= Virial pressure from pair correlations
{title2=$p=nk_BT-(2\pi n^2/3)\int_0^\infty r^3v'(r)g(r)\,dr$}
For a differentiable central <pair potential> in a classical homogeneous three-dimensional fluid, the mechanical <virial theorem> gives the displayed pressure. The <radial distribution function> accounts for the probability of each pair separation. In the dilute limit, $g(r)=e^{-v(r)/(k_BT)}+O(n)$, and integration by parts yields $B_2(T)=-2\pi\int_0^\infty[e^{-v(r)/(k_BT)}-1]r^2dr$. Thus the <second virial coefficient> controls $p/(nk_BT)=1+B_2n+\cdots$. A few virial terms are a low-density approximation, not generally a converged description of a dense liquid. A discontinuous hard core needs its contact form or a limiting argument.
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