Kac master equation (${\partial }_t{F}_N=N(Q-I)F_N$)

The jump process with total collision rate $N$ and normalized pair-angle operator $Q$ evolves by ${\partial }_t{F}_N=N(Q-I)F_N$. It preserves mass and the total squared velocity. Its one-coordinate distribution satisfies the Kac marginal evolution equation, and its Gaussian relative entropy in Kac's model decreases at the rate given by Kac entropy production.

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.