A Kac system interacting with two heat reservoirs: the shearing case

We study a system formed by $M$ particles moving in 3 dimensions and interacting with two heat reservoirs, each with $N\gg M$ particles. The system and the reservoirs interact via random collisions and thus evolve via a Kac-type master equation. The initial state of the reservoirs is given by two non-centered Maxwellian distributions; they have temperature $T_+$ and $T_-$ and have average velocity $\vec p_+$ and $\vec p_-$, respectively. We prove that, for times shorter than $\sqrt{N}/M$, the interaction with the two reservoirs is well-approximated by the interaction with two shearing {\it dynamic} Maxwellian thermostats (i.e. heat reservoirs with $N=\infty$). As a byproduct of our analysis, we obtain a uniform in time approximation when $T_+=T_-$ and $\vec p_+=\vec p_-$.