Flux enhancement and multistability induced by time delays in a feedback controlled flashing ratchet
Feedback controlled ratchets are thermal rectifiers that use information on the state of the system to operate. We study the effects of time delays in the feedback for a protocol that performs an instantaneous maximization of the center-of-mass velocity in the many particle case. For small delays the center-of-mass velocity decreases for increasing delays (although not as fast as in the few particle case). However, for large delays we find the surprising result that the presence of a delay can improve the flux performance of the ratchet. In fact, the maximum flux obtained with the optimal periodic protocol is attained. This implies that the delayed feedback protocol considered can perform better than its non-delayed counterpart. The improvement of the flux observed in the presence of large delays is the result of the emergence of a new dynamical regime where the presence of the delayed feedback stabilizes quasiperiodic solutions that resemble the solutions obtained in a certain closed-loop protocol with thresholds. In addition, in this new regime the system presents multistability, i.e. several quasiperiodic solutions can be stable for a fixed time delay.
Brownian motors or ratchets are mechanisms that induce transport rectifying the motion of Brownian particles through the introduction of a time-dependent perturbation that drives the system out of equilibrium rei02 (). This type of systems allow to get insight in non-equilibrium processes rei02 (). In addition, they are also important due to their applications to many fields as nanotechnology and biology rei02 (); lin02 ().
The two main types of ratchets are rocking ratchets mag93 (); ast94 () and flashing ratchets ajd93 (); ast94 (). In rocking ratchets (also called tilting ratchets) the perturbation acts as a time-dependent additive driving force, which is unbiased on the average, while in flashing ratchets (also called pulsating ratchets) the time-dependent perturbation changes the potential shape without affecting its spatial periodicity. An example of a flashing ratchets is a ratchet that operates switching on and off a spatially periodic asymmetric potential. In this particular case it can be seen that a simple periodic or random switching can rectify thermal fluctuations and produce a net current of particles.
A new class of ratchets that use information on the state of the system to operate have been introduced in Ref. cao04 (). These feedback ratchets (or closed-loop ratchets) are able to increase the net current and the power output of collective Brownian ratchets cao04 (); din05 (); fei06 (); fei07 (). Feedback can be implemented monitoring the positions of the particles (see for example Refs. rou94 (); mar02 ()) and subsequently using the information gathered to decide whether to switch on or off the ratchet potential according to a given protocol. In addition, feedback ratchets have been recently suggested as a mechanism to explain the stepping motion of the two-headed kinesin bie07 ().
The first feedback protocol proposed was the so-called instantaneous maximization of the center-of-mass velocity cao04 (), which switches on the potential only if switching on would imply a positive displacement for the center-of-mass position (i.e., if the net force with the potential on would be positive). The instantaneous maximization protocol gives the maximum current in the case of one particle and performs better than any open-loop protocol for few particles. However, it has a very low performance in the many particle case given an average center-of-mass velocity smaller than that obtained with an optimal periodic protocol. (We call many particle case the case when the fluctuations of the net force are smaller than its maximum absolute value.) An improvement of the instantaneous maximization protocol is the threshold protocol din05 (), which consist on introducing two threshold values in order to switch the potential before the net force reaches a zero value. In this way, there is an increase of the performance for many particles up to velocity values equaling the ones of the optimal open-loop periodic protocol.
In order to check if it is experimentally feasible to obtain the increase of performance theoretically predicted for the few particle case one important question is to check the effects of time delays in the feedback that would be present in any experimental implementation. These time delays in the feedback come from the fact that the measure, transmission, processing, and action steps take a finite time interval ste94 (); bec05 (). Time delays in the feedback also appear naturally in complex systems with self regulating mechanisms boc00 (); fra05b (). Recently, we have investigated the effects that the delay has in the operation of feedback controlled ratchets in the few particle case fei07b (). We have found that even in the presence of time delays feedback controlled ratchets can give better performance than the corresponding optimal open-loop ratchet, although time delays decrease the performance.
