  We study a system of $N$ particles interacting through the Kac collision,
with $m$ of them interacting, in addition, with a Maxwellian thermostat at
temperature $\frac{1}{\beta}$. We use two indicators to understand the approach
to the equilibrium Gaussian state. We prove that i) the spectral gap of the
evolution operator behaves as $\frac{m}{N}$ for large $N$ ii) the relative
entropy approaches its equilibrium value (at least) at an eventually
exponential rate $\sim \frac{m}{N^2}$ for large $N$. The question of having
non-zero entropy production at time $0$ remains open. A relationship between
the Maxwellian thermostat and the thermostat used in Bonetto, Loss,
Vaidyanathan (J. Stat. Phys. 156(4):647-667, 2014) is established through a van
Hove limit.
