Ratchet effect on a relativistic particle driven by external forces

arXiv:1106.4861v1 [nlin.PS] 23 Jun 2011 Ratchet effect on a relativistic particle driven by external forces Niurka R. Quintero1, Renato Alvarez-Nodarse2and Jos´e A. Cuesta3 1Departamento de F´ısica Aplicada I, E. S. P., Universidad de Sevilla, C/ Virgen de ´ Africa 7, E-41011, Sevilla, Spain 2Departamento de An´alisis Matem´atico, Universidad de Sevilla, apdo. 1160, E-41080, Sevilla, Spain 3Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matem´aticas, Universidad Carlos III de Madrid, avda. de la Universidad 30, E-28911 Legan´es, Madrid, Spain E-mail: [email protected],[email protected],[email protected] Abstract. We study the ratchet effect of a damped relativistic particle driven by both asymmetric temporal bi-harmonic and time-periodic piecewise constant forces. This system can be formally solved for any external force, providing the ratchet velocity as a non-linear functional of the driving force. This allows us to explicitly illustrate the functional Taylor expansion formalism recently proposed for this kind of systems. The Taylor expansion reveals particularly useful to obtain the shape of the current when the force is periodic, piecewise constant. We also illustrate the somewhat counterintuitive effect that introducing damping may induce a ratchet effect. When the force is symmetric under time-reversal and the system is undamped, under symmetry principles no ratchet effect is possible. In this situation increasing damping generates a ratchet current which, upon increasing the damping coefficient eventually reaches a maximum and decreases toward zero. We argue that this effect is not specific of this example and should appear in any ratchet system with tunable damping driven by a time-reversible external force. PACS numbers: Submitted to: J. Phys. A: Math. Gen. Ratchet effect on a relativistic particle 2 1. Introduction The ratchet effect is identified with the motion of particles or solitons induced by zero-average periodic forces [1, 2], sometimes in the presence of thermal fluctuations. The effect arises as a subtle interplay between nonlinearities in the system and broken symmetries. Ratchets appear in many fields of physics, where net motion is generated either by an asymmetric, periodic, spatial potential [3, 4, 5, 6, 7, 8, 9, 10], or by an asymmetric temporal forcing [10, 11, 12, 13, 14, 15, 16, 17, 18]. In both cases the ratchet effect can be regarded as an application of Curie’s symmetry principle, which states that a symmetry transformation of the cause (forces) is also a symmetry transformation of the effect (ratchet velocity) [19, 20]. Most studies of ratchets driven by temporal forces employ a bi-harmonic forcing f(t) = ǫ1cos(qωt +φ1) + ǫ2cos(pωt +φ2),(1) where pand qare positive integers which, without loss of generality, can be taken coprime (otherwise common factors can be absorbed in the frequency ω) and ǫ1,ǫ2are small non-zero parameters. If both pand qare odd, the force (1) exhibits the shift symmetry (Sf)(t) = f(t+T/2) = −f(t), where T= 2π/ω. In systems invariant under time translations this implies that both, f(t) and −f(t) generate the same ratchet current (or velocity) defined as [21, 22] v= lim t→+∞ 1 tZt 0 ˙x(τ)dτ = lim t→+∞ x(t) t,(2) where x(t) is the position of the particle, soliton, or localized structure. If reversing the force changes the sign of the current, this current must be zero. So shift-symmetric bi-harmonic forces cannot induce a ratchet effect. In contrast, if pand qhave different parity, shift symmetry is broken by f(t) so the force can induce a nonzero net current [23]. Many attempts have been made to determine quantitatively the dependence of the ratchet