- #1
Sat D
- 11
- 3
- Homework Statement
- In Torquato's Random Heterogeneous Materials, he has written
##\frac{p}{\rho kT} = 1+2^{d-1}\eta g_2 (D^{+}) = 1+2^{d-1}\eta [c(D^+)-c(D^-)]##
which he arrives to using the Ornstein-Zernike equation.
How does he reach this conclusion?
- Relevant Equations
- where ##g_2(D^+)## is the contact value from the right-side of the radial distribution function, and $\eta$ is a dimensionless reduced density.
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.
##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.