Overdamped deterministic ratchets driven by multifrequency forces

Vol. 37 (2006) ACTA PHYSICA POLONICA B No 5 OVERDAMPED DETERMINISTIC RATCHETS DRIVEN BY MULTIFREQUENCY FORCES∗ David Cubero, Jesús Casado-Pascual, Azucena Alvarez Manuel Morillo Universidad de Sevilla, Facultad de Física Apdo. Correos 1065, Sevilla 41080, Spain Peter Hänggi Institut für Physik, Universität Augsburg Universitätsstraße 1, 86135 Augsburg, Germany (Received February 21, 2006) We investigate a dissipative, deterministic ratchet model in the overdamped regime driven by a rectangular force. Extensive numerical calculations are presented in a diagram depicting the drift velocity as a function of a wide range of the driving parameter values. We also present some theoretical considerations which explain some features of the mentioned diagram. In particular, we proof the existence of regions in the driving parameter space with bounded particle motion possessing zero current. Moreover, we present an explicit analytical expression for the drift velocity in the adiabatic limit. PACS numbers: 05.60.–k, 05.45.Pq, 05.45.Ac, 05.45.Xt 1. Introduction and model set-up Directed current in ratchet systems have received much attention over recent years [1–5]. One of the reasons to study these type of systems is motivated by the attempt to understand the physical mechanism of motion of molecular motors in biological systems [2,6] and to investigate its role in the design of new material properties [3,4]. The dynamics of a particle moving in a periodic potential under the action of an applied time periodic term is rather complex and rich, displaying typically even chaotic behavior [7–10]. A feature of particular interest is the emergence of directed currents in the ∗Presented at the XVIII Marian Smoluchowski Symposium on Statistical Physics, Zakopane, Poland, September 3–6, 2005. (1467) 1468 D. Cubero et al. system response to an external time-periodic driving with zero time-averaged value. Though most of the works about ratchet systems consider the presence of noise, this phenomenon may also arise in deterministic systems, both in the overdamped [11–13] and underdamped [7, 8] regimes. Moreover, the phenomenon of anticipated synchronization occurring in inertial deterministic ratchets have been studied recently [14]. In particular, in the overdamped regime, the one-dimensional particle dynamics x(t)usually considered is governed by a first order differential equation of the type ˙x(t) = −U′[x(t)] + F(t),(1) where the dot and the prime denote time and spatial derivatives, respectively, U(x)is a periodic potential with spatial period λ[i.e.,U(x+λ)=U(x)], and F(t)is a time-periodic driving force with period T[i.e.,F(t+T)=F(t)]. In this type of systems, the current is defined as the average velocity v= lim t→∞ x(t)−x(0) t.(2) As shown in Ref. [11] co-existing attractors can exist for large driving strengths, which, however, are not current-carrying. A finite current v, possessing an unbounded x(t)-trajectory, is consequently independent of the initial condition x(0). It can also be shown that a directed current (v6= 0) is only possible if at least one of the following symmetries is broken [2,4,5,15]: ∃x0∈ ℜ such that U(x0−x) = U(x0+x),∀x∈ ℜ,(3) F(t+T/2) = −F(t),∀t∈ ℜ.(4) Our main interest in this paper is to gain a deeper insight into these type of systems exploring some quantitative and qualitative aspects of its very rich dynamics. Specifically, we will consider the same sawtooth potential as in Refs. [11,16] U(x) = −1 2πsin(2πx) + 1 4sin(4πx),(5) which has a spatial period λ= 1. In addition, rather than using a sinusoidal driving force F(t), in this work we will consider a multifrequency timeperiodic force given by F(t) = (Afor 0≤t < T 2 −Afor T 2≤t < T, (6) with Abeing a constant. Since the potential (5) breaks the spatial symmetry (3), a directed current is possible, even though the time symmetry (4) is fulfilled. Overdamped Deterministic Ratchets Driven by . . . 