String ratchets: ac driven asymmetric kinks

PHYSICAL REVIEW E, VOLUME 65, 051103

String ratchets: ac driven asymmetric kinks G. Costantini,1 F. Marchesoni, 1,2,3 and M. Borromeo3,4 1

2

Istituto Nazionale di Fisica della Materia, Universita` di Camerino, I-62032 Camerino, Italy Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, Michigan 48109-1120 3 Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, I-06123 Perugia, Italy 4 Dipartimento di Fisica, Universita` di Perugia, I-06123 Perugia, Italy 共Received 18 January 2002; published 26 April 2002兲

We simulated numerically the time evolution of a one-kink bearing, damped elastic string sitting on noiseless periodic substrates of two types: 共I兲 asymmetric, time independent, 共II兲 symmetric, periodically deformable. An asymmetric kink subjected to an ac drive is shown to drift steadily with finite average speed independent of its initial kinetic conditions. In the overdamped regime the resulting net kink transport can be attributed to the rectification of the Brownian motion of a pointlike particle with oscillating mass. For intermediate to low damping completely different features show up, due to the finite size of the objects being transported; in particular, the kink current hits a maximum for an optimal value of the damping constant, resonates at the kink internal-mode frequency and, finally, reverses sign within a certain range of the drive parameters. DOI: 10.1103/PhysRevE.65.051103

PACS number共s兲: 05.60.Cd, 05.50.⫹q, 11.27.⫹d

I. INTRODUCTION

A transport mechanism of potential relevance both to applied physics and nanobiology is the so-called ratchet effect 关1兴. In its simplest instance a ratchet device can be assimilated to a Brownian particle with coordinate x(t) moving in an asymmetric periodic potential V(x), with V(x⫹a) ⫽V(x), subjected to viscous damping and ac drive 共rocked ratchet 关2兴兲: The natural direction and the intensity of the ratchet current 具 x˙ 典 results from a rather intricate interplay of particle inertia, spatial asymmetry, and time correlation of the forcing terms 共including fluctuations when present兲. A similar mechanism is expected to operate also when instead of a pointlike Brownian particle one considers a fluctuating 共one-dimensional兲 elastic string ␾ (x,t) 关3兴 and replaces, accordingly, V(x) with a periodic substrate potential V 关 ␾ 兴 , such that V 关 ␾ ⫹a 兴 ⫽V 关 ␾ 兴 . The kinks and antikinks born by the string as it connects adjacent substrate valleys tend to glide apart, so that the string center of mass advances effectively in the natural ratchet direction 关4兴. In rocked ratchets noise is required either to aid the escape of the Brownian particles over the potential barriers or to nucleate kinkantikink pairs along the string, directed parallel to the substrate valleys. In both cases a sufficiently large drive amplitude can activate a net ratchet current even in the absence of fluctuations. In the present paper we address the ratchet dynamics of an elastic string diffusing on a periodically tilted substrate. In order to catch the essence of the mechanism at work we ignore the spatiotemporal fluctuations responsible for the thermalization of unperturbed string 关3兴; instead, we impose that the string always bears at least one kink, that is ␾ (⬁,t)⫺ ␾ (⫺⬁,t)⫽a. The initial problem is thus reduced to the question as how each individual damped kink 共antikink兲 responds to an external periodic drive. In the case of a symmetric substrate (V 关 ⫺ ␾ 兴 ⫽V 关 ␾ 兴 , like in the sineGordon theory兲 and a sinusoidal tilt, it has been noticed 关5兴 1063-651X/2002/65共5兲/051103共7兲/$20.00

