Time-delayed feedback control of a flashing ratchet

Closed-loop or feedback control ratchets use information about the state of the system to operate with the aim of maximizing the performance of the system. In this paper we investigate the effects of a time delay in the feedback for a protocol that performs an instantaneous maximization of the center-of-mass velocity. For the one and the few particle cases the flux decreases with increasing delay, as an effect of the decorrelation of the present state of the system with the information that the controller uses, but the delayed closed-loop protocol succeeds to perform better than its open-loop counterpart provided the delays are smaller than the characteristic times of the Brownian ratchet. For the many particle case, we also show that for small delays the center-of-mass velocity decreases for increasing delays. However, for large delays we find the surprising result that the presence of the delay can improve the performance of the nondelayed feedback ratchet and the flux can attain the maximum value obtained with the optimal periodic protocol. This phenomenon is the result of the emergence of a dynamical regime where the presence of the delayed feedback stabilizes one quasiperiodic solution or several (multistability), which resemble the solutions obtained in the so-called threshold protocol. Our analytical and numerical results point towards the feasibility of an experimental implementation of a feedback controlled ratchet that performs equal or better than its optimal open-loop version.

Figure 1

(a) “Smooth” potential [Eq. (5)] for $V_0=5k_{B}T$. (b) “Sawtooth” potential [Eq. (4)] for $V_0=5k_{B}T$ and $a=1∕3$. Units: $L=1$ and $k_{B}T=1$.

Figure 2

Effective force for small delays for potential height $V_0=5k_{B}T$ and asymmetry $a=1∕3$ in the one particle case [Eq. (12)]. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 3

Effective potential [Eq. (15)] for potential height $V_0=5k_{B}T$ and asymmetry $a=1∕3$ for time delays $\tau =0$, ${10}^{-4}$, ${10}^{-3}$, and ${10}^{-2}$. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 4

One particle flux vs delay. The numerical result (solid line) is compared with the estimations using the effective potential [Eq. (15)] (dashed line) and using $C\left(\tau \right)$ (dotted line). $V_0=5k_{B}T$ and $a=1∕3$. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 5

Covariance $\tilde{C}$ [Eq. (17)] as a function of the time delay for potential heights $V_0=k_{B}T$, $5k_{B}T$, and $10k_{B}T$. Asymmetry parameter $a=1∕3$. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 6

Effective force (from numerical simulations) $(N=1)$ for several time delays. $V_0=5k_{B}T$ and $a=1∕3$. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 7

Center-of-mass velocity as a function of the time delay for number of particles $N=1$, 10, 50, and 100. Parameters of the potential: $V_0=5k_{B}T$ and $a=1∕3$. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 8

Many particle case: Center-of-mass velocity as a function of the delay (for large delays only the first branch is represented here) and comparison with the results obtained with the threshold protocol and with the large delay approximation Eq. (33). For the “smooth” potential [Eq. (5)] with $V_0=5k_{B}T$ and $N={10}^5$ particles. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 9

Many particle case: Evolution of the net force with zero delay $(\tau =0)$ for the “smooth” potential Eq. (5) with $V_0=5k_{B}T$ and $N={10}^5$ particles. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 10

Many particle case: Evolution of the net force with a small delay $(\tau =0.02)$ for the “smooth” potential Eq. (5) with $V_0=5k_{B}T$ and $N={10}^5$ particles. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 11

Many particle case: Evolution of the net force with a large delay $(\tau =0.12)$ for the “smooth” potential Eq. (5) with $V_0=5k_{B}T$ and $N={10}^5$ particles. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 12

Many particle case: First three branches of stable solutions for the “smooth” potential [Eq. (5)] with $V_0=5k_{B}T$ and $N={10}^5$ particles. Units: $L=1$, $D=1$, and $k_{B}T=1$.

Figure 13

Many particle case: Period $\mathcal{T}$ of the quasiperiodic solutions for small delays and for large delays. For large delays only the first three branches of stable solutions are shown. “Smooth” potential [Eq. (5)] with $V_0=5k_{B}T$ and $N={10}^5$ particles. Units: $L=1$, $D=1$, and $k_{B}T=1$.