June 2007 EPL, 78 (2007) 50004 doi: 10.1209/0295-5075/78/50004
www.epljournal.org
Pulsating potential ratchet P. Reimann and M. Evstigneev Universit¨ at Bielefeld, Fakult¨ at f¨ ur Physik - Universit¨ atsstr. 25, 33615 Bielefeld, Germany received 7 February 2007; accepted in final form 20 April 2007 published online 22 May 2007 PACS PACS PACS
05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion 02.50.Ey – Stochastic processes 05.60.Cd – Classical transport
Abstract – We consider Brownian motion in a periodic array of potential wells. The potential barriers between adjacent wells are spatially symmetric and pulsating periodically in time. While the modulation amplitude is the same for all barriers, the spatial symmetry is broken by sequentially alternating the respective pulsation frequencies among three different incommensurate values. The result is a net motion (ratchet effect), whose direction depends in an intriguing way on the detailed choice of parameters. c EPLA, 2007 Copyright
Introduction. – Our present theoretical study is mainly motivated by closely related ongoing experiments with colloidal particles by C. Bechinger and collaborators [1,2]. The general context are ratchet effects, i.e. directed transport in anisotropic periodic systems far from thermal equilibrium [3–7]. To the best of our knowledge, our present variant of the effect has not been considered before and brings along several quite interesting new facets. Conceptually, it may also be of some interest with respect to intracellular transport and ion pumps [3,4,7], though the details of those biological systems are clearly quite different. While a ratchet effect is not possible in spatially symmetric systems, we investigate a model where this symmetry is broken dynamically [8]. We consider the motion of a Brownian particle in a potential modulated in a specific way: while the time-averaged potential remains symmetric and spatially periodic, its different maxima are changing in time with different frequencies. Experimentally, this can be realized, e.g., in a system of identical colloidal particles positioned along two parallel straight lines at equal distances by means of laser traps [2]. For one of the straight lines, particles can be somewhat displaced in the perpendicular direction independently of one another. Hence, a further colloidal particle confined in between these two straight lines effectively experiences such a symmetric periodic potential with varying potential barriers. An alternative experimental realization employs a single colloidal particle in a periodic array of equally spaced laser traps, whose intensities can be independently modulated in time [9].
Model. – The above-mentioned experimental setup can be theoretically described as one-dimensional Brownian motion of an overdamped particle with coordinate x(t) in a pulsating periodic potential V (x, t), η x(t) ˙ =−
∂ V (x(t), t) + 2ηkB T ξ(t). ∂x
(1)
Here, η is the viscous friction coefficient, thermal fluctuations are modeled by unbiased δ-correlated Gaussian noise ξ(t), T stands for the temperature, and kB for Boltzmann’s constant. The “backbone” of the pulsating potential V (x, t) consists of its time-averaged static part V0 (x) := t limt→∞ t−1 0 V (x, t ) dt, which is assumed to be spatially periodic and symmetric, i.e. V0 (x + L) = V0 (x) and V0 (−x) = V0 (x + ∆x) for some ∆x. Hence, if only this average potential were acting in (1), both spatial directions would be equivalent and the average transport current 1 t x ˙ := lim x(t ˙ ) dt (2) t→∞ t 0 would be zero. Next, the time dependence of the potential V (x, t) is assumed to amount to variations of the potential barriers between any pair of neighboring potential wells of V0 (x), see fig. 1. While the temporal variations may be different for any barrier, they are assumed to always maintain the symmetry about the potential maxima. Due to this symmetry, one might naively expect that, e.g., in fig. 1 transitions from x = 0 to x = L are equally likely as those from x = L to x = 0, and similarly for the
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P. Reimann and M. Evstigneev Next, the full pulsating potential V (x, t) is obtained by modulating V0 (x) between any pair of neighboring minima at x = (i − 1)L and x = iL with a time dependent multiplicative factor fi (t), cf. fig. 1. In the simplest case, every fi (t) switches periodically between the same two values 0 and 2 with some frequencies ωi , i.e. the potential barrier between x = (i − 1)L and x = iL is either “on” or “off”, and on the average V0 (x) is reproduced. Finally, we assume that every third barrier is pulsating at the same frequency, Fig. 1: Sketch of the pulsating potential V (x, t). Solid line: time-averaged potential V0 (x). The dashed lines and the arrows indicate the time-dependent variations of V (x, t) about V0 (x). The modulations of the potential barriers all have the same amplitudes but are otherwise independent of each other.