In this paper we investigate the effects of time delays in the instant maximization protocol for the many particle case. We find that for small delays the asymptotic average center-of-mass velocity decreases for increasing delays (although not as fast as in the few particle case). However, if we continue increasing the time delay the average velocity starts to increase up to the value obtained for an optimal open-loop protocol. This surprising result makes that for many particles the instant maximization protocol gives greater average velocities in the presence of delay than in its absence. In Sec. II we present the evolution equations of the system. In the next section, Sec. III, we briefly review the results for zero delays that will be useful, and thereafter we expose the results in the two dynamical regimes: small delays and large delays. Finally, in Sec. IV we summarize and discuss the results.
Ii The model
The feedback ratchet we consider consists of Brownian particles at temperature in a periodic potential . The force acting on the particles is , where the prime denotes spatial derivative. The state of this system is described by the positions of the particles satisfying the overdamped Langevin equations
where is the friction coefficient (related to the diffusion coefficient through Einstein’s relation ) and are Gaussian white noises of zero mean and variance . The control policy uses the sign of the net force per particle,
as follows: The controller measures the sign of the net force and, after a time , switches the potential on () if the net force was positive or switches the potential off () if the net force was negative. Therefore, the delayed control protocol considered is
with the Heaviside function [ if , else ].
As ratchet potential we consider the ‘smooth’ potential of period and height (Fig. 1 a),
We have also verified that analogous results are obtained for the ‘saw-tooth’ potential of period , i.e. , height , and asymmetry parameter (Fig. 1 b),
The height of the potential is the potential difference between the potential at the minimum and at the maximum, while is the distance between the minimum and the maximum positions. In view of this definition, the ‘smooth’ potential (4) has asymmetry parameter .
Throughout the rest of this paper, we will use units where , , and .
We consider in this paper the many particles case that is characterized by the fact that the typical fluctuations of the net force are smaller than the maximum values of its absolute value.
Iii Delayed many particle feedback ratchet
We study the effects of time delays in the previous feedback controlled Brownian ratchets in the many particles case, considering both the ‘smooth’ potential and the ‘saw-tooth’ potential for various potential heights and different initial conditions.
We find that the system presents two regimes separated by a delay for which the center-of-mass velocity has a minimum; see Fig. 2. In the small delay regime () the flux decreases with increasing delays as one could expect. On the contrary, in the large delay regime () we have observed and explained a surprising effect, namely, the center-of-mass velocity increases for increasing delays and the system presents several stable solutions. We have found that this critical time delay is inversely proportional to the potential height with a proportionality constant that mildly depends on the number of particles.
iii.1 Zero delay
The many particle ratchet in absence of delay (i.e., in the model of Sec. II) have been studied in Ref. cao04 (). It has been shown that the net force per particle exhibits a quasideterministic behavior that alternates large periods of time with (on dynamics) and large periods of time with (off dynamics). The center-of-mass velocity can be computed as
where and are obtained fitting the displacement during the ‘on’ evolution for an infinite number of particles (see Ref. fei06 () for details).
On the other hand, for many particles the fluctuations of the net force are smaller than the maximum value of the net force. This allows the decomposition of the dynamics as the dynamics for an infinite number of particles plus the effects of the fluctuations due to the finite value of . The late time behavior of the net force for an infinite number of particles is given for the on and off dynamics by cao04 (),
The coefficients , , and can be obtained fitting this expression with the results obtained integrating a mean field Fokker-Planck equation obtained in the limit and without delay; see Refs. cao04 (); fei06 () for details. For a finite number of particles the fluctuations in the force induce switches of the potential and the times on and off are computed equating to the amplitude of the force fluctuations, resulting cao04 ()
with and .
iii.2 Small delays
For small delays, , we observe that the flux decreases with the delay. See Fig. 2. We have seen that this decrease is slower than that found for the few particle case fei07b (), and that the expressions derived to describe this decrease in the few particles case does not hold here. However, the decrease observed here can be understood by the fact that the delay implies an increase of the time interval between switches, which makes the tails of longer than for no delay and the form of less smooth than for no delay. See Fig. 3. The main effect of the delay is to stretch the ‘on’ and ‘off’ times of the dynamics; then, using the many particle approximation cao04 () we can write
where we have found that the increase of the length of the on-off cycle is proportional to the delay .
iii.3 Large delays
After the minimum flux is reached for , the flux begins to increase with the time delay (see Fig. 2). This increase is due to a change in the dynamical regime: for the present net force starts to be nearly synchronized with the net force a time ago. This selfsynchronization gives rise to a quasiperiodic solution of period . Note that there is not a strict periodicity due to stochastic fluctuations in the ‘on’ and ‘off’ times. Looking at the dependence, Fig. 4, we see that the solutions stabilized by the selfsynchronization are similar to those obtained with the threshold protocol din05 (); fei06 (). In Fig. 2 we show that the threshold protocol that has the same period gives similar center-of-mass velocity values, confirming the picture. (Differences are due to the fact that we have considered for the threshold protocol simulations with on and off thresholds of the same magnitude, while Fig. 4 shows that the effective thresholds are different.)
