from this?

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

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 $$\frac{p}{\rho kT} = 1+2^{d-1}\eta [c(D^+)-c(D^-)]$$

How does he reach this conclusion?

Ornstein-Zernike states that $$h(r_{12}) = c(r_{12}) + \rho \int d\mathbf{r}_3 c(r_{13})h(r_{32})$$ which after a Fourier transform becomes $$\hat{C} (\mathbf{k}) = \frac{\hat{H}(\mathbf{k})}{1+\rho \hat{H}(\mathbf{k})}$$

However, I don't see how to simplify this to the second equation he has. I would appreciate any advice you have.

What I want to do is tabulate values of $$g(r)$$, for different values of $$r$$ greater than $$\sigma$$. Using $$g(r)$$ I want to calculate reduced density, $$p/\rho k T$$, and compare it to the values I get from Stirling-Carnahan for the reduced density.

Finding the explicit analytic form of $$g(r)$$ for all distances is feasible but not straightforward. The basic step is in Wertheim's solution ( Wertheim, M. S. (1963). Exact solution of the Percus-Yevick integral equation for hard spheres. Physical Review Letters, 10(8), 321 ).
However, if the problem is only the contact value of $$g(r)$$, the solution is much simpler. It is based on the fact that, although $$g(r)$$ and $$c(r)$$ are discontinuous at the diameter distance $$r=\sigma$$, their difference must be continuous. This is a trivial consequence of the Ornstein-Zernike equation: $$h(r)-c(r)$$ is a convolution of two functions with a discontinuity at $$\sigma$$. Thus, it must be continuous at $$\sigma$$ (a possible way to get convinced of this fact is from the Fourier representation of OZ showing that the leading term of the asymptotic behavior of $$\hat{C} (\mathbf{k})$$ and $$\hat{H} (\mathbf{k})$$ must be the same).
Therefore, $$g(\sigma^+)-g(\sigma^-)= c(\sigma^+)-c(\sigma^-)$$. But, since $$g(\sigma^-)=0$$ (core condition) and $$c(\sigma^+)=0$$ (Percus-Yevick approximation), from the knowledge of $$c(r)$$ inside the core it is possible to get the contact value of $$g(r)$$.
• Thank you for your response @GiorgioP. What I want to do is tabulate values of $g(r)$, and see how the value of $g(r)$ behaves inside the core $r<\sigma$. Using $g(r)$ I want to calculate reduced density, $p/\rho k T$, and compare it to the values I get from Stirling-Carnahan. I have updated the question. – megamence Dec 13 '20 at 13:21
• @megamence I do not understand what you want to do. By construction, the $g(r)$ corresponding to the PY solution is zero inside the core (that condition is used to derive the form of $c(r)$ for $r<0$). On the other hand, the argument I gave above is enough to get $g(\sigma^+)=-c(\sigma^+)$. Therefore, it is enough to evaluate $-c(\sigma^+)$ to get the compressibility factor $p/\rho k T$. – GiorgioP Dec 13 '20 at 15:11
• I apologize for the confusion, it's been a couple long days. I want to find a function of $r$ and see how it behaves for different values of $\eta$. Once I have $g(r)$, I want to compute reduced density. – megamence Dec 13 '20 at 15:39
• @megamence In Wertheim's paper I cited you may find also the expression for $g(r)$ for $r>\sigma$. It is not possible to discuss the details of the algorithm and its implementation in a comment and even a specific question would be probably considered off topics (homework-like) on this site. – GiorgioP Dec 13 '20 at 16:13