velocity, v, on the parameters of the bi-harmonic force (1) [11, 24, 25]. Invariably, the analysis performed in these works rests on the so-called method of moments, where it is assumed that the average ratchet velocity can be expanded as a series of the odd moments of f(t), i.e. P∞ k=1h[f(t)]2k+1iwith hh(t)i=RT 0dt h(t). This method seemed to work for some systems but not for others without a clear reason and with no known criterion to tell ones from the others. We have recently shown that the moment method relies on an assumption that almost never holds, and have provided an alternative procedure that yields the correct result regardless of the system [23]. The aim of this paper is to provide explicit examples which illustrates this otherwise abstract method —the functional expansion of vin terms of f— using a working example for which an analytic solution can be found. The system represents the motion of a damped, relativistic particle under the effect of two different forces: a bi-harmonic force Ratchet effect on a relativistic particle 3 like (1), and a time-periodic piecewise constant force like f(t) =      ǫ1if 0 < t < Tl, 0 if Tl< t < T −Tl, −ǫ1if T−Tl< t < T. (3) To this purpose we introduce the model as well as its analytic solution in section 2. In section 3 we discuss the phenomenon of damping-induced ratchets. The formalism developed in [23] is fully illustrated for this problem in section 4. For these two specific driving forces it is also shown that the method of moments is valid only when the dynamics of the relativistic particle is overdamped, and fails otherwise. Conclusions are summarized in section 5. 2. Motion of a relativistic particle driven by a bi-harmonic force The equation of motion of a relativistic particle with mass M > 0, whose position and velocity at time tare denoted x(t) and u(t), respectively, is dx dt =u(t), x(0) = x0,(4a) Mdu dt =−f(t)(1 −u2)3/2−γu(1 −u2), u(0) = u0,(4b) where x0and u0are the initial conditions, γ > 0 represents the damping coefficient and f(t) is a T-periodic driving force. Notice that if the force f(t) satisfied (Sf)(t) = f(t+T/2) = −f(t), then (4b) would be invariant under a combination of shift symmetry (S:t7→ t+T/2) and the sign change x7→ −x. Changing the variable u(t) by the momentum P(t) = Mu(t) p1−u2(t)(5) transforms (4b) into the linear equation dP dt =−βP −f(t), P(0) = P0=Mu0 p1−u2 0 ,(6) where β=γ/M. Equation (6) is easily solved to give P(t) = P0e−βt −Zt 0 dzf(z)e−β(t−z).(7) From (5) one obtains u(t) = ∞ X k=0 −1 2k(2k−1)!! k!P(t) M2k+1 .(8) Let us now focus our attention on the T-periodic driving force f(t) given by (1) with p= 2 and q= 1 (the most common choice of parameters [26, 22, 14, 27]). Substituting (1) into (7) leads to P(t) = e P0e−βt −˜ǫ1cos(ωt +φ1−χ1)−˜ǫ2cos(2ωt +φ2−χ2), Ratchet effect on a relativistic particle 4 with e P0=P0+ ˜ǫ1cos(φ1−χ1) + ˜ǫ2cos(φ2−χ2), ˜ǫ1=ǫ1(β2+ω2)−1/2, χ1= tan−1(ω/β), ˜ǫ2=ǫ2(β2+ 4ω2)−1/2, χ2= tan−1(2ω/β). As t→ ∞ the momentum P(t) behaves, for any β > 0, as P(t)∼ −˜ǫ1cos(ωt +φ1−χ1)−˜ǫ2cos(2ωt +φ2−χ2), thus the term P(t)2k+1 in (8) is O(ǫr 1ǫs 2) with r+s= 2k+ 1. Since the time average of P(t) is zero, the leading term of (2) in powers of ǫ1and ǫ2will be −1 2M3lim t→∞ 1 tZt 0 P(τ)3dτ=3 2M3T˜ǫ2 1˜ǫ2ZT 0 cos(ωτ +φ1−χ1)2cos(2ωτ +φ2−χ2) dτ =3 8M3˜ǫ2 1˜ǫ2cos(2φ1−φ2+χ2−2χ1). Therefore, the rachet velocity (2), for small amplitudes ǫ1and ǫ2, is given by v=Bǫ2 1ǫ2cos(2φ1−φ2+θ0),(9) with B=3 8M3(β2+ω2)pβ2+ 4ω2, θ0=χ2−2χ1=−tan−12ω3 β(β2+ 3ω2),(10) in agreement with the result reported in [23]. Notice that in the undamped limit γ→0 (equivalently β→0) the parameters (10) become B=3 16M3ω3, θ0=−π 2, whereas in the overdamped limit M→0 (and therefore β→ ∞ with Mβ =γ) we get B=3 8γ3, θ0= 0, both limits agree with the predictions of [23]. 