Power and exponential asymptotic forms of correlation functions

Using the Ornstein–Zernike equation, we obtain two asymptotic equations, one describing the exponential asymptotic behavior and the other describing the power asymptotic behavior of the total correlation function $h(r)$. We show that the exponential asymptotic form is applicable only on a bounded distance interval $l<r<L$. The power asymptotic form is always applicable for $r>L$ and reproduces the form of the interaction potential. In this case, as the density of a rarified gas decreases, $L\to l$, the exponential asymptotic form vanishes, and only the power asymptotic form remains. Conversely, as the critical point is approached, $L\to\infty$, and the applicability domain of the exponential asymptotic form increases without bound.