transitions over any other potential barrier. Accordingly, one might expect that the average current (2) will be zero, independently of any further details of the model. In the remainder of the paper we will demonstrate and explain why this is not true and that the direction of the current (2) depends in a very complicated way on various details and quantitative parameter values of the model. To begin with, we note the following three general symmetry properties: i) Replacing the pulsating potential V (x, t) in (1) by V (x + ∆x, t + ∆t) + ∆V (t) does not change the average current in (2) for arbitrary ∆x, ∆t, and any time-dependent function ∆V (t). ii) Replacing V (x, t) by V (−x, t) is tantamount to inverting the x-axis in (1) and hence the current x ˙ in (2) will change its sign. iii) By introducing z(t) := x(−t) we can infer [10] that z(t) ˙ = −x(−t) ˙ and hence by averaging that z ˙ = −x. ˙ On the other hand, one readily sees that z(t) satisfies the same dynamics as x(t) in (1) except that V (x, t) has to be replaced by −V (x, −t) and ξ(t) by −ξ(−t). Since the latter two stochastic processes are statistically equivalent, we can conclude that turning the potential “upside down” according to V (x, t) → −V (x, −t) implies x ˙ → −x. ˙ In particular, if there exist ∆x, ∆t, and ∆V (t) such that −V (x, −t) = V (x + ∆x, t + ∆t) + ∆V (t), then we can infer from i) and iii) that x ˙ = 0 [10]. Furthermore, ii) and iii) imply that V (x, t) → −V (−x, −t) leaves x ˙ invariant. In this sense, the case of pulsating potential barriers, on which we focus henceforth, is equivalent to the case of “pulsating potential wells”, which may be a more realistic model, e.g., in the above-mentioned experiment with a single particle in an array of laser traps with periodically modulated intensities. In the remainder of the paper we focus on the simplest example of a purely sinusoidal V0 (x) which takes the value zero at its minima at x = iL, i ∈ Z, and with potential barriers of height A, V0 (x) = A
1 − cos(2πx/L) . 2
(3)
V (x, t) = V0 (x) fi (t) for x ∈ [(i − 1)L, iL],
(4)
fi (t) = 1 + sign{sin(ωi t)}, ωi = ωi+3 .
(5)
In principle, one could also include phases ϕi into the periodic functions fi (t). However, we will always tacitly restrict ourselves to the generic case that the three frequencies ω1 , ω2 , ω3 are incommensurate and hence the time averaged current in (2) is the same for any choice of those phases. Alternatively, in the case of commensurate frequencies an average over those phases is tacitly included in (2). Without such an average, the quite intriguing situation of a qualitatively different behavior of the current (2) would arise whenever the frequencies change from commensurate to incommensurate. On the other hand, trivial transport effects would occur, which we want to avoid. The simplest example consist in the choice ω1 = ω2 = ω3 and ϕi = 2πi/3, for which V (x, t) basically becomes a “traveling wave potential” which drags along the particle x(t) in (1) in a rather trivial, pumplike manner [5]. In practice, those phase-dependences are usually far beyond the accuracy of numerical simulations or experimental measurements unless the frequency ratios are rationals n/m with small integers n, m. Hence, if apparently rational frequency ratios appear in the following, one of the frequencies is tacitly understood to be multiplied by a factor very close to unity, e.g. 1.0001 in the actual numerical simulations, so that the ratios are always “practically irrational”. Basic physical mechanism. – In the following, we focus on the specific model (1), (3)-(5) with the ratio A/kB T so large that thermally activated transitions over any barrier can safely be neglected while this barrier is “on” (i.e. fi (t) = 2 in (5)). The reason is that for this special case the basic physical mechanisms are particularly easy to understand and the resulting effects are quantitatively most pronounced. We start with the example of a particle in well 0 (see fig. 1) with both adjacent potential barriers being “on”. Since A/kB T is large, the particle remains within a very small vicinity of x = 0 with very high probability. Now, assume the right potential barrier is switched “off” for a relatively short time, i.e. the frequency ω1 in (5) is relatively large. To escape from the original potential well at x = 0, the particle must diffusively travel at least half the distance to the next well at x = L while the barrier is