1469 The paper is organized as follows. In Sec. 2, we perform a detailed numerical study of Eq. (1) for a wide range of the driving parameter values A and ω= 2π/T. In particular, we present a colored phase-diagram vversus A and ω, which is inspired by the figure provided by Prof. Peter Talkner and collaborators in [16]. In order to explain some features of this diagram we propose in Sec. 3 some simple theoretical considerations. Finally, in the last section, we summarize our findings. 2. Dynamical regimes and numerical evaluation of the current Regardless of the value of the driving frequency, there exist two critical amplitudes A∗ 1= 3/4and A∗ 2= 3/2separating three different regimes. For amplitudes A∈[0, A∗ 1], both potential states, U(x) + Ax and U(x)−Ax, possess a periodic array of equilibrium points (see Fig. 1). Since the inertial term m¨x(t)is absent, the particle cannot cross these equilibrium points, and consequently, it remains trapped between them, leading to a zero current. For A∈(A∗ 1, A∗ 2], the potential state U(x) + Ax retains its equilibrium points, while U(x)−Ax does not have any, that allowing non-bounding motion in the positive direction. Finally, when A > A∗ 2, neither of the potential states possesses equilibrium points, and the particle motion is not bounded in either direction. -3 -2 -1 0 1 2 3 x -1 0 1 2 instantaneous velocity Fig. 1. Plots of the instantaneous velocities −U′(x)+A(solid line) and −U′(x)−A (dashed line) as a function of xfor a driving force with A= 0.5. (All quantities in dimensionless units.) In order to go further in our analysis, we have resorted to a numerical treatment of our model using the freely available integrator RKSUITE [17]. Wide regions of parameter space have been explored. In Fig. 2 we have used a color code to represent the drift velocity vfor a rectangular driving force of varying amplitude Aand frequency ω. The regions in black correspond to bounded, time-periodic particle motion with zero current. More precisely, the velocity in those regions is smaller than 10−6. The diagram has a rich and interesting structure. Regions of zero drift velocity are intermingled with regions with finite values, giving rise to a finger-shaped structure. 1470 D. Cubero et al. Notice that for the parameter values considered, the drift velocity is always positive or zero, showing the largest magnitude in the intermediate region A∈(A∗ 1, A∗ 2]. This last feature could be understood by taking into account that in this regime the motion in the positive direction is never compensated by motion in the opposite direction. 0 2 46 8 Ω 0 2 4 6 8 A 0 0.1 0.2 0.3 0.4 0.5 0.6 Fig. 2. The “Talkner”-current phase diagram. The drift velocity as a function of the driving parameters Aand ωis represented using a color density plot. Black color has been used for velocities less than 10−6. (All quantities in dimensionless units.) 0 2 4 6 8 ω 0 0.1 0.2 0.3 0.4 0.5 v Fig. 3. Drift velocity v(solid line) as a function of the driving frequency ωfor A= 1.4<A∗ 2. The dashed line shows the adiabatic value. The arrows depicts the zerovelocity bands given by the theory in the text. (All quantities in dimensionless units.) Overdamped Deterministic Ratchets Driven by . . . 1471 In Fig. 3 we present a section of the diagram for a driving amplitude in the intermediate regime A= 1.4. A series of peaks corresponding to the fingers in Fig. 2 are observed. Fig. 4 shows a representative section for A > A∗ 2. By contrast with Fig. 3, it presents an intermediate gap of particle localization. If we further increase the value of A, more intermediate gaps appear, as can be seen in the phase diagram. 0 2 4 6 8 ω 0 0.1 0.2 0.3 v Fig. 4. The same as in Fig. 3 but for A= 2 > A∗ 2. 3. Some theoretical results Even though the nonlinearity of the system precludes a complete and detailed analytical solution of the problem, it is possible to explain some features of the phase diagram by simple considerations. 