Kac chaos (${\Pi }_k(F_N)\rightharpoonup f^{\otimes k}$)

A permutation-symmetric sequence of probability laws is chaotic with one-particle law $f$ when, for every fixed $k$, its $k$-coordinate marginal distributions converge weakly to the product law $f^{\otimes k}$ as $N\to \infty$. This is an asymptotic independence condition, not a consequence of symmetry alone. A rigorous kinetic limit also requires propagation of this property under the dynamics and sufficient convergence to pass collision integrals.