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Pulsating potential ratchet “off”. Denoting the probability for doing so by pesc (ω1 ), one readily finds that (6) ωi /ω ∗ , pesc (ωi ) = κ erfc ω ∗ := 16πkB T /ηL2 , (7) −1/2 ∞ −y 2 where erfc(x) := 2π e dy is the complementary x error function. In doing so, we have tacitly assumed that i) the barrier to the left of x = 0 is always “on”, hence x = 0 can be approximately treated as a reflecting boundary, and ii) the effect of all further barriers beyond x = L is negligible. Under these assumptions the factor κ in (6) is exactly unity. If only ii) is satisfied, i.e. the barrier to the left of x = 1 happens to be “off”, one finds κ = 1/2. Fig. 2: The function k(ω) from eq. (10). ω1 and ω ˜ 1 exemplify ω1 ). Adopting the very crude approximation that the barrier two different frequencies with the same rate k(ω1 ) = k(˜ to the left is “on” and “off” with probability 1/2, we obtain κ ≈ 3/4 .
(8)
Assumption ii) breaks down, i.e. whether or not the subsequent barrier between x = L and x = 2L is “on” or “off” starts to matter, very roughly speaking if (6) with (8) lead to escape probabilities exceeding 1/2. Observing that erfc(0.48) 0.5 and 0.482 0.23, it follows that the argument ωi in (6) must be restricted to the regime ωi > ωmin ≈ 0.23 ω ∗ .
(9)
The actual time-dependent transition probability from potential well 0 to well 1 (see fig. 1) is quite complicated. It is almost zero while the barrier is “on” and then gradually increases during the “off”-phase. By integration over the entire driving period τ1 := 2π/ω1 one recovers pesc (ω1 ) from (6). As far as the occupation probability of potential well 0 is concerned, the actual time-dependent escape rate can be approximated by its time-average k(ω1 ) := pesc (ω1 )/τ1 under the proviso that the total escape probability pesc (ω1 ) itself remains reasonably small, say smaller than 1/2. As seen above, this condition is equivalent to (9). Similar considerations apply to the escape probability from any potential well at x = (i − 1)L to the adjacent minimum at x = iL and also for the backtransitions from iL to (i − 1)L. The main conclusion is the following approximation for the escape rate k(ωi ) across the i-th pulsating barrier within the frequency range (9):
ωi kT ωi k(ωi ) = 6 2 ∗ erfc . (10) ηL ω ω∗ The most prominent feature of this function —see fig. 2— is its non-monotonicity with a maximum at ωmax 0.7 ω ∗.
(11)
The basic physical reason for this non-monotonicity is that the thermally driven escape probability per unit time goes to zero both for very short and very long “on-off” cycles. We emphasize that this basic mechanism is very robust
and general, it is by no means restricted to our specific pulsating potential from (3)-(5). We remark that the non-monotonicity of the escape rate in fig. 2 is related to the so-called resonant activation effect [11,12], though the details of the considered systems and the essential physical mechanisms are somewhat different in the two cases. Next we analyze the average particle current (2) as generated by the ratchet model (1) with three pulsating potentials (3)-(5). A particularly interesting situation arises in the case that two barriers are pulsating rather fast and one rather slowly, say ω1 , ω2 > ωmin, ω3 < ωmin .
