Effect of time delay on feedback control of a flashing ratchet

It was recently shown that the use of feedback control can improve the performance of a flashing ratchet. We investigate the effect of a time delay in the implementation of feedback control in a closed-loop collective flashing ratchet, using Langevin dynamics simulations. Surprisingly, for a large ensemble, a well-chosen delay time improves the ratchet performance by allowing the system to synchronize into a quasi-periodic stable mode of oscillation that reproduces the optimal average velocity for a periodically flashing ratchet. For a small ensemble, on the other hand, finite delay times significantly reduce the benefit of feedback control for the time-averaged velocity, because the relevance of information decays on a time scale set by the diffusion time of the particles. Based on these results, we establish that experimental use of feedback control is realistic. Introduction. – Flashing ratchets rectify the thermal motion of diffusive particles by exposing them to a time-dependent, spatially periodic and asymmetric potential [1–5]. These systems are attracting significant interest [6] because they may be applied, for example, in particle separation [7, 8] or as a power source for polymer motors [9]. In addition, flashing ratchets are one of the simplest realizations of Brownian motors in general [5], and are therefore of fundamental interest in non-equilibrium statistical mechanics and as models for synthetic [10] or biological [11] molecular motors. In most studies of flashing ratchets, the potential is switched periodically or randomly, without regard to the particle distribution. Recently, a flashing ratchet with feedback control was introduced, where the external potential depends on the state of the system [12]. The instantaneous center-of-mass velocity was maximized by turning the potential on only when the ensemble-averaged force exerted by the potential would be positive. This strategy maximizes the time-averaged center-of-mass velocity for one particle (N = 1) and performs better than a periodically flashing ratchet for N < 10 − 10. However, because fluctuations are needed to trigger the next switching event, the time-averaged velocity goes to zero for large N where center-of-mass fluctuations become rare. In an improved feedback strategy [13], the potential is switched on (off) whenever the average force increases (decreases) through set thresholds, eliminating the need to wait for fluctuations. This protocol performs as well as the feedback strategy in [12] for N = 1, and yields approximately the same current as optimal periodic flashing for large N . In an experiment designed to implement such feedback strategies, there will be a finite time lag between the collection of information and any feedback to the system because of the time it takes to acquire an image of particles, and to determine the particle positions and ensemble averages. Delayed feedback has been demonstrated to produce complicated dynamics in chaotic, inertial ratchets [14–16], stochastic systems that display a Hopf bifurcation (oscillatory instability) in the absence of delay [17] and biological systems [18–20], and some analytical methods have recently been developed to study time delay in stochastic systems [21–23]. Here we study the impact of time delay on the effectiveness of the feedback control strategy introduced in [12]. For a small number of particles (less than N ≈ 300), we find that finite delay times in the feedback significantly reduce the average velocity, because the quality of the collected information decays on the scale of the system’s characteristic diffusion time. This is consistent with the result [24] that loss of information about the system limits the improvement of the flux for the feedback ratchet strategy in [12] over a periodically flashing ratchet. On the other hand, for the quasi-deterministic case of N > 10 − 10, we find that a well-chosen delay time effectively reproduces the advantageous threshp-1 ar X iv :0 70 7. 09 01 v2 [ co nd -m at .s ta tm ec h] 9 N ov 2 00 7