= Interaction pressure tensor
{title2=$P^\phi_{ij}$}
For a central differentiable <pair potential>, an exact local representation of the interaction <momentum flux> is
$$
P^\phi_{ij}(\mathbf x)=-\frac12\int d^3R\,\frac{R_iR_j}{R}\phi'(R)\int_0^1n_2(\mathbf x+(s-1)\mathbf R,\mathbf x+s\mathbf R)\,ds.
$$
Here $n_2$ is the <two-particle number density>. The line integral assigns each pair's exchanged <momentum> along the segment joining the particles. Translating both endpoints gives $\partial_s n_2=\mathbf R\cdot\nabla_{\mathbf x}n_2$. Integrating over $s$ and using pair-exchange symmetry proves that $-\nabla\cdot\mathbf P^\phi$ is the interaction <force density>. Its homogeneous isotropic specialization gives the interaction part of the <virial pressure from pair correlations>. The kinetic <pressure tensor> accounts separately for random particle motion.
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