that an isolated frictionless kink may travel in either direction depending on the tilt parameters and, most importantly, on its initial conditions 共momentum modulus and phase兲. Such an effect follows from the spontaneous symmetry breaking induced by the external tilt; indeed, on averaging over all initial conditions the kink current vanishes, as one would expect on the ground of simple symmetry arguments. Furthermore, adding damping, no matter how small, makes this effect vanish completely. As stated in the earlier ratchet literature 关1兴, the onset of a kink ratchet current requires a sufficient amount of asymmetry in the system. In this paper we address two classes of asymmetric kink dynamics: 共I兲 Ratchet Potential (RP): V 关 ␾ 兴 is intrinsically asymmetric and time independent; the kink is asymmetric also in the absence of a tilt 共unperturbed kink兲 due to the asymmetry of the barrier separating any two adjacent potential valleys; 共II兲 Deformable Potentials (DP): V 关 ␾ 兴 is symmetric at all times; its shape is modulated so that its valleys broaden and shrink 共with constant height兲 periodically in time 关6兴; the tilt is phase locked to the substrate modulation, thus providing for an effective cycle asymmetry. To avoid further complications, we assume that the strings of both classes bear one species of kink, only; this implies that all V 关 ␾ 兴 valleys are degenerate and have the same curvature. Entropic rectification effects like those described in Refs. 关7,8兴 are thus ruled out. The main conclusion of our study is that strings of classes I and II possess sufficient asymmetric coupling to the ac drive to sustain steady kink transport even in the presence of finite damping. The rather complicated dependence of the kink ratchet current on the damping constant and the tilt parameters makes a detailed analysis of the system at hand worthy our extensive numerical simulation effort reported on here. Our presentation is organized as follows. In Sec. II we introduce an example for each class of asymmetric strings RP and DP; we then estimate the effect of a small periodic tilt on the relevant kink dynamics in the adiabatic limit, thus explaining why we expect a finite kink current. In Sec. III we

65 051103-1

©2002 The American Physical Society

G. COSTANTINI, F. MARCHESONI, AND M. BORROMEO

PHYSICAL REVIEW E 65 051103

present the outcome of our simulation work and focus on the kink current dependence on the damping constant and the tilt frequency; intriguing effects like non-Smoluchowski overdamped laws, optimal ratchet damping and parametric resonant current inversions are thus revealed. Finally, in Sec. IV we outline a summary of the results and conclusions, as well as an outlook of potential extensions of this work. II. ASYMMETRIC SUBSTRATES

A damped elastic string ␾ (x,t) moving on a periodic substrate is described by the classical field equation 关9–11兴

␾ tt ⫺c 20 ␾ xx ⫹ ␻ 20 V ⬘ 关 ␾ 兴 ⫽⫺ ␣␾ t ⫹F 共 t 兲 .

共1兲

Here, c 0 and ␻ 0 are the parameters of the unperturbed string equation; the potential V 关 ␾ 兴 is periodic in ␾ , i.e., V 关 ␾ ⫹a 兴 ⫽V 关 ␾ 兴 , and its amplitude was set to one following an appropriate choice of ␻ 0 ; the potential minima 共valleys兲 are located at ␾ ⫽0共mod a) and separated by barriers centered at ␾ ⫽ ␾ 0 共mod a) with 0⬍ ␾ 0 ⭐a depending on the degree of asymmetry of the substrate; ␣ denotes the string damping constant and the periodic tilt F(t) is taken sinusoidal, i.e., F(t)⫽F 0 sin(⍀t), for simplicity. The unperturbed string 关 F(t)⫽0,␣ ⫽0兴 bears both extended 共phonons兲 and localized solutions 共solitons兲 关9–11兴. Localized solutions can be conveniently approximated to lin(0) (0) and antikinks ␾ ⫺ , ear superpositions of moving kinks ␾ ⫹

