  We study the time-evolution of cumulants of velocities and kinetic energies
in the stochastic Kac model for velocity exchange of $N$ particles, with the
aim of quantifying how fast these degrees of freedom become chaotic in a time
scale in which the collision rate for each particle is order one. Chaos here is
understood in the sense of the original Sto\ss zahlansatz, as an almost
complete independence of the particle velocities which we measure by the
magnitude of their cumulants up to a finite, but arbitrary order. Known
spectral gap results imply that typical initial densities converge to uniform
distribution on the constant energy sphere at a time which has order of $N$
expected collisions. We prove that the finite order cumulants converge to their
small stationary values much faster, already at a time scale of order one
collisions. The proof relies on stability analysis of the closed, but
nonlinear, hierarchy of energy cumulants around the fixed point formed by their
values in the stationary spherical distribution. It provides the first example
of an application of the cumulant hierarchy method to control the properties of
a microscopic model related to kinetic theory.
