= Pair potential
{title2=$u(r)$}
{wiki}
A <pair potential> assigns interaction <potential energy> to two particles as a function of their separation, for example $u(r)$ for an isotropic interaction. A pairwise-additive model then has $U=\sum_{i<j}u(r_{ij})$, counting each unordered pair once. This reduces a many-particle energy calculation to repeated evaluation of one two-particle law.
A <hard sphere> model takes $u(r)=\infty$ below the core diameter and zero outside it, forbidding overlap without adding attraction. A more general <pair potential> can contain both repulsion and attraction, producing competition between packing and binding. The associated radial <force> is $-du/dr$ where the derivative exists. Pairwise additivity is a modeling assumption: the interaction between two particles can be modified by their environment, requiring effective state-dependent interactions or explicit many-particle terms. Knowing $u(r)$ supplies an input to <statistical mechanics>, not an immediate solution for every collective phase.
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