冉 冊 冕

x⫺X 共 t 兲 ⫽⫾ 1⫺

u2

1/2

c0

␻0

c 20

(0) ␾⫾ „x⫺X(t)…

␾0

d␾

冑2V 关 ␾ 兴

,

共2兲

provided that the separation between their centers of mass X(t)⫽x 0 ⫹ut(x 0 ⫹ and u are integration constants兲 is very 2 large compared with their size c 0 / ␻ min with ␻ min 2 ⫽␻0V ⬙ 关0兴 共dilute gas approximation兲. As anticipated in Sec. I we ignore the kink-antikink pair nucleation mechanism, where thermal fluctuations would play a central role 关3兴, and focus on the response of a single preexisting 共geometrical兲 (0) subjected to the perturbation on the right-hand side kink ␾ ⫹ of Eq. 共1兲, i.e., in the absence of noise. Note that the ratchet (0) (0) current of ␾ ⫺ is necessarily opposite to that of ␾ ⫹ ; moreover, the string center of mass advances by one full step a to the right as a kink travels all the way from X⫽⬁ to X⫽ ⫺⬁ 共or vice versa for an antikink兲. A static (u⫽0) perturbed kink ␾ ⫹ (x,t) can be regarded as an extended quasiparticle 关12兴 of radius c 0 / ␻ min and mass M 共 t 兲⫽

冕

⬁

⫺⬁

关 ␾ ⫹ 共 x,t 兲兴 2x dx

共3兲

共throughout the present paper 关 . . . 兴 x denotes a spatial derivative; here the t dependence accounts for the kink shape modulation兲. The time dependence of M (t) is induced by the periodic tilt F(t); however, it takes an intrinsic asymmetry or a tilt-dependent deformation of the substrate ␻ 20 V 关 ␾ 兴 for the kink mass modulation to cause a rectification of the kink dynamics as shown in the following section.

FIG. 1. Substrates 共a兲 RP of Eq. 共4兲; 共b兲 DP of Eq. 共7兲 in the absence of tilt, F 0 ⫽0. In the snapshots of 共b兲 the deformation parameter s(t), Eq. 共11兲, has constant amplitude s 0 ⫽0.5 and decreasing phase, corresponding to s⫽0.5,0,⫺0.5, respectively. Note that the natural ratchet direction of the double-sine potential plotted in 共a兲 is positive 共see text兲. A. Ratchet potentials

The RP model used in our simulations is the benchmark potential of the current literature on rocked ratchets 关1,4兴, namely,

冋

V 关 ␾ 兴 ⫽k sin

册

冋

册

2␲ 4␲ k ¯ 兲 ⫹ sin ¯兲 , 共␾⫺␾ 共␾⫺␾ a 4 a

共4兲

¯ ⫽(a/2␲ ) and ␾ with k ⫺1 ⫽(3⫹ 冑3)( 冑3/2) 1/2a/8␲ ⫺1 ⫻cos 关(⫺1⫹冑3)/2兴 . Substrate 共4兲 can be regarded as the most straightforward variation of the sine-Gordon 共SG兲 theory that allows for spatial anisotropy. Such an occurrence has been considered, for instance, in the context of dislocation theory 关4,13兴 and long Josephson junction array design 关14,15兴. In Fig. 1共a兲 and throughout the present paper we set a ⫽2␲. The asymmetry of the substrate 共4兲 is apparent as the ¯ potential barriers are located at ␾ 0 共mod 2 ␲ ) with ␾ 0 ⫽2 ␾ (0) and 0⬍ ␾ 0 ⬍ ␲ . The resulting unperturbed kink solution ␾ ⫹ is stable under small perturbations due to the degeneracy of the V 关 ␾ 兴 minima; its mass M 0 is a constant inverse proportional to the kink size c 0 / ␻ min . In the presence of the tilt F(t), the minima of the effective potential ␻ 20 V 关 ␾ 兴 ⫺F(t) ␾ shift back and forth around their unperturbed position 0共mod 2 ␲ ); as a consequence, the 2 (t) oscillates in time with curvature of the tilted valleys ␻ min amplitude proportional to F 0 . An explicit calculation carried out in the adiabatic limit ⍀→0 and for small tilt amplitudes F 0 Ⰶ ␻ 0 yields

