Useful work versus dissipation in weakly coupled systems at different temperatures

We consider a pair of systems which are coupled to a reversible work source to represent a heat engine. The coupling between these systems, initially at different temperatures, is made by a weak interaction. If the relaxation time scales associated with these systems are very different an extra amount of work can be obtained. This additional work gained due to the direct coupling can be greater than the additional dissipation. We show that depending on the nature of the interacting systems it is necessary to look at the second or even higher order contributions in the coupling parameter to observe this effect. Applications are presented for the ideal gases coupling, harmonic oscillators with quartic and a kind of Henón–Heiles couplings.