This paper studies a mathematical gas model in which a small system of M particles moves in three dimensions and collides at random with two much larger heat baths (reservoirs). Each reservoir has N particles with N≫M. The reservoirs start in Maxwellian states — bell‑shaped velocity distributions — but their average velocities (centers of mass) need not be zero. When the two reservoirs have different average velocities, their motion produces a shear in the small system. The authors ask when the effect of the two finite reservoirs can be replaced, for the system, by two idealized infinite reservoirs.

The authors work with a Kac‑type master equation. In that framework collisions are random binary events that conserve momentum and energy. They write down the full generator of the combined evolution of the system plus the two reservoirs, and they compare it to an effective evolution where the system collides with two Maxwellian thermostats. A Maxwellian thermostat here means an infinite reservoir that supplies velocities drawn from a Gaussian (Maxwellian) law. The novelty is that the thermostats are made “dynamic”: their temperatures and center‑of‑mass velocities are allowed to evolve in time in the same way those quantities would evolve for the finite reservoirs.

The main rigorous result is that, for times shorter than about sqrt(N)/M, the true evolution of the system plus finite reservoirs is well approximated by the evolution with the two dynamic Maxwellian thermostats. In other words, if the reservoirs are much larger than the system, then on a relatively short time window the system sees the reservoirs as if they were infinite, provided one lets the thermostat parameters (temperature and mean velocity) change in time. The proof uses a Duhamel expansion (a way to compare two evolutions) and a functional inequality previously introduced by the authors. They measure closeness of probability laws with a metric (the GTW d2 metric) that requires the compared distributions to have the same average velocity.