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冋 冉 冊

2 2 ␻ min 1⫹ 1⫺ 共 t 兲 ⫽ ␻ min

1

F0

冑3

␻ 20 k

册

⫹••• ,

共5兲

STRING RATCHETS: ac DRIVEN ASYMMETRIC KINKS

PHYSICAL REVIEW E 65 051103

2 with ␻ min ⫽␻20k(3冑3/2) 1/2. Accordingly, being M (t) ⬀ ␻ min(t), one sets the maximum amplitude of the kink mass oscillation to

␦M M0

⫽

1 4

冉冑 冊 3 2

1/2

F0

␻ 20

共6兲

.

关Notice that for a regular SG potential the time modulation 2 (t) would be quadratic in F(t).兴 Our derivation of Eq. of ␻ min 共6兲 clearly hints at a coupling mechanism between kink center of mass and kink internal mode, as invoked in Ref. 关15兴. B. Deformable potentials

The class of the soliton-bearing DP strings was introduced by Peyrard and Remoissenet 关6兴 as an improved model of real one-dimensional atomic chains where typical potential wells and barriers have different curvature, as opposed to the oversimplified Frenkel-Kontorova 共or discrete SG兲 model 关11兴, which is invariant under ‘‘inversion,’’ i.e., V 关 ␾ ⫹a/2兴 ⫽⫺V 关 ␾ 兴 . Following Ref. 关6兴 with minor notation changes, we define 关Fig. 1共b兲兴: V 关 ␾ ;s 兴 ⫽

冋

册

1 共 1⫹s 兲 2 共 1⫺cos ␾ 兲 ⫺1 , 2 共 1⫺s 兲 2 ⫹2s 共 1⫺cos ␾ 兲

共7兲

with 兩 s 兩 ⬍1 and string constant a⫽2 ␲ . The curvature of the substrate valleys and barriers are, respectively,

冉 冊

␻ 20 1⫹s 2 ␻ min 共 s 兲 ⫽ ␻ 20 V ⬙ 关 0 兴 ⫽ 2 1⫺s

M0

⫽

␮共 s 兲

冑兩 ␮ 共 s 兲 2 ⫺1 兩

再

共8兲

tanh⫺1 冑兩 ␮ 共 s 兲 2 ⫺1 兩 ,

s⭐0,

tan⫺1 冑兩 ␮ 共 s 兲 2 ⫺1 兩 ,

s⭓0,

共9兲

where ␮ (s)⬅ ␻ min(s)/␻min , ␻ min⬅␻min(0) and M 0 is the (0) (x)⬅ ␾ ⫹ (x;0), i.e., M 0 ⫽M (0) mass of the SG kink ␾ ⫹ ⫽8 ␻ min /c0. On assuming that 兩 s 兩 Ⰶ1 the cumbersome expression 共9兲 can be approximated to

冉

冊

2 M 共 s 兲 ⫽M 0 1⫹ s⫹••• . 3

共10兲

In our simulations—we depart here from the modelization of Ref. 关6兴—it was assumed that the deformation parameter s oscillates in time, s→s(t), with amplitude s 0 ⬍1 and angular frequency ⍀. Moreover, the relative phase of the potential modulation s(t) and the external tilt F(t) was set arbitrarily in such a way that, when V 关 ␾ ;t 兴 ⬅V 关 ␾ ;s(t) 兴 is tilted to the right 关 F(t)⬎0 兴 , its valleys 共barriers兲 get narrower 共broader兲 and vive versa, namely, s 共 t 兲 ⫽s 0 sin共 ⍀t 兲 ,

C. Kink ratcheting

We are now in the position to argue why we expect a nonvanishing kink ratchet current for both potentials 共4兲 and 共7兲. A simple perturbation approach 关12,16兴 valid in the adiabatic limit ⍀→0, only, leads to state that a single kink 共antikink兲 moving along the elastic string 共1兲 obeys the noiseless Langevin equation 共LE兲 p˙ ⫽⫺ ␣ p⫿aF 共 t 兲 ,

