Carnot Cycle at Finite Power: Attainability of Maximal Efficiency

We want to understand whether and to which extent the maximal (Carnot) efficiency for heat engines can be reached at a finite power. To this end we generalize the Carnot cycle so that it is not restricted to slow processes. We show that for realistic (i.e. not purposefully-designed) engine-bath interactions, the work-optimal engine performing the generalized cycle close to the maximal efficiency has a long cycle time and hence vanishing power. This aspect is shown to relate to the theory of computational complexity. A physical manifestation of the same effect is the Levinthal's paradox in the protein folding problem. The resolution of this paradox for realistic proteins allows to construct engines that can extract at a finite power 40% of the maximally possible work reaching 90% of the maximal efficiency. For purposefully designed engine-bath interactions, the Carnot efficiency is achievable at a large power.