  We investigate the construction of chaotic probability measures on the
Boltzmann's sphere, which is the state space of the stochastic process of a
many-particle system undergoing a dynamics preserving energy and momentum.
Firstly, based on a version of the local Central Limit Theorem (or Berry-Esseen
theorem), we construct a sequence of probabilities that is Kac chaotic and we
prove a quantitative rate of convergence. Then, we investigate a stronger
notion of chaos, namely entropic chaos introduced in \cite{CCLLV}, and we
prove, with quantitative rate, that this same sequence is also entropically
chaotic. Furthermore, we investigate more general class of probability measures
on the Boltzmann's sphere. Using the HWI inequality we prove that a Kac chaotic
probability with bounded Fisher's information is entropically chaotic and we
give a quantitative rate. We also link different notions of chaos, proving that
Fisher's information chaos, introduced in \cite{HaurayMischler}, is stronger
than entropic chaos, which is stronger than Kac's chaos. We give a possible
answer to \cite[Open Problem 11]{CCLLV} in the Boltzmann's sphere's framework.
Finally, applying our previous results to the recent results on propagation of
chaos for the Boltzmann equation \cite{MMchaos}, we prove a quantitative rate
for the propagation of entropic chaos for the Boltzmann equation with
Maxwellian molecules.
