= Kac marginal evolution equation
{c}
{title2=$\partial_t\Pi_1=\pi^{-1}\int(\Pi_2\circ R_\theta-\Pi_2)\,dv_2d\theta$}
For a permutation-symmetric density in the <Kac master equation>, integrate over all but the first velocity. Collisions among integrated coordinates cancel. The $N-1$ pairs containing the first coordinate give equal terms, with coefficient $N(N-1)/(2\pi\binom N2)=1/\pi$. The resulting exact evolution depends on the two-coordinate <marginal distribution>; it is not a closed equation for the first marginal without additional asymptotic independence.
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