Unifying Three Perspectives on Information Processing in Stochastic Thermodynamics

So far, feedback-driven systems have been discussed using (i) measurement and control, (ii) a tape interacting with a system, or (iii) by identifying an implicit Maxwell demon in steady-state transport. We derive the corresponding second laws from one master fluctuation theorem and discuss their relationship. In particular, we show that both the entropy production involving mutual information between system and controller and the one involving a Shannon entropy difference of an information reservoir like a tape carry an extra term different from the usual current times affinity. We, thus, generalize stochastic thermodynamics to the presence of an information reservoir.

Figure 1

Two-state model with feedback at fixed time intervals interpreted as a tape interacting with a heat bath and a work reservoir. The full arrow represents an interaction time interval $t$, and the dotted arrow represents an instantaneous effective transition due to the new incoming bit.

Figure 2

Representation of the formal duplication of the system. The full circles represent generic states $i$ different from $u$ and $d$; the curved lines represent links between states with nonzero transition rates $W_{ij}$ and $W_{ji}$; the arrows represent the transition rates between the replicas, with the full arrows representing $\gamma \epsilon$ and the dotted arrows representing $\gamma (1-\epsilon )$.