2

2 2 (s)⫽␻20兩V ⬙ 关␲兴兩⫽␻min (⫺s). Note that the DP 共7兲 coand ␻ max incides with a SG potential for s⫽0 and is ␾ →⫺ ␾ symmetric for any allowed value of s. The mass of the DP kink ␾ ⫹ (x;s) has been derived in the original papers 关6兴; in our notation

M共s兲

Accordingly, for s 0 Ⰶ1 the kink mass M (t)⫽M „s(t)… oscillates around its unperturbed value M 0 with maximal amplitude ␦ M /M 0 ⫽2s 0 /3. Of course the s-F phase may be changed thus altering the kink ratchet current in a predictable fashion 共see, for instance, Sec. III C, bottom兲; some of the simulation results reported in Sec. III for our DP model may depend indeed on the choice 共11兲, whereas the underlying physical interpretation does not. More importantly, one should notice that a time-dependent DP model may have practical applications, for instance, in dislocation theory. Equation 共1兲 is known to describe the motion of a noiseless linear defect 共or dislocation兲 ␾ gliding on a Peierls-Nabarro substrate ␻ 20 V 关 ␾ 兴 under the action of an external uniform periodic stress field F(t) 关13兴. If one assumes that the applied stress can perturb periodically the lattice substrate, too, then compressions and dilatations are likely to affect the curvature of the PeierlsNabarro barriers differently, as suggested in Ref. 关6兴.

F 共 t 兲 ⫽F 0 sin共 ⍀t 兲 .

共11兲

共12兲

where a⫽2 ␲ , p⫽M (t)c 0 / 冑1⫺(u/c 0 ) 2 denotes the kinetic momentum of the kink and u⫽X˙ is the velocity of its center of mass. For F 0 Ⰶ ␻ 0 and ␣ Ⰷ ␻ 0 the modulus of u(t) is much smaller than c 0 and, therefore, the LE 共12兲 can be rewritten in nonrelativistic form, p⯝M (t)u and M 共 t 兲 X¨ ⫽⫺ ␣ M 共 t 兲 X˙ ⫿aF 共 t 兲 .

共13兲

It follows immediately that kinks and antikinks are pulled apart with stationary speed u(t)⫽⫿aF(t)/ ␣ M (t); in particular, when the substrate is tilted to the right, F(t)⬎0, the kinks move to the left with negative velocity and vice versa. In the same parameter regime, F 0 Ⰶ ␻ 0 , ␣ Ⰷ ␻ 0 , and ⍀ →0, the mass M (t) of both the RP and DP kinks oscillates with time according to Eqs. 共5兲,共6兲, and 共10兲,共11兲, respectively: A kink with a negative velocity, i.e., for F(t)⬎0, is more massive, or slower, than a kink with positive velocity, i.e., for F(t)⬍0; hence, ¯u ⫽ 具 u 共 t 兲 典 ⫽

1 T⍀

冕

T⍀

0

u 共 t 兲 dt⯝

aF 0 2 ␣ M 20

␦M,

共14兲

with T ⍀ ⫽2 ␲ /⍀ and ␦ M /M 0 ⯝0.23(F 0 / ␻ 20 ) for the RP 共4兲 and ␦ M /M 0 ⯝0.67s 0 for the DP 共7兲. In Fig. 2共a兲 we display the average ratchet velocity ¯u F 0 for a kink of either class: as predicted in Eq. 共14兲, ¯u ⬀F 0␤ with ␤ ⫽1 共DP兲 and ␤ ⫽2 共RP兲. We make now an important remark. As mentioned earlier in this section, by pulling the kinks to the right, X(t)→⬁, and the antikinks to the left, X(t)→⫺⬁, the string center of

