Mayer f-function

Remarks

Always $f \ge -1$, and $f(x) \to 0$ as $\abs{x} \to \infty$ when $\varphi(x) \to 0$. Example: the hard-sphere potential in $\R^3$, with $\varphi(x) = +\infty$ for $\abs{x} < \sigma$ and $\varphi(x) = 0$ otherwise, gives $f = -\mathbf{1}_{\set{\abs{x} < \sigma}}$. Its second virial coefficient is $B_2 = -\frac{1}{2} \int_{\R^3} f(x) \, dx = \frac{2 \pi \sigma^3}{3}$. Non-example: the Boltzmann factor $e^{-\beta \varphi}$ itself is not the Mayer function; it tends to $1$ rather than $0$ at large distances. The identity $\prod_{i < j} e^{-\beta \varphi(x_i - x_j)} = \prod_{i < j} (1 + f_{ij})$ expands into a sum over graphs on the particles.