A Kac system interacting with two heat reservoirs

We study a system formed by $M$ particles moving in 3 dimension and interacting with 2 heat reservoirs with $N>>M$ particles each. The system and the reservoirs evolve and interact via random collision described by a Kac-type master equation. The initial state of the reservoirs is given by 2 Maxwellian distributions at temperature $T_{+}$ and $T_{-}$. We show that, for times much shorter than $\sqrt{N}$ the interaction with the reservoirs is well approximated by the interaction with 2 Maxwellian thermostats, that is, heat reservoirs with $N=\mathrm{\infty }$. As a byproduct, if $T_{+}=T_{-}$ we extend the results in \cite{BLTV} to particles in 3 dimension.