(12)
Hence, the transitions across barriers 1 and 2 can be approximately described in terms of the average rates from (10), whereas the effect of barrier 3 can be understood as follows. While 3 is “on”, thermally activated transitions across this high barrier are extremely rare and hence no significant particle current is possible. Since the average transition rates in both directions are approximately equal across both barrier 1 and barrier 2, all three potential wells are approximately equally populated at the end of the rather long “on-phase” of barrier 3. When barrier 3 turns “off”, the previous wells at x = 2L and x = 3L merge into one extended lowland and their previously rather sharply peaked probability distributions go over into a much more diluted, approximately constant distribution between x = 2L and x = 3L. Hence, the transitions out of well 1 across barriers 1 and 2 are unchanged just after barrier 3 has disappeared, while the back-transitions into well 1 are drastically reduced. Thus, a net transport of particles results in one or the other direction until well 1 is so much depleted that a new steady-state situation is approached at the end of the “off-phase” of barrier 3. The direction of this transport is given by the direction of escapes out of well 1 across that barrier with the larger rate (10). Finally, barrier 3 returns to “on” so that initially the wells 2 and 3 are equally populated, while well 1 has a much lower population. Due to this symmetry
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P. Reimann and M. Evstigneev between 2 and 3, no transient particle flux arises during the subsequent convergence towards the equally populated steady state at the end of the “on-phase”. Theoretical predictions and comparison with numerical results. – The upshot of our preceding line of reasoning is the prediction of a current reversal [13] by means of changing one of the control parameters of the system: under the condition (12), the direction of the net particle current (2) will be positive if the escape rate k(ω2 ) over the barrier to the right of the well 1 is larger than the competing rate k(ω1 ), and negative otherwise. To be specific, we focus on the case that ω1 > ωmax and ω3 < ωmin are fixed, while ω2 is considered as a control parameter. If ω2 coincides with one of the other two frequencies then for symmetry reasons we can infer that x ˙ = 0. Furthermore, according to the above prediction we ˙ > 0 for can read off from fig. 2 that x ˙ < 0 for ω2 > ω1 , x ω1 , ω1 ), and x ˙ < 0 for ω2 ∈ (ωmin , ω ˜ 1 ), where ω ˜ 1 is ω2 ∈ (˜ implicitly defined via k(˜ ω1 ) = k(ω1 ). In other words, we predict three sign changes of x ˙ as a function of ω2 , two symmetry-induced ones at ω1 and ω3 , and a further one in between at ω ˜ 1 without any underlying symmetry reason. Regarding quantitative predictions, our system is difficult to treat analytically, because it is far from thermal equilibrium, does not possess a small parameter, and is characterized by the coexistence of several competing effects favoring opposite current directions. For this reason we have not been able to provide an analytic approximation for the current. Leaving this interesting problem for future studies, we verify our qualitative predictions by numerically simulating the Langevin equation (1). In doing so, special care has been taken to verify that our simulation results are robust with respect to further reduction of the time step and small variations of the system’s parameters. The results of our numerical simulations presented in fig. 3 are seen to fully confirm the qualitative predictions above. Three further basic features of fig. 3 can also be intuitively understood as follows. i) For asymptotically large ω2 the particle is no longer able to resolve the very fast pulsations of the second barrier and effectively is left with the time averaged potential barrier as described by V0 (x). Since its amplitude A in (3) is much larger than the thermal energy kB T , transitions over that barrier become very rare and hence the particle current x ˙ approaches a very small negative limit for ω2 → ∞. ii) The case ω2 < ωmin but ω2 > ω3 can be understood as follows. If barrier 3 is “on”, there is no appreciable particle current. If barrier 3 is “off” and barrier 2 is “off” as well, a practically uniform probability distribution is approached between x = L and x = 3L with equal transition probabilities in both directions across barrier 1. The moment barrier 2 turns “on”, this balance is disturbed, resulting in an excess of escapes in the negative direction. The overall result is x ˙ < 0. In the opposite case ω2 < ω3 one readily concludes in the same way that x ˙ > 0. iii) For ω2 → 0, barrier 2
Fig. 3: Average current x ˙ vs. frequency ω2 from numerical simulations of the model (1), (3)-(5) in dimensionless units with L = 1, η = 1, kB T = 1, ω1 = 50, ω3 = 2. According to (7) we thus have ω ∗ 50. For example, one time unit then translates into one second and one length unit into 1 µm for a spherical colloidal particle in water at room temperature, if its diameter is about 0.4 µm.
