Propagation and generation of "chaos" is an important ingredient for
rigorous control of applicability of kinetic theory, in general. Chaos
is here understood as sufficient statistical independence of random
variables related to the "kinetic" observables of the system. Cumulant
hierarchy of these random variables thus often gives a way of
controlling the evolution and degree of such independence, i.e., the
degree of "chaos" in the system. In this talk, we will consider two,
qualitatively different, example cases for which kinetic theory is
believed to be applicable: the stochastic Kac model with random velocity
exchange and the discrete nonlinear Schrodinger evolution (DNLS) with
suitable random, spatially homogeneous initial data. In both cases, we
set up suitable random variables and propose methods to control the
evolution of their cumulant hierarchies. The talk is based on joint
works with Sakari Pirnes and Aleksis Vuoksenmaa, and earlier works with
Matteo Marcozzi, Alessia Nota, and Herbert Spohn.