Stochastic thermodynamics with information reservoirs

We generalize stochastic thermodynamics to include information reservoirs. Such information reservoirs, which can be modeled as a sequence of bits, modify the second law. For example, work extraction from a system in contact with a single heat bath becomes possible if the system also interacts with an information reservoir. We obtain an inequality, and the corresponding fluctuation theorem, generalizing the standard entropy production of stochastic thermodynamics. From this inequality we can derive an information processing entropy production, which gives the second law in the presence of information reservoirs. We also develop a systematic linear response theory for information processing machines. For a unicyclic machine powered by an information reservoir, the efficiency at maximum power can deviate from the standard value of $1/2$. For the case where energy is consumed to erase the tape, the efficiency at maximum erasure rate is found to be $1/2$.

Figure 1

Two-state system interacting with a bit. If a thermal transition happens from $d$ ($u$) to $u$ ($d$) the bit changes its state from 0 (1) to 1 (0).

Figure 2

Possible effective transitions generated by the new incoming bit. Case 1 corresponds to the system being in state $u$ and the new incoming bit in state 0, leading to extracting a quantity $E$ of work. In case 2 the system is in state $d$ and the new incoming bit is in state 1, which leads to a quantity $E$ of work entering the system from the work reservoir. Case 3 (4) corresponds to the system being in state $d$ ($u$) and the the new incoming bit in state 0 (1), which does not involve an exchange of energy with the work reservoir. The letter $b$ in the tape represents a bit that can be in either state 0 or state 1.

Figure 3

Transition rates for the four-state model. The thermal transitions take place with rates $k_{+}$ and $k_{-}$. Transitions between states with a different subscript are related to the tape moving forward and a new bit coming to interact with the system. The solid arrows represent transitions with rate $\gamma (1-\epsilon$) and the dashed arrows transitions with rate $\gamma \epsilon$.

Figure 4

Two-state reduction of the four-state model from Fig. 3.

Figure 5

Sketch of a system interacting with standard reservoirs with inverse temperature ${\beta }_{\nu }$ and field $f_{\nu }$. Information reservoirs are characterized by ${\epsilon }_n$, the probability of a bit in state 1. The additional work reservoir, related to transitions mediated by the information reservoir for which the internal energy of the system changes, is indicated by $WR$.

Figure 6

Three-state model. The rates $W_{ac}=\gamma \epsilon$ and are $W_{ca}=\gamma (1-\epsilon )$ are relate to the information reservoir. They are the same for both the refrigerator powered by a tape (Sec. 4a) and the thermoelectric machine interacting with a tape (Sec. 4b).

Figure 7

Phase diagram of the three-state model from Fig. 6 with particle exchange with the reservoirs. The signs of the triplet $({\dot{q}}_1,{\dot{q}}_2,\dot{w})$ are $(-,+,-)$ in $IA$, $(-,-,-)$ in $IIA$, $(+,-,-)$ in $IIIA$, $(+,-,+)$ in $IB$, $(+,+,+)$ in $IIB$, and $(-,+,+)$ in $IIIB$. The differences between the phases are explained in the text.

Figure 8

Unicyclic model. The transition rates between states 0 and 1 are associated with an information reservoir, whereas the other transition rates are related to a standard reservoir. Note that we have a cyclic system with $N+1$ being the 0 state again.