Fig. 4: Same as fig. 3 but keeping ω2 = 10 fixed and varying η instead.
is half of the time “on”, admitting no current, and half of the time “off”, resulting according to ii) in a finite current x ˙ > 0, which is constant since ω1 and ω3 are kept constant. The result is a finite positive net current x ˙ in the limit ω2 → 0. iv) According to ii) and iii) we expect no further sign change of x ˙ in the regime ω2 < ωmin apart from that at ω2 = ω3 . Two particularly interesting effects are obtained by fixing ω2 to the value at the intermediate sign change of x, ˙ e.g. ω2 10 in fig. 3. Since this current inversion is not connected with any underlying symmetry, it follows that x ˙ will also exhibit a sign change when keeping ω2 fixed and varying instead another parameter of the system [5]. As a first example, variations of the friction coefficient η are illustrated in fig. 4. For instance, in the case of colloidal particles, the friction coefficient η
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Pulsating potential ratchet to be rather small in order to achieve appreciable currents x, ˙ see fig. 3. Alternative experimental realizations which naturally are conducted on much smaller length scales and therefore will not have those problems, may be cold atoms in resonance with laser induced optical lattices [14,15]. The same applies to molecular motors and ion pumps in biological systems, for which the basic physical mechanism of our pulsating potential ratchet possibly may play a role as well [3–7] in the modified version of randomly rather than periodically pulsating potentials. ∗∗∗
Fig. 5: Same as fig. 3 but keeping ω2 = 10 fixed and varying ω1 instead.
We thank C. Bechinger, S. Bleil, R. Eichhorn, and C. Schmitt for stimulating discussions. This work was supported by Deutsche Forschungsgemeinschaft under SFB 613, RE 1344/3-1, and RE 1344/4-1.
depends, among others, on size and shape of the particles. REFERENCES Hence, different types of particles are predicted to move into opposite directions in the same pulsating potential [1] For a recent review, see Babic D., Schmitt C. and ratchet device if the parameters are fine-tuned according Bechinger C., Chaos, 15 (2005) 026114. to fig. 4. As a second example, variations of ω1 are shown [2] Bleil S., Reimann P. and Bechinger C., Phys. Rev. E, 75 (2007) 031117. in fig. 5. Quite remarkably, the current x ˙ now changes its ¨licher F., Ajdari A. and Prost J., Rev. Mod. Phys., [3] Ju sign five times, twice due to symmetry reasons, and three 69 (1997) 1269. times without any underlying symmetry. The theoretical ´nyi I., Eur. Biophys. J., 27 [4] Astumian R. D. and Dere explanation of this behavior remains as an open problem (1998) 474. for future studies. Our above-developed analysis of the [5] Reimann P., Phys. Rep., 361 (2002) 57. basic physical mechanisms cannot be applied in this case [6] Linke H. (Special issue Guest Editor), Appl. Phys. A, 75 for two reasons. First, k(ω2 ) and k(ω3 ) by definition (2002) 167. will coincide within this approximation, implying x ˙ =0 [7] Klafter J. and Urbakh M. (Special issue Guest independently of any other parameter value. Second, ω2 Editors), J. Phys.: Condens. Matter, 17 (2005) S3661. practically coincides with ωmin , hence neither the rate [8] Wonneberger W., Solid State Commun., 30 (1979) 511; description for quickly pulsating barriers nor our “on-off” Neumann E. and Pikovsky A., Eur. Phys. J. B, 26 (2002) 219; Breymayer H.-J., Risken H., Vollmer considerations for slowly pulsating potentials apply. Conclusions. – The basic ingredient for a ratchet effect in a pulsating potential and its intriguing parameter dependence, as exemplified in figs. 3-5, is the nonmonotonicity of the average escape rate across a single pulsating potential barrier, see fig. 2. Since such a nonmonotonicity is a quite generic feature, our main findings are independent of many special properties of the considered model (1), (3)-(5), in particular the assumptions that the dynamics is one-dimensional and that the potential barriers jump discontinuously between rather high values (compared to the thermal energy) and a completely “flat” configuration. However, smooth and/or less pronounced time variations and/or smaller maximum potential barriers will lead to quantitatively smaller currents under otherwise unchanged conditions. On the other hand, even stronger potential modulations, such that barriers and wells even exchange their roles in the course of time, will further accelerate the particle current, but are also more difficult to realize experimentally. Apart from that, a further experimental difficulty in the case of colloidal particles may be that they are required
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