3.1. Proof of existence of regions displaying particle localization Due to the truncation implicit in any numerical calculation, the simulations reported above are not able to distinguish between a situation of exact particle localization or a very small drift, lower than the tolerance we chose to determine the black regions in Fig. 2. Furthermore, a simple explanation of the existence of such regions in the non-trivial regime A > A∗ 1would be desirable. Let us assume that the particle starts at x0at the beginning of a driving period. It then moves under a force −U′(x) + Afor half a period until it reaches the position x1> x0. Then the driving force switches sign so the total force on the particle is now −U′(x)−A, which makes it (in general) move backwards up to a position x′ 0after another T/2. If x′ 0=x0, the particle has returned to the initial position, and consequently, the process is repeated successively, leading to a drift velocity strictly equal to zero. For this situation to happen the following equations must hold 1472 D. Cubero et al. T 2= x1 Z x0 dx −U′(x) + A(7) T 2= x0 Z x1 dx −U′(x)−A.(8) Subtracting both equations we arrive at the condition G(x0) = G(x1),(9) where G(x) = x Z 0 d˜xU′(˜x) A2−U′(˜x)2.(10) Therefore, we can determine the set of pairs (x0, x1)with zero current for a given driving strength Aby plotting the function G(x)versus x(see Fig 5). The intersection of a horizontal line with G(x)in this plot provides the possible values of the pair (x0, x1). The period Tassociated with the pair is then given by Eq. (7) or (8). Since the drift velocity vis independent of the initial conditions [15], it would be exactly zero for those driving parameter values Aand T. In Fig. 5, we plot G(x)for the same value of A= 2(> A∗ 2)as in Fig. 4. A horizontal line crosses G(x)at the points A, B, C, and D, providing the -2 -1 0 1 2 3 x -0.2 -0.1 0 0.1 0.2 G(x) A’ A B C D A’’ A’’’ B’ Fig. 5. Determination of zero-velocity bands with x(t+T) = x(t). Solid line depicts G(x), defined in Eq. (10), as a function of xfor the same value of Aas in Fig. 4. Overdamped Deterministic Ratchets Driven by . . . 1473 coordinates xA, xB, xC, and xD. If we choose x0=xA, and x1as any of the other points, we obtain three pairs of points with driving periods TAB, TAC, and TAD. Since the integrand in Eq. (7) is always positive, the further away x1is from x0, the larger the period, which implies TAB < TAC < TAD. Before we proceed in analyzing the plot in greater detail, let us study the symmetries of G(x). Since it is an integral of a space-periodic function [see Eq. (10)] we have G(x+ 1) = G(x) + φ , (11) where φ=G(1). In addition, taking into account that U′(x)is an even function, necessarily G(−x) = −G(x).(12) Property (11) leads to the fact that we only need to vary x0within a spatial interval of length unity, whereas property (11) combined with (12) implies that G(x)has an inversion center at x=n, with nany integer. The driving period T, when viewed as a function of x1or x0[see Eq. (7)], also obeys these properties. Because of these symmetry properties, choosing the point A as x0in Fig. 5 is equivalent to choosing any of the points A′, A′ ′ and A′′′, in the sense that they lead to the same driving periods. Let us consider a horizontal line between points A and A′. As this line gets closer to A, two of the intersection points approach each other, and consequently, the driving period associated to them tends to zero. Therefore, T= 0 (ω=∞) corresponds to particle localization. Moving up the line continuously we obtain a whole band of driving periods starting from zero up to a maximum value given by the pair (xB′, xA′)when the line crosses A′. This pair is equivalent to (xA, xB). Two more intermediate periods in the band can be obtained from the pairs (xB, xC)and (xC, xD)in the figure. The next value of the driving period obtained from Fig. 5 is the one given by the pair (xA, xC). This pair marks the start of a second band that ends at the maximum value given by