Stochastic thermodynamics of bipartite systems: transfer entropy inequalities and a Maxwell's demon interpretation

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.