 Mean-field closure for velocity alignment

Neglect of pair correlations closes the first marginal equation of the dissipative Liouville equation. For unit-normalized marginal distributions, the acceleration is $(N-1)\mathbf{L}[p_1]$. With number-normalized $f=N{p}_1$, it is $(1-1/N)\mathbf{L}[f]$. The usual mean-field alignment equation neglects this relative $1/N$ correction. Independence at one time does not imply that interactions preserve independence.