Hard spheres

Example

The hard-sphere fluid is a classical system of $N$ identical balls of diameter $\sigma > 0$ in a region of volume $V$ in $\R^3$. The balls interact through the pair potential $$\phi(r) = \begin{cases} +\infty & \text{if } r < \sigma, \\ 0 & \text{if } r \ge \sigma, \end{cases}$$ where $r$ is the distance between two centres. In the canonical ensemble the Boltzmann factor $e^{-\beta \sum_{i<j} \phi(\abs{x_i - x_j})}$ is $1$ if no two balls overlap and $0$ otherwise. So the configurational distribution is uniform on the set of non-overlapping configurations, for every temperature.

Check with the second virial coefficient: the Mayer f-function is $f(r) = e^{-\beta\phi(r)} - 1$, equal to $-1$ for $r < \sigma$ and $0$ for $r \ge \sigma$. Hence $$B_2 = -\frac12 \int_{\R^3} f(\abs{x}) \, d^3 x = \frac12 \cdot \frac{4\pi}{3} \sigma^3 = \frac{2\pi \sigma^3}{3},$$ which does not depend on the temperature, and the low-density pressure is $P \approx n k_B T (1 + B_2 n)$ with $n = N/V$.