When a reciprocating heat engine is started it eventually settles to a stable
mode of operation. The approach of a first principle quantum heat engine toward
this stable limit cycle is studied. The engine is based on a working medium
consisting of an ensemble of quantum systems composed of two coupled spins. A
four stroke cycle of operation is studied, with two {\em isochore} branches
where heat is transferred from the hot/cold baths and two {\em adiabats} where
work is exchanged. The dynamics is generated by a completely positive map. It
has been shown that the performance of this model resembles an engine with
intrinsic friction. The quantum conditional entropy is employed to prove the
monotonic approach to a limit cycle. Other convex measures, such as the quantum
distance display the same monotonic approach. The equations of motion of the
engine are solved for the different branches and are combined to a global
propagator that relates the state of the engine in the beginning of the cycle
to the state after one period of operation of the cycle. The eigenvalues of the
propagator define the rate of relaxation toward the limit cycle. A longitudinal
and transverse mode of approach to the limit cycle is identified. The entropy
balance is used to explore the necessary conditions which lead to a stable
limit cycle. The phenomena of friction can be identified with a zero change in
the von Neumann entropy of the working medium.Comment: 29 pages and six figure
