  Mark Kac introduced what is now called 'the Kac Walk' with the aim of
investigating the spatially homogeneous Boltzmann equation by probabilistic
means. Much recent work, discussed below, on Kac's program has run in the other
direction: using recent results on the Boltzmann equation, or its
one-dimensional analog, the non-linear Kac-Boltzmann equation, to prove results
for the Kac Walk. Here we investigate new functional inequalities for the Kac
Walk pertaining to entropy production, and introduce a new form of
`chaoticity'. We then show how these entropy production inequalities imply
entropy production inequalities for the Kac-Boltzmann equation. This results
validate Kac's program for proving results on the non-linear Boltzmann equation
via analysis of the Kac Walk, and they constitute a partial solution to the
`Almost' Cercignani Conjecture on the sphere.