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⫽0 and ␾ N ⫽2 ␲ to accommodate one kink and long enough for the moving kink not to experience boundary forces 关3兴. Details of the integration code employed in our numerical study are reported in Refs. 关16,17兴. Here we limit ourselves to noticing that, due to the presence of the viscous damping ˙ t , the kink ratchet velocity ¯u does not depend term ⫺ ␣␾ appreciably on its initial conditions. This might be an issue at extremely small ␣ values, where the string dynamics is expected to turn chaotic. In such a limit, however, the string configuration would be no longer separable into a linear superposition of stable kinks and antikinks and the present approach would become untenable 共see Sec. III B兲. A. The overdamped regime

FIG. 2. Kink ratchet velocity ¯u in the overdamped regime: 共a兲 dependence on the tilt amplitude F 0 at ␣ ⫽50; 共b兲 dependence on the damping constant ␣ at F 0 ⫽0.5. Other simulation parameters: c 0 ⫽50 and ⍀⫽8⫻10⫺3 . The asymptotic laws 共dashed lines兲 have been drawn for the reader’s convenience.

˙ 典典 ⬍0, mass drifts with negative average velocity 具具 ␾ 具具 ••• 典典 denoting the double average of ␾ (x,t) with respect to time and space. A negative string current is to be expected for the RP of Fig. 1共a兲: On replacing ␾ with x in Eq. 共4兲 one obtains the standard double-sine potential of Ref. 关2兴, which in the overdamped regime corresponds to a rocked ratchet with negative natural direction. The case of the DP in Fig. 1共b兲 is more intriguing. A particle with coordinate x moving along the periodic potential V(x;s), obtained by setting ␾ →x in Eq. 共7兲, and subject to the modulation of Eq. 共11兲, would obey the equation of motion x¨ ⫽⫺ ␣ x˙ ⫺V ⬘ 共 x;s 兲 ⫹ 共 s/ 兩 s 兩 兲 F 0 .

共15兲

As V(x;⫺s)⫽⫺V(x⫺ ␲ ;s), one can easily prove that in the stationary regime x˙ (t;s)ª⫺x˙ (t;⫺s) „i.e., x(t;s) and ⫺x(t;⫺s) obey the same equation of motion…, for any constant value of s, and therefore, after averaging over an entire deformation cycle of s(t), ¯u ⫽ 具 x˙ „s(t)…典 ⫽0. Note that this property of Eq. 共15兲 depends crucially on the choice 共11兲 of the s⫺F phase. The observed net string ratchet current ˙ 典 ‹⫽0 should remind us that in our DP model asymmetry Š具 ␾ comes into play only because of the finite extension of the ratchetet objects, namely, the kinks and antikinks born by the string. III. NUMERICAL ANALYSIS

We simulated 关17兴 a one-kink bearing chain (x→i⌬x)

␾¨ i ⫺c 20 ⌬ 2 ␾ i ⫹V ⬘ 关 ␾ i 兴 ⫽F⫺ ␣ 0 ␾˙ i ⫹ ␨ i 共 t 兲 ,

共16兲

with i⫽1,2, . . . ,N, where ⌬x⫽1, ⌬ 2 ␾ i ⫽ ␾ i⫹1 ⫹ ␾ i⫺1 ⫺2 ␾ i , and V 关 ␾ 兴 denotes either on-site potential 共4兲 or 共7兲. The chain 兵 ␾ i 其 is free-end ( ␾ 0 ⫽ ␾ 1 , ␾ N⫹1 ⫽ ␾ N ) with ␾ 1

