Fast and slow degrees of freedom coupling two different reservoirs

We considered a Hamiltonian system that can be described by two generalized variables. One of them relaxes quickly when the system is in contact with a heat bath at fixed temperature, while the second one, the slow variable, mimics the interaction of the system with another heat bath at a lower temperature. The coupling between these variables leads to an energy flow between the heat baths. Allahverdyan and Nieuwenhuizen [Phys. Rev. E 62, 845 (2000)] proposed a formalism to deal with such problem and calculated the steady states of the system and some related properties as entropy production, energy dissipation, etc. In this work we applied the formalism to a coupled system of ideal gases and also to an ideal gas interacting with a harmonic oscillator. If the temperatures of the heat baths are not too close, the Onsager relations do not apply.