Information in feedback ratchets

Feedback control uses the state information of the system to actuate on it. The information used implies an effective entropy reduction of the controlled system, potentially increasing its performance. How to compute this entropy reduction has been formally shown for a general system and has been explicitly computed for spatially discrete systems. Here, we address a relevant example of how to compute the entropy reduction by information in a spatially continuous feedback-controlled system. Specifically, we consider a feedback flashing ratchet, which constitutes a paradigmatic example for the role of information and feedback in the dynamics and thermodynamics of transport induced by the rectification of Brownian motion. A Brownian particle moves in a periodic potential that is switched on and off by a controller. The controller measures the position of the particle at regular intervals and performs the switching depending on the result of the measurement. This system reaches a long-time dynamical regime with a nonzero mean particle velocity, even for a symmetric potential. Here, we calculate the efficiency at maximum power in this long-time regime, computing all the required contributions. We show how the entropy reduction can be evaluated from the entropy of the non-Markovian sequence of control actions, and we also discuss the required sampling effort for its accurate computation. Moreover, the output power developed by the particle against an external force is investigated, which—for some values of the system parameters—is shown to become larger than the input power provided by the switching of the potential. The apparent efficiency of the ratchet thus becomes higher than one, if the entropy reduction contribution is not considered. This result highlights the relevance of including the entropy reduction by information in the thermodynamic balance of feedback-controlled devices, specifically when writing the second principle. The inclusion of the entropy reduction by information leads to a well-behaved efficiency over all the range of parameters investigated.

Figure 1

Entropy reduction by information gathered by the feedback ratchet controller. Additional information on the state of the system further determines the macrostate by reducing the number of compatible microstates [13] and, thus, also reduces entropy. In the top panel, the probability distribution of the particle microstates prior to the measurement is plotted, this macrostate has entropy $S$. The measurement has two possible outputs: the particle is found at the left (of the vertical dashed line), with probability $p$ and associated entropy $S_p$, or at the right, with probability $q=1-p$ and associated entropy $S_q$. The additional information concentrates the probability in the microstates compatible with the result of the measurement, reducing the entropy of the resulting macrostate. Therefore, the average entropy after the measurement $p{S}_p+q{S}_q$ is lower than the entropy before the measurement $S$ [see Eq. (9c)].

Figure 2

Potential (top panel) and effective potential emerging in the limit $\mathrm{\Delta }{t}_m\to 0$ (bottom panel), as a function of the position of the particle. In the limit $\mathrm{\Delta }{t}_m\to 0{}^{+}$, the control acts continuously and thus the potential $V\left(x\right)$ is switched on (off) for positions giving a positive (negative) force $-V^{\prime }\left(x\right), -V^{\prime }\left(x\right)>0$ ($-V^{\prime }\left(x\right)<0$). This yields the effective potential $V_{\text{eff}}$ shown in the bottom panel.

Figure 3

Conceptual scheme of the feedback ratchet for the one-particle system considered in this work. The Brownian particle moves in a periodic potential $V\left(x\right)$, given by Eq. (3), that can be either off (dashed lines) or on (solid lines), and it is also coupled to a thermal bath at temperature $T$. A feedback controller measures the state of the system at time $t_k$, i.e., $x\left(t_k\right)$, and sets the next bit of the controller memory ($C_k=0,1$) accordingly, following Eq. (5), so that the potential acting on the particle for $t=t{}_k{}^{+}$ is $C_{k}V\left(x\right)$. In the specific trajectory portrayed here, the potential was off at time $t{}_k{}^{-}$ and, since $x\left(t{}_k{}^{-}\right)$ lies on an interval where the force $-V^{\prime }$ is positive, the potential is switched on for $t=t{}_k{}^{+}$.

Figure 4

Convergence of the estimator for the entropy reduction per measurement $\mathrm{\Delta }{\tilde{S}}_{\text{info}}$. On the left panel, the entropy of sequence $H_{\text{s}}$ of control actions in the long-time regime as a function of the length of the sequence $M$ is displayed, for data corresponding to one trajectory, i.e., $N_{\text{sim}}=1$. Data for three different values of the time between control updates $\mathrm{\Delta }{t}_m$ are presented, $\mathrm{\Delta }{t}_m=0.02$ (circles), $\mathrm{\Delta }{t}_m=0.002$ (squares), and $\mathrm{\Delta }{t}_m=0.0002$ (triangles), with two values of the number of control actions $N_a$ for each case, $N{}_a=10{}^5$ (open blue symbols) and $N{}_a=10{}^8$ (filled red symbols). On the right panel, the corresponding slope $H_{\text{s}}^{\prime }$ for $\mathrm{\Delta }{t}_m=0.002$ is shown, together with the one corresponding to $\mathrm{\Delta }{t}_m=0.0002$ (inset), using the same symbol and color code. The linear behavior of $H_{\text{s}}$, corresponding to a roughly constant value of $H_{\text{s}}^{\prime }$, is neatly observed, beyond $M\ge M_0\approx 4$ for $\mathrm{\Delta }{t}_m=0.002$, and $M\ge M_0\approx 10$ for $\mathrm{\Delta }{t}_m=0.0002$. System parameters are $a=1/3$ and $V_0=5$ (for the potential), and $F_{\text{ext}}=0$ (no external force). For these values of the parameters, the characteristic times of the system are: 0.5 (diffusion time over $L$), 0.066 (drag time in the $(0,a)$ interval), 0.133 (drag time in the $(a,L)$ interval). (Units: $L=1, k_{B}T=1, \gamma =1$.)

Figure 5

Time evolution of the particle's position and its average value. Specifically, we plot $\langle x\left(t\right)\rangle$ (blue solid line), averaged over $N_{\text{sim}}={10}^5$ trajectories, and $x\left(t\right)$ (red squares) for a single trajectory. Due to the action of the feedback controller, the particle flows to the right. The system parameters are $\mathrm{\Delta }{t}_m=0.02,F_{\text{ext}}=0,V_0=5,a=1/3$. (Units: $L=1, k_{B}T=1, \gamma =1$.)

Figure 6

Time evolution of the average internal energy $\langle U\rangle$ and the probability $P(C=1)$ of finding the potential switched on. The left axis corresponds to $\langle U\rangle$ (blue line) and the right one to $P(C=1)$ (red diamonds). Both quantities have been evaluated averaging over $N_{\text{sim}}={10}^5$ trajectories. The emergence of the long-time regime is neatly observed: on the one hand, $\langle U\rangle$ shows a periodic behavior; on the other hand, $P(C=1)$ tends to a constant value. System parameters and units are the same as in Fig. 5.

Figure 7

Entropy reduction per measurement $\mathrm{\Delta }{\tilde{S}}_{\text{info}}$ as a function of the time between control updates $\mathrm{\Delta }{t}_m$. The inset shows the entropy reduction rate ${\sigma }_{\text{info}}$. System parameters and units are the same as in Fig. 4, $V{}_0=5, a=1/3$, and $F_{\text{ext}}=0$. $\mathrm{\Delta }{\tilde{S}}_{\text{info}}$ monotonically increases, reaching the asymptotic value $\mathrm{\Delta }{S}_{\text{asy}}$ in Eq. (42) (dashed line) when $\mathrm{\Delta }{t}_m$ becomes of the order of the longest characteristic time—see Fig. 4 for the values of the characteristic times.

Figure 8

Entropy reduction per measurement $\mathrm{\Delta }{\tilde{S}}_{\text{info}}$ as a function of the potential asymmetry $a$. The entropy reduction vanishes for both $a\to 0$ and $a\to 1$, when the off and on control actions dominate, respectively. Three sets of points, corresponding to different values of the time between control updates, $\mathrm{\Delta }{t}_m=0.2$ (black triangles), $\mathrm{\Delta }{t}_m=0.02$ (red squares), and $\mathrm{\Delta }{t}_m=0.002$ (blue diamonds), are shown. Vertical dashed lines indicate maxima position. Other system parameters are $V_0=5$ and $F_{\text{ext}}=0$.

Figure 9

Entropy reduction per measurement $\mathrm{\Delta }{\tilde{S}}_{\text{info}}$ as a function of the potential height $V_0$. The entropy reduction increases, saturating for potential height values $V_0\gtrsim 30$. System parameters are $\mathrm{\Delta }t{}_{\text{m}}=0.02, a=1/3$, and $F_{\text{ext}}=0$.

Figure 10

Entropy reduction per measurement $\mathrm{\Delta }{\tilde{S}}_{\text{info}}$ as a function of the external force $F_{\text{ext}}$. System parameters are $\mathrm{\Delta }{t}_m=0.02, a=1/3$, and $V_0=5$. Increasing the external force decrease the entropy reduction, particularly for values $F_{\text{ext}}>1$—the force unit is given by the quotient of energy and length units, i.e., the force unit is $k_{B}T/L$. The dashed vertical line corresponds to $F_{\text{stop}}$.

Figure 11

Power and efficiency as a function of the external force $F_{\text{ext}}$. On the left panel, we plot the input and output powers $P_{\text{in}}$ (blue stars) and $|P_{\text{ext}}|$ (red circles, recall that $P_{\text{ext}}<0$), averaged over one trajectory corresponding to a long time with $N_a={10}^8$ control updates. On the right panel, we show both the open-loop efficiency ${\eta }_0$ (red circles) and the actual efficiency $\eta$ (blue diamonds). System parameters are identical to those in Fig. 10, for which $F_{\text{stop}}\approx 2.75$.

Figure 12

Same as in Fig. 11, but for a smaller time between control updates. Specifically, we are showing data corresponding to $\mathrm{\Delta }{t}_m=0.002$, for which $F_{\text{stop}}\approx 4.2$. For $F_{\text{ext}}\gtrsim 3.5, P_{\text{in}}<\left|P_{\text{ext}}\right|$ and thus the open-loop efficiency takes unphysical values, ${\eta }_0>1$. However, the actual closed-loop efficiency $\eta$ properly accounts for the reduction of the particle entropy due to the information gathered by the control, and thus it is consistently smaller than unity.