Second Law of Thermnodynamics for Active Heat Engines

Cyclic heat engines have been a major motivation for the emergence of thermodynamics. In the last decade, cyclic heat engines that have large fluctuations and operate at finite time were studied within the more modern framework of stochastic thermodynamics. The second law for such heat engines is the standard second law of thermodynamics, which states that the efficiency of the heat engine cannot be larger than the Carnot efficiency. The concept of active cyclic heat engines for a system in the presence of hidden dissipative degrees of freedom, also known as a nonequilibrium or active reservoir, has also been studied in theory and experiment. Such active engines show rather interesting behavior such as an "efficiency" larger than the Carnot bound. They are also likely to play an important role in future developments, given the ubiquitous presence of active mediums. However, a general second law for cyclic active heat engines was still lacking. Here we obtain a second law for active heat engines. Two main features of our second law are: it does not involve the energy dissipation of the hidden degrees of freedom, which is typically much larger than the extracted work, and it is expressed in terms of quantities that can be measured directly from the observable degrees of freedom. Besides heat and work our second law contains an information-theoretic term, which constitute an extra resource that active heat engines can explore to extract work.