This picture allows to understand the increase of velocity for increasing delay, and the presence of a maximum. This maximum is related with the optimal values of the thresholds that have been shown in fei06 () to give a quasiperiodic solution of period , with and the optimal ‘on’ and ‘off’ times of the periodic protocol. Therefore, if we know the values of and for the optimal periodic protocol [ and ] we can predict that the maximum of the center-of-mass velocity is reached for a delay
and has a value
with the center-of-mass velocity for the optimal open-loop protocol. Thus, this expression gives the position and height of the maximum of the delayed feedback control protocol in terms of the characteristic values of the optimal open-loop control. In particular, it implies that the position and height of the maximum for the flux is independent of the number of particles.
As an example we can apply these expressions to the ‘smooth’ potential with that for the optimal periodic protocol gives for and , so we obtain in agreement with Fig. 2.
For values of the delay of the order of or larger than quasiperiodic solutions of other periods start to be stable; see Fig. 5. The periods for the net force that are found are those that fit an integer number of periods inside a time interval , verifying that the present net force is synchronized with the net force a time ago, that is, the quasiperiodic solutions have periods , , In addition, it can be seen that the center-of-mass velocity of the branch whose has period is related with that of the branch through
We highlight that several branches can be stable for the same time delay . Whether the system finally goes to one or another stable solution depends on the initial conditions and on the particular realization of the noise. See Fig. 5. For these branches we have found initial conditions that goes to these solutions and that remain in them during several thousands of periods, indicating that they are stable solutions or at least metastable solutions with a large lifetime.
The analogy with the threshold protocol allows to use the analytic results of fei06 () to get further insight in the numerical results. The behavior for large delays for the branch can be obtained using the relation
On the other hand, for very large values of the delays of the first branch the solutions in a given branch start to become unstable, what can be understood noting that this happens when the fluctuations of the net force become of the order of the absolute value of the net force. Thus, the maximum delay that gives a stable solution in the first branch is
where and are determined as in Eq. (9). For example, for the ‘smooth’ potential with , which has and , we obtain for particles the value in accordance with the numerical results shown in Figs. 2 and 5.
We have studied the effects of time delays in the many particle case, where surprising and interesting results arise. Although in the many particle case without delay the instantaneous maximization protocol performs worst than the optimal open-loop protocol, the introduction of a delay can increase the center-of-mass velocity up to the values given by the optimal open-loop control protocol.
For small delays the asymptotic average velocity decreases for increasing delays, until it reaches a minimum. After this minimum, a change of regime happens and the system enters a selfsynchronized dynamics with the net force at present highly correlated with the delayed value of the net force used by the controller. This selfsynchronization stabilizes several of the quasiperiodic solutions that can fit an integer number of periods in a time interval of the length of the time delay. The stable quasiperiodic solutions have an structure similar to those solutions appearing in the threshold protocol. This analogy has allowed us to make numerical and analytical predictions using the previous results for the threshold protocol fei06 (). In particular, we have established the location and value of the maximum, and also the value of the time delay beyond which a quasiperiodic solution becomes unstable. The results obtained shown that for most time delays several solutions are stable and therefore the systems presents multistability; which stable solution is reached depends on the past history of the system.
The possibility to choose the quasiperiod of the solution we want to stabilize just tuning the time delay can have potential applications to easily control the particle flux. Note that we can even leave some branch just going to time delays where the branch is already unstable, and force the system to change to another branch of solutions.
Acknowledgements.We acknowledge financial support from the MEC (Spain) through Research Projects FIS2005-24376-E and FIS2006-05895, and from the ESF Programme STOCHDYN. In addition, MF thanks the Universidad Complutense de Madrid (Spain) for support through grant “Beca Complutense”.
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