3. Ratchet induced by damping The depence of von parameters of the system like the damping coefficient (through β=γ/M) shown in (9) and (10) reveals an interesting effect. If we take φ1=φ2= 0 in f(t) and do some algebra, the ratchet velocity (for small amplitudes of the force) turns out to be v=ǫ2 1ǫ2 ω3M3V(β/ω), V (x) = 3x(x2+ 3) 8(x2+ 1)2(x2+ 4).(11) Function V(x) is depicted in Figure 1. The most remarkable observation is that the current increases up to a maximum with increasing damping before it begins to show the expected decay. Intuition dictates that the current should decrease with damping Ratchet effect on a relativistic particle 5 012 3 4 5 β/ω 0 0.02 0.04 0.06 0.08 0.1 v (Mω)3/ε1 2ε2 Figure 1. Plot of the current velocity v, in units of ǫ2 1ǫ2/(Mω)3, vs. the damping coefficient β, in units of the frequency, ω, induced by a biharmonic force like (1) with φ1=φ2= 0. Notice that this force is time reversible, i.e., f(−t) = f(t). because friction opposes movement, so the fast increase it reveals for small damping is counterintuitive. The cause of this effect is the interplay between the breaking of the time-reversal symmetry R:t7→ −tthat generates the ratchet current, and the damping that hinders it [27]. In the limit β→0 the system (2) is invariant under Rand a sign change of u, because for φ1=φ2= 0 the force (1) satisfies f(−t) = f(t). Accordingly v= 0 in this limit. But introducing damping breaks the symmetry of the equation and induces a net movement of the particle. For small damping, the higher the damping coefficient βthe larger v. If we keep on increasing βeventually the friction it introduces in the movement of the particle causes the decay of vas β−3. This argument makes it clear that in any ratchet system with a tunable damping and undergoing the action of a time-reversible bi-harmonic force, the ratchet effect can be generated upon increasing damping above zero. 4. Ratchet velocity as a functional of the force The starting point to obtain formula (9) for a ratchet system is to realize that vis a functional of f(t) and that, under certain regularity assumptions, one such functional can be expanded as a functional Taylor series [28, 29, 30] as v[f] = X nodd ZT 0 dt1 T···ZT 0 dtn Tcn(t1,...,tn)f(t1)···f(tn),(12) where the kernels cn(t1, . . . , tn) are proportional to the nth functional derivatives of the functional v[f]. These kernels can be taken T-periodic in each variable and totally symmetric under any exchange of variables. Only odd terms appear in this expansion as a cosequence of the symmetry v[−f] = −v[f] that these systems have. Ratchet effect on a relativistic particle 6 That vis indeed a functional of f(t) in this example is obvious from equations (2)– (7). The aim of this section is to determine explicitly the expansion (12) for this exactly solvable example. Let us start off by rewriting the integral in (7) as Zt 0 dzf(z)e−β(t−z)=I1(t) + I2(t),(13) I1(t) = n(t) X k=1 ZT 0 dzf(z)e−β(t−z−(k−1)T), I2(t) = Zα(t) 0 dzf(z)e−β(α(t)−z), where α(t) = t−n(t)Tand n(t) = [t/T] ([X] denoting the integer part of X). Notice that α(t+T) = α(t). Now, since S(t)≡ n(t) X k=1 eβ(k−1)T=eβnT −1 eβT −1,(14) then I1(t) = e−βt C S(t), with C=ZT 0 dzf(z)[eβz −1].(15) Using this form in (7) we can write P(t) = Ae−βt +e P(t),(16) where A=P(0) + C(eβT −1)−1and e P(t) is the T-periodic function e P(t) = −1 eβT −1ZT 0 dyf(y)e−βα(t)[eβy −1] −Zα(t) 0 dyf(y)e−β(α(t)−y). It is thus enough to obtain e P(t) in the interval 0 ≤t < T , where it adopts the compact form e P(t) = −ZT 0 dyf(y)e−β(t−y)χ(y, t),(17) defining χ(y, t) = 1−e−βy eβT −1+ Θ(t−y) (18) (as it is customary, Θ(x) denotes the Heaviside function, which is 1 if x > 0 and 0 otherwise). Equations (17)–(18) have a well defined β→0+limit, namely e P(t) = −ZT 0 dyf(y)χ1(y, t), χ1(y, t) = y T+ Θ(t−y).