The numerical results of Fig. 2共b兲 already show that in the limit ␣ →⬁ the kink ratchet velocity ¯u decays like ␣ ⫺(1⫹2 ␤ ) with ␤ ⫽1 共DP兲 and ␤ ⫽2 共RP兲, as opposed to the standard Smoluchowski law ␤ ⫽0 applicable to stationary dynamics. A qualitative explanation of such an asymptotic behavior lies beyond the reach of the adiabatic limit ⍀→0 adopted so far. For small amplitude modulations the kink mass M (t) oscillations are expected to conform to the linear response theory, i.e., M 共 t 兲 ⯝M 0 ⫹ ␦ M 共 ⍀ 兲 sin关 ⍀t⫺ ␸ 共 ⍀ 兲兴 ,

共17兲

where ␸ (⍀→0)⫽0⫹ is a phase lag and

冋

册

␤ ␦M共⍀兲 1 ⬃ , ␦M共0兲 1⫹ 共 ⍀ ␶ 兲 2

共18兲

2 and ␦ M (0)⬅ ␦ M given in Eq. 共6兲 for the RP with ␶ ⬅ ␣ / ␻ min kink and in Eq. 共11兲 for the DP kink. The result of Eq. 共18兲 was derived through the following perturbation argument 关9,18兴. Let ␦ ␾ (x,t)⫽O(F 0 ) quantify (0) (x) underwent the amount of deformation the static kink ␾ ⫹ (0) (x,t) because of the perturbation F(t), i.e., ␾ ⫹ (x,t)⫽ ␾ ⫹ ⫹ ␦ ␾ (x,t). On employing definition 共3兲 the perturbed kink mass reads

M 共 t 兲 ⯝M 0 ⫹2 ⫹

冕

⬁

⫺⬁

冕

⬁

⫺⬁

(0) 关␾⫹ 共 x 兲兴 x 关 ␦ ␾ 共 x,t 兲兴 x dx

关 ␦ ␾ 共 x,t 兲兴 2x dx.

共19兲

The O关 ␦ ␾ 兴 term on the rhs of Eq. 共19兲 is of the same order (0) as the potential deformation factor 2( ␻ 0 /c 0 ) 2 兰 (V 关 ␾ ⫹ 兴 ⫺V 关 ␾ ⫹ 兴 )dx. Recalling that for the nondeformable RP 共4兲 this quantity is O(F 20 ) 关18兴, we conclude that ␦ M ⫽O( 兩 ␦ ␾ 兩 ␤ ). In order to estimate the magnitude of ␦ ␾ one can replace (0) ␾ in Eq. 共1兲 with ␾ ⫹ ⫹ ␦ ␾ and then solve the ensuing phonon equation 关9兴

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(0) ␦ ␾ tt ⫺c 20 ␦ ␾ xx ⫽⫺ ␻ 20 V ⬙ 关 ␾ ⫹ 共 x 兲兴 ␦ ␾ ⫺ ␣ ␦ ␾ t ⫹F 共 t 兲 .

共20兲

STRING RATCHETS: ac DRIVEN ASYMMETRIC KINKS

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FIG. 3. Kink ratchet velocity ¯u vs ␣ 共logarithmic scale兲 on the DP substrate 共7兲 for different values of the ac frequency ⍀. Dashed curve: adiabatic approximation 共21兲 with M (t) given by Eq. 共10兲 with s(t) of Eq. 共11兲. Other simulation parameters: s 0 ⫽0.5, c 0 ⫽50, and F 0 ⫽0.3.

The kink deformations that contribute the most to the actual value of M (t) are the so-called kink internal mode共s兲 关15兴, whose characteristic frequency is typically close to ␻ min . In the overdamped limit, ␦ ␾ tt ⯝0, and for ⍀Ⰶ ␻ 0 , one immediately recovers the linear response formula 兩 ␦ ␾ (⍀) 兩 / 兩 ␦ ␾ (0) 兩 ⬃ 关 1⫹(⍀ ␶ ) 2 兴 ⫺1 and, eventually, Eq. 共18兲 for ␦ M (⍀). Finally, inserting Eq. 共18兲 into the velocity expression 共14兲 关with ␦ M → ␦ M (⍀)兴 leads to the asