Efficiency at maximum power of a discrete feedback ratchet

Efficiency at maximum power is found to be of the same order for a feedback ratchet and for its open-loop counterpart. However, feedback increases the output power up to a factor of five. This increase in output power is due to the increase in energy input and the effective entropy reduction obtained as a consequence of feedback. Optimal efficiency at maximum power is reached for time intervals between feedback actions two orders of magnitude smaller than the characteristic time of diffusion over a ratchet period length. The efficiency is computed consistently taking into account the correlation between the control actions. We consider a feedback control protocol for a discrete feedback flashing ratchet, which works against an external load. We maximize the power output optimizing the parameters of the ratchet, the controller, and the external load. The maximum power output is found to be upper bounded, so the attainable extracted power is limited. After, we compute an upper bound for the efficiency of this isothermal feedback ratchet at maximum power output. We make this computation applying recent developments of the thermodynamics of feedback-controlled systems, which give an equation to compute the entropy reduction due to information. However, this equation requires the computation of the probability of each of the possible sequences of the controller's actions. This computation becomes involved when the sequence of the controller's actions is non-Markovian, as is the case in most feedback ratchets. We here introduce an alternative procedure to set strong bounds to the entropy reduction in order to compute its value. In this procedure the bounds are evaluated in a quasi-Markovian limit, which emerge when there are big differences between the stationary probabilities of the system states. These big differences are an effect of the potential strength, which minimizes the departures from the Markovianicity of the sequence of control actions, allowing also to minimize the departures from the optimal performance of the system. This procedure can be applied to other feedback ratchets and, more in general, to other control systems.

Figure 1

Scheme of the dynamics of the particle in the ON and OFF potentials and of the information gathering and actuation of the controller. The system evolves with the ON or the OFF potential during a time interval $\mathrm{\Delta }t=n\delta t$. After this time, the controller measures the position of the particle. If it is at a site $A$ or $B$, the controller switches the potential ON, whereas if the particle is at site $C$, the potential is switched OFF. Then the particle evolves again during another time interval $\mathrm{\Delta }t$ until the next measurement.

Figure 2

(a) Force that gives the maximal output $F^{*}$ and (b) maximal power output $P_{\max }$, both as functions of the potential height $V_0$ for different temporal discretizations $\delta t$ for the case $n=1$, i.e., when the time between the controller's actions $\mathrm{\Delta }t$ coincides with the discretization time $\delta t,\mathrm{\Delta }t=n\delta t=\delta t$. Solid lines correspond to the analytical approximations for the limiting cases $V\ll 1$ and $V\gg 1$ [respectively, Eqs. (20) and (21)].

Figure 3

(a) Force that gives the maximal output $F^{*}$ and (b) maximal power output $P_{\max }$, both as functions of the potential height $V_0$ for different measurement times $\mathrm{\Delta }t$. In both figures, we have taken values of $\delta t\ll \mathrm{\Delta }t$ (exactly, $\delta t=\mathrm{\Delta }t/100$), so the discretization effects are negligible. (The limit $\delta t\to 0$ and $n\to \infty$ with fixed $\mathrm{\Delta }t=n\delta t$ corresponds to the continuous time limit.) In the limit $V_0\to \infty$ the maximal power output reaches a constant value which depends on $\mathrm{\Delta }t$ (see Sec. 3); i.e., the particle cannot extract an arbitrary large power with this protocol. Moreover, the power output reaches its maximum values for $V_0\gtrsim 10$ whereas the force that maximizes this power output is $F\lesssim 3.8$. Hence, the limit $a\gg b^3$ is justified when the system works at maximum power output. Solid lines represent the analytical expressions obtained for $F^{*}$ and $P_{\max }$ in the limit $a\gg b^3$ (see Sec. 3). For big enough potential heights, they work well for small enough $\mathrm{\Delta }t$, whereas for $\mathrm{\Delta }t\gtrsim {10}^{-2}$ the differences between the numerical results and the analytical expressions become greater. The analytical limit for $\mathrm{\Delta }t={10}^{-1}$ is not represented since it differs greatly from the numerical result.

Figure 4

Lower bounds of entropy reduction rate per control action ${\overline{H}}_{\mathrm{low}}$ in panel (a) and per time unit ${\overline{H}}_{\mathrm{low}}/\mathrm{\Delta }t$ in panel (b) as functions of the time between the controller's actions $\mathrm{\Delta }t$ for $V_0=20$ and $F=F^{*}(V_0,\mathrm{\Delta }t)$ (force that maximizes the power output). The time step has been chosen small enough ($\delta t=\mathrm{\Delta }t/100$) to make the discretization effects vanish. (The limit $\delta t\to 0$ and $n\to \infty$ with fixed $\mathrm{\Delta }t=n\delta t$ corresponds to the continuous time limit.) The upper bounds, not represented, present a maximal deviation from the lower ones of $2\times {10}^{-8}\%$ (because the evolution is very close to be Markovian). We can see how in the limit $\mathrm{\Delta }t\to 0$ (continuous control) the entropy reduction rate per controller's action goes to zero, whereas the entropy reduction rate per unit time diverges logarithmically. Moreover, for $\mathrm{\Delta }t\gtrsim 1$ the entropy reduction per control action reaches a constant value, since in this case the system reaches equilibrium before the controller acts and then the value is the same regardless of the time interval between control actions; and the entropy reduction per unit time goes to zero due to the large time interval between consecutive controller's actions.

Figure 5

Upper bound of the efficiency, ${\eta }_{\mathrm{up}}$, at maximal power as function of the potential height $V_0$ for different values of $\mathrm{\Delta }t$. The time step has been chosen small enough ($\delta t=\mathrm{\Delta }t/100$) to avoid discretization effects.

Figure 6

Maximal efficiency at maximal power output, ${\eta }_{\sup }$, as function of $\mathrm{\Delta }t$. For each $\mathrm{\Delta }t$ the discretization time step $\delta t$ is two orders smaller, in order to avoid discretization effects. Near $\mathrm{\Delta }t\sim {10}^{-2}$ there exists a maximum of efficiency, with a value 2.7%. On the other hand, for big enough $\mathrm{\Delta }t$ the maximal efficiency is zero, which means that the time between control actions is so large that the controller is not able to force the particle to move against the external force $F$.

Figure 7

Maximal power output $P_{\max }$ as a function of the potential height $V_0$ for the open-loop flashing ratchet with a temporal discretization of $\delta t=5\times {10}^{-8}$, and for the feedback flashing ratchet with measurement interval of $\mathrm{\Delta }t={10}^{-2}\text{and}{10}^{-3}$. We chose these measurement time intervals because for the open-loop protocol the maximal power output occurs at $t_{\mathrm{off}}\sim t_{\mathrm{on}}\sim {10}^{-3}$, and the maximum efficiency for the feedback ratchet was achieved at $\mathrm{\Delta }t\sim {10}^{-2}$ (see Fig. 6). In this figure it is shown that the feedback protocol increases notably the maximal output power for the system.

Figure 8

Efficiencies $\eta$ for the cases plotted in Fig. 7. This figure shows that the efficiencies for the open-loop protocol are similar to the efficiencies for the feedback protocol.

Figure 9

Input energy rate $-\mathrm{\Delta }{U}_{\mathrm{cont}}/\mathrm{\Delta }t$ and contribution to the free energy rate from the entropy reduction $T\mathrm{\Delta }{S}_{\mathrm{cont}}/\mathrm{\Delta }t$ for the feedback ratchet compared with the input energy rate for the open-loop ratchet $-\mathrm{\Delta }{U}_{\mathrm{cont}}/(t_{\mathrm{on}}+t_{\mathrm{off}})$, as a function of the potential strength $V_0$ for the cases plotted in Figs. 7 and 8.

Figure 10

Potentials and jumps considered in Appendix pp1. In panel (a), a scheme of the double well is shown. In panel (b), a scheme of the finite jump potential is plotted.