(19) On the other hand, for zero-average forces f(t) the kernel χ(t, z) can be further simplified to e P(t) = −ZT 0 dyf(y)e−β(t−y)χ2(y, t), χ2(y, t) = 1 eβT −1+ Θ(t−y).(20) Ratchet effect on a relativistic particle 7 Whatever the form, it should be periodically extended beyond the interval [0, T). It is then clear that (2) and (8) boil down to v= ∞ X k=0 −1 2k(2k−1)!! k!M2k+1 ZT 0 dτ Te P(τ)2k+1.(21) A direct comparison of this equation with the functional Taylor series (12) yields c2k(t1,...,t2k) = 0,(22a) c2k+1(t1, . . . , t2k+1) = −1 2k(2k−1)!! k!M2k+1 T2ka2k+1(t1,...,t2k+1),(22b) where am(t1,...,tm) = ZT 0 dτe−βm(τ−¯ t) m Y k=1 χ(tk, τ),¯ t=1 m m X k=1 tk.(23) As expected [23], functions am(t1,...,tm) are, by construction, T-periodic in each variable and symmetric under any exchange of their arguments. The integral in (23) can be performed integrating by parts and taking into account that d dτ χ(y, τ) = δ(τ−y) (a Dirac delta). The result is am(t1,...,tm) = eβm¯ t βm (m Y k=1 χ(tk,0) − m Y k=1 χ(tk+T, 0) + m X j=1 e−βmtj m Y k=1, k6=j χ(tk, tj)),(24) where we have used the fact that χ(tk, T)e−βT =χ(tk+T, 0). As usual, empty products are assumed to be 1 (the case of the last term for m= 1). The limit β→0 of this expression is better obtained by replacing χ(y, t) by χ1(y, t) in (23) and integrating by parts again. This results in am(t1,...,tm) = T m Y k=1 χ1(tk, T)− m X j=1 tj m Y k=1, k6=j χ1(tk, tj).(25) Finally, in the overdamped case (M→0, β→ ∞), instead of (6) the evolution of Pis given by P(t) = −(1/β)f(t), so vcan be expressed simply as v=− ∞ X k=0 −1 2k(2k−1)!! k!γ2k+1 1 TZT 0 dt f(t)2k+1.(26) From (12) and (26) it follows that c2k(t1,...,t2k) = 0 and c2k+1(t1,...,t2k+1) = −−T2 2k(2k−1)!! k!γ2k+1 δ(t1−t2)···δ(t2k−t2k+1).(27) 4.1. Forcing with a time-periodic piecewise constant force The expansion (12) with kernels (22b) and (23) turns out to be useful to analyze different types of forcing. For instance, another standard choice in the literature (see [1] and references therein), alongside with the bi-harmonic force (1), has been the time-periodic piecewise constant force defined in (3). This force is shift-symmetric only for Tl=T/2, so any other value Tl< T/2 breaks this symmetry and induces a ratchet current. Ratchet effect on a relativistic particle 8 In order to ascertain the effect of this force in system (4b) for small amplitudes ǫ1≪1, we will compute the first nonzero term in the expansion (12). To that purpose we need to evaluate (c.f. equation (23)) Km≡ ham(t1,...,tm)f(t1)···f(tm)i=ZT 0e−βτ I(τ)mdτ, (28a) I(τ)≡1 TZT 0 eβtχ2(t, τ)f(t)dt, (28b) where the choice χ2(t, τ) instead of χ(t, τ) is made because f(t) in (3) has zero average. According to (20) χ2(t, τ) = (1 −e−βT )−1¯χ2(t, τ), where ¯χ2(t, τ) = (1 if t < τ, e−βT if t > τ.(29) Substitution into (28b) yields I(τ) = ǫ1 βT 4 1−e−βT sinh2βTl 2+Q(τ),(30a) Q(τ) =      eβτ −eβTlif 0 < τ < Tl, 0 if Tl< τ < T −Tl, eβ(T−Tl)−eβτ if T−Tl< τ < T. (30b) It is straightforward to check that K1in (28a) vanishes, so the first term that may not be zero is K3. Lengthy calculations lead to K3=−32ǫ3 1 β4T3 eβT (eβT −1)2sinh2β(T−2Tl) 2sinh4βTl 2,(31) that is to say v=4ǫ3 1 (βM)3βT sinh2β(T−2Tl) 2sinh4(βTl/2) sinh2(βT/2) +o(ǫ3 1).(32) It is interesting to noticing that K3= 0 if Tl=T/2 because in that case the timeperiodic piecewise constant force (3) is shift-symmetric. On the other hand, we can determine the value of Tlfor which the ratchet effect is maximum by differetiating (32). This leads to sinh β(T−3Tl) 2sinh β(T−2Tl) 2sinh3βTl 2= 0.