  We consider the stationary state of a Markov process on a bipartite system
from the perspective of stochastic thermodynamics. One subsystem is used to
extract work from a heat bath while being affected by the second subsystem. We
show that the latter allows for a transparent and thermodynamically consistent
interpretation of a Maxwell's demon. Moreover, we obtain an integral
fluctuation theorem involving the transfer entropy from one subsystem to the
other. Comparing three different inequalities, we show that the entropy
decrease of the first subsystem provides a tighter bound on the rate of
extracted work than both the rate of transfer entropy from this subsystem to
the demon and the heat dissipated through the dynamics of the demon. The latter
two rates cannot be ordered by an inequality as shown with the illustrative
example of a four state system.
