  We consider solutions to the Kac master equation for initial conditions where
$N$ particles are in a thermal equilibrium and $M\le N$ particles are out of
equilibrium. We show that such solutions have exponential decay in entropy
relative to the thermal state. More precisely, the decay is exponential in time
with an explicit rate that is essentially independent on the particle number.
This is in marked contrast to previous results which show that the entropy
production for arbitrary initial conditions is inversely proportional to the
particle number. The proof relies on Nelson's hypercontractive estimate and the
geometric form of the Brascamp-Lieb inequalities due to Franck Barthe. Similar
results hold for the Kac-Boltzmann equation with uniform scattering cross
sections.