(33) The only three solutions to this equation are Tl= 0, Tl=T/2 and Tl=T/3. The first two do not produce any ratchet current (with Tl= 0 f= 0 whereas for Tl=T/2 the force is shift-symmetric), therefore the last one provides its maximum value. As a final remark, expression (32) has well defined overdamped (M→0, β→ ∞, with finite γ=βM) and undamped (β→0) limits. In fact, the undamped limit of (32) yields v=ǫ3 1 4(MT)3T4 l(T−2Tl)2+o(ǫ3 1),(34) Ratchet effect on a relativistic particle 9 whereas the overdamped produces v=o(ǫ3 1). Indeed, since f(t)2k+1 =ǫ2k 1f(t), in the overdamped case Eq. (26) immediately implies v= 0. This is in marked contrast with the overdamped deterministic dynamic of a particle in a sinusoidal potential driven by a bi-harmonic force [31]. In this case, the zero ratchet velocity can be explained as a symmetry effect. Indeed, notice that f(t) = −f(−t) when f(t) is given by (3) (something that only happens for the bi-harmonic force (1) for specific choices of the phases), and that the overdamped limit of Eq. (4b) remains invariant under a simultaneously action of time-reversal and a sign change of xand u(see [23] for further details). 5. Discussion We have studied the dynamics of a damped relativistic particle under two zero average T-periodic forces which breaks the shift-symmetry f(t+T/2) = −f(t). This nonlinear system can be explicitly solved through a transformation that renders it linear. Therefore, the ratchet average velocity, v, is exactly obtained for any arbitrary force f(t). This result allows us to show, first of all, that the ratchet velocity cannot be obtained in general by using the method of moments (according to which vis obtained as a series of the odd moments of f(t)). And secondly, that vis a functional of f(t), i.e. v[f]. Indeed, for any T-periodic force we have explicitly found the coefficients of the functional Taylor expansion (12). In particular, this expansion shows that the method of moments is only justified in the strict overdamped limit (see Eqs. (26) and (27)). Due to the symmetry v[f] = −v[−f] only odd terms contribute to the Taylor expansion. Besides, since the ratchet velocity is translationally invariant, the kernel c1(t1) must be a constant. So the first order term vanishes because the force has zero average. Therefore, the first term in the expansion (12) that is not necessarily zero is the third one, irrespective of the kind of nonlinearity of the system. We have chosen to illustrate this functional representation the bi-harmonic force (1) (with p= 2 and q= 1) as well as a time-periodic piecewise constant force (3). We have obtained the leading term of the average velocity for both these forces. They are given by equations (9)–(10) and (32), respectively. It is worth emphasizing that the method of moments always predicts a zero ratchet velocity when the system is driven by a timeperiodic piecewise constant. This is to be compared with the result (32) obtained here. We have discussed the two limiting dynamics: undamped and overdamped. In these two limits the system remains invariant if the driving force has the symmetries f(t) = f(−t) and f(t) = −f(−t), respectively. If the relativistic particle is driven by a bi-harmonic force v∼ǫ2 1ǫ2cos(2φ1−φ2) in the overdamped limit, whereas v∼ǫ2 1ǫ2sin(2φ1−φ2) in the undamped limit. In the latter case this means no ratchet current if we set φ1=φ2= 0. The unexpected consequence of this is that introducing damping generates a ratchet current, whose intensity grows up to a maximum before it drops to zero upon a further increase of the damping. This effect is a result of a trade off between symmetry effects and friction and our prediction is that should be observed in any system with damping and forced with a time-reversible external force.