Kac marginal evolution equation (${\partial }_t{\Pi }_1={\pi }^{-1}\int ({\Pi }_2\circ R_{\theta }-{\Pi }_2)d{v}_{2}d\theta$)

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 $N(N-1)/(2\pi (\genfrac{}{}{0ex}{}{N}{2}))=1/\pi$. 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.