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For a central differentiable pair potential, an exact local representation of the interaction momentum flux isHere $n_2$ 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 $-\nabla \cdot {\mathbf{P}}^{\varphi }$ 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.

Spatial homogeneity makes $\rho$ independent of position and makes the two-particle number density depend only on the separation. Isotropy removes its directional dependence. ConsequentlyHere $g_2$ is a radial distribution function, with the finite-$N$ normalization inherited from a; statistical independence is unnecessary. In particular, the translated pair in c has separation $\mathbf{R}$ for every $s$, so its $s$ integral is simply ${\rho }^2{g}_2(R,t)$.

Isotropy of the random velocity $\mathbf{c}=\mathbf{v}-\mathbf{u}$ makes its second velocity moment proportional to the identity. Taking its trace fixes the proportionality constant:Thus $\theta$ is the one-component random velocity variance; for unit mass in thermal equilibrium, $\theta =k_{B}T$. A Maxwell-Boltzmann velocity distribution is not required for the pressure tensor to be isotropic.

For the interaction contribution, the angular integral is $\int d\Omega {\hat{R}}_i{\hat{R}}_j=(4\pi /3){\delta }_{ij}$, by isotropy and its trace. With $d^{3}R=R^{2}dRd\Omega$, c becomesCombining the two pressure tensors gives the virial pressure from pair correlations:For a repulsive pair potential, the outward radial force is $-{\varphi }^{\prime }(R)\ge 0$. Since $g_2\ge 0$, the second term in $p$ is nonnegative, and positive when occupied separations feel nonzero repulsion. It expresses additional momentum transfer through interactions. The displayed integral assumes a differentiable pair potential and convergence; a hard core requires a contact formula or a limiting argument.

Structure through pair correlations. For a homogeneous isotropic monatomic liquid of number density $n=N/V$, the radial distribution function $g(r)$ is defined byIt compares the conditional density around a tagged particle with the bulk density. A repulsive core makes close neighbours unlikely; a first peak identifies a preferred coordination shell, and damped peaks at larger radii describe short-range order. Far away, $g(r)\to 1$ in the bulk limit. Unlike a crystal, an ordinary liquid has no indefinitely repeated positional order. An ideal uncorrelated gas has $g(r)=1$ away from the tagged particle. The shell factor $4\pi r^2$ must not be confused with $g(r)$ itself.

The same pair structure enters the static structure factor measured through scattering. In the bulk isotropic convention,Thus real-space coordination and reciprocal-space scattering peaks describe the same correlations.

Energy from the structure. If interactions are sums of a central pair potential $v(r)$, the equipartition theorem gives the classical kinetic energy $3N{k}_{B}T/2$. Count the interaction of a tagged particle with its neighbours and divide by two to avoid counting each pair twice. The pair-correlation expression for internal energy isThis makes structure thermodynamically useful: short-distance repulsion and the attractive region contribute with different weights. Additional internal molecular modes or many-body interactions require extra terms. The formula is a classical bulk pair-potential result, not a universal energy formula for every liquid.

Pressure and the dilute expansion. The mechanical virial theorem relates the pair forces to pressure. The virial pressure from pair correlations isAt low density, $g(r)\simeq e^{-\beta v(r)}$, giving the virial expansionThe second virial coefficient has units of volume per particle; higher coefficients encode additional cluster correlations. A hard-sphere exclusion diameter $d$ gives $B_2=2\pi d^3/3>0$. Attractions contribute negative terms and can make $B_2$ negative. Although the pair-correlation formulas can describe a dense liquid when its actual $g(r)$ is known, truncating the virial series after a few terms is normally a dilute-fluid approximation. It should not be silently extrapolated through condensation or into a dense liquid.