A Kac System Interacting with Two Heat Reservoirs

Abstract

Abstract We study a system formed by M particles moving in 3 dimension and interacting with 2 heat reservoirs with $$N>>M$$ 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 $$T_+$$ T + and $$T_-$$ T - . We show that, for times much shorter than $$\sqrt{N}/M$$ N / M the interaction with the reservoirs is well approximated by the interaction with 2 Maxwellian thermostats, that is, heat reservoirs with $$N=\infty $$ N = ∞ . As a byproduct, if $$T_+=T_-$$ T + = T - we extend the results in [3] to particles in 3 dimension.