Stochastic thermodynamics of finite-tape information ratchet

Abstract

In this paper, we explore into the stochastic thermodynamics of a finite-tape information ratchet system. By determining the entropy production of the tape-ratchet system, we derive the information processing second law obeyed by the system. We found that the entropy production takes the form of the non-adiabatic component in both the transient and stationary regime, with the adiabatic component vanishes. The out-of-equilibrium stochastic behaviour is thus driven by non-adiabatic entropy production, with the occurrence of spontaneous relaxation from nonequilibrium initial state and the switching operation of the finite-tape information ratchet system. The observed nonequilibrium processes include (1) work extraction by assimilating excess heat from the heat reservoir when the information ratchet functions as an engine, or (2) dissipating work as excess heat into the heat reservoir when the information ratchet acts as a Landauer eraser. In particular, we observe the phenomenon of irreversible work conversion into excess heat as the information ratchet operates in the nonequilibrium stationary state.

Appendix A mathematical proofs

1.1 A.1 Positivity of the adiabatic and non-adiabatic entropy production

By using $-\ln x \ge 1 - x$, we see that

$$\begin{aligned} \Delta _{a} S= & {} \sum _{i,j} M_{ji} p_i \ln \left( \frac{M_{ji} p_i^{\text {st}}}{M_{ij} p_j^{st}}\right) \nonumber \\= & {} -\sum _{i,j} M_{ji} p_i \ln \left( \frac{M_{ij} p_j^{st}}{M_{ji} p_i^{\text {st}}}\right) \nonumber \\\ge & {} \sum _{i,j} M_{ji} p_i \left( 1 - \frac{M_{ij} p_j^{st}}{M_{ji} p_i^{\text {st}}}\right) \nonumber \\= & {} \sum _{i,j} M_{ji} p_i - \sum _{i,j} p_i \frac{M_{ij} p_j^{st}}{p_i^{\text {st}}}\nonumber \\= & {} \sum _i p_i \sum _j M_{ji} - \sum _i \frac{p_i}{p_i^{\text {st}}}\sum _j M_{ij} p_j^{\text {st}} \nonumber \\= & {} \sum _i p_i - \sum _i \frac{p_i}{p_i^{\text {st}}} p_i^{\text {st}}\nonumber \\= & {} 1 - \sum _i p_i\nonumber \\= & {} 1 - 1 \nonumber \\= & {} 0, \end{aligned}$$

where we have used $\sum _i p_i = 1$ and $\sum _{j} M_{ji} =1$. Similarly, we observe that

$$\begin{aligned} \Delta _{\text {na}} S= & {} \sum _{i,j} M_{ji} p_i \ln \left( \frac{p_j^{\text {st}} p_i}{\tilde{p}_j p_i^{\text {st}}}\right) \nonumber \\= & {} -\sum _{i,j} M_{ji} p_i \ln \left( \frac{\tilde{p}_j p_i^{\text {st}}}{p_j^{\text {st}} p_i}\right) \nonumber \\\ge & {} \sum _{i,j} M_{ji} p_i \left( 1 - \frac{\tilde{p}_j p_i^{\text {st}}}{p_j^{\text {st}} p_i}\right) \nonumber \\= & {} \sum _{i,j} M_{ji} p_i - \sum _{i,j} M_{ji} \frac{\tilde{p}_j p_i^{\text {st}}}{p_j^{\text {st}}}\nonumber \\= & {} \sum _i p_i \sum _j M_{ji} - \sum _j \frac{\tilde{p}_j}{p_j^{\text {st}}}\sum _i M_{ji} p_i^{\text {st}} \nonumber \\= & {} \sum _i p_i - \sum _j \frac{\tilde{p}_j}{p_j^{\text {st}}} p_j^{\text {st}}\nonumber \\= & {} 1 - \sum _j \tilde{p}_j\nonumber \\= & {} 1 - 1 \nonumber \\= & {} 0. \end{aligned}$$

1.2 A.2 $\Delta E = W = \Delta H + \Delta I$

First, we note that

$$\begin{aligned} \Delta E = \sum _{ij} M_{ji} p_i (E_j - E_i) . \end{aligned}$$

(A1)

Because $E_j - E_i = \ln \left( M_{ij}/M_{ji}\right)$ according to Eq. (8), we have

$$\begin{aligned} \Delta E = \sum _{ij} M_{ji} p_i \ln \left( \frac{M_{ij}}{M_{ji}}\right) = W , \end{aligned}$$

(A2)

based on Eq. (10). Also,

$$\begin{aligned} \Delta E= & {} \sum _{ij} M_{ji} p_i (E_j - E_i) \nonumber \\= & {} \sum _{ij} M_{ji} p_i E_j - \sum _{ij} M_{ji} p_i E_i \nonumber \\= & {} \sum _{ij} M_{ij} p_j E_i - \sum _{ij} M_{ji} p_i E_i \nonumber \\= & {} \sum _i \sum _j M_{ij} p_j E_i - \sum _i \sum _j M_{ji} p_i E_i \nonumber \\= & {} \sum _i \tilde{p}_i E_i - \sum _i p_i E_i \nonumber \\= & {} - \sum _i \tilde{p}_i \ln \left( \frac{e^{-E_i}}{Z_0}\right) + \sum _j p_j \ln \left( \frac{e^{-E_j}}{Z_0}\right) \nonumber \\= & {} - \sum _i \tilde{p}_i \ln p^{\text {eq}}_i + \sum _j p_j \ln p^{\text {eq}}_j , \end{aligned}$$

(A3)

where $p^{\text {eq}}_i = e^{-E_i}/Z_0$ with $Z_0= \sum _i e^{-E_i}$.

From Eq. (40), we have

$$\begin{aligned} \Delta I= & {} \sum _i \tilde{p}_i \ln \left( \frac{\tilde{p}_i}{p^{\text {eq}}_i}\right) - \sum _j p_j \ln \left( \frac{p_j}{p^{\text {eq}}_j}\right) \nonumber \\= & {} \sum _i \tilde{p}_i \ln \tilde{p}_i - \sum _i \tilde{p}_i \ln p^{\text {eq}}_i - \sum _j p_j \ln p_j + \sum _j p_j \ln p^{\text {eq}}_j \nonumber \\= & {} \sum _i \tilde{p}_i \ln \tilde{p}_i - \sum _j p_j \ln p_j - \sum _i \tilde{p}_i \ln p^{\text {eq}}_i + \sum _j p_j \ln p^{\text {eq}}_j \nonumber \\= & {} -\Delta H + \Delta E , \end{aligned}$$

(A4)

where we have used Eqs. (12) and (A3). Thus, we have

$$\begin{aligned} \Delta E = W = \Delta H + \Delta I . \end{aligned}$$

(A5)