the pair (xA, xD). Moving up the line up to A′ ′ gives all the intermediate values in the band. Clearly, the first band is associated with movement within a spatial interval less than unity (i.e.,0≤x1−x0<1), whereas the second band is related to displacements two times that distance (1≤x1−x0<2). In Fig. 4 we have indicated with arrows the calculated frequency bands. It can be seen that they do not cover the entire region with numerically evaluated zero velocity. This is due to the fact that the above discussed mechanism is just the simplest one leading to zero current. The drift velocity can also vanish because the particle returns to its initial position after two or more driving periods, instead of after the first one. 1474 D. Cubero et al. The same analysis can be carried out for a subthreshold driving A∈ (A∗ 1, A∗ 2]. In this case G(x)presents singularities at the equilibrium points, which prevents the particle from crossing those points when F(t) = −A. This leads to a single band which is near the zero-current region observed in the simulations, as shown in Fig. 3. 3.2. Adiabatic limit In this section we will obtain an analytical expression for the current in the adiabatic limit ω→0 vad(A) = lim ω→0v(A, ω).(13) Specifically, as proved later on, vad(A)is given by the average value vad(A) = 1 2[v+(A) + v−(A)] ,(14) where v+(A)and v−(A)are, respectively, the velocities in the presence of the static forces −U′(x) + Aand −U′(x)−A. Obviously, v+(A) = 0 for A∈[0, A∗ 1]and v−(A) = 0 for A∈[0, A∗ 2], as the particle ends up being trapped by an equilibrium point. For A > A∗ 1, v+(A) = 1 τ+(A),(15) where τ+(A) = 1 Z 0 dx A−U′(x)=2q2A+p−3 + 4A(1 + A) p(−1 + 2A)(3 + 2A)(−3 + 4A)(16) is the time taken for the particle to travel a distance equal to 1(the spatial period) in the presence of the static force −U′(x) + A. Analogously, for A > A∗ 2, v−(A) = −1 τ−(A),(17) where τ−(A) = 1 Z 0 dx A+U′(x)=p4A+ 2√3 + 4A+p4A−2√3 + 4A p(−3 + 2A)(1 + 2A)(3 + 4A)(18) is defined as τ+(A)but replacing −U′(x) + Aby −U′(x)−A. Overdamped Deterministic Ratchets Driven by . . . 1475 In order to prove Eq. (14), let us consider separately the three regimes A∈[0, A∗ 1],A∈(A∗ 1, A∗ 2], and A∈(A∗ 2,∞). For A∈[0, A∗ 1]the result in Eq. (14) is trivial, since vad(A) = v−(A) = v+(A) = 0. In the intermediate regime A∈(A∗ 1, A∗ 2], let us assume that the particle is initially located at a minimum of the potential U(x) + Ax (as we have mentioned before, the drift velocity does not depend on this particular initial condition). If we choose the time period of F(t)as T= 2Nτ+(A), with N= 1,2,3,..., then, during the first half-period, Nτ+(A), the particle stays trapped at the initial location. After the second half-period, 2Nτ+(A), the particle arrives at a new minimum of U(x)+Ax separated from the initial location by a distance N. During the third half-period, 3Nτ+(A), the particle stays trapped at that minimum, and so on. Thus, for a given A∈(A∗ 1, A∗ 2], all the frequencies ωN(A) = π Nτ+(A)(19) lead to the same value of the drift velocity v[A, ωN(A)] = N 2Nτ+(A)=1 2τ+(A)=v+(A) 2.(20) Consequently, taking into account that limN→∞ ωN(A) = 0, it follows that vad(A) = lim N→∞ v[A, ωN(A)] = v+(A) 2.(21) This proves Eq. (14) for A∈(A∗ 1, A∗ 2], since in this regime v−(A) = 0. Finally, for A∈(A∗ 2,∞), it is easy to prove that τ+(A)/τ−(A)is a continuous strictly increasing function of Awhich takes values in the interval (0,1). Let us assume first that for a given A∈(A∗ 2,∞)the ratio τ+(A)/τ−(A)is a rational number p/q, with p < q. Then, if we choose the time period of F(t)as T= 2Nqτ+(A) = 2Npτ−(A), with N= 1,2,3,..., the particle travels a distance equal to N(q−p)every time period. Consequently, all the frequencies ωN(A) = π Nqτ+(A)=π Npτ−(A)(22) lead to the same value of the drift velocity v[A, ωN(A)] = N(q−p) 2Nqτ+(A)=1 2τ+(A)−1 2τ−(A)=v+(A)+v−(A) 2.(23) Taking into account once again that limN→∞ ωN(A) = 0, it follows that vad(A) = lim N→∞ v[A, ωN(A)] = v+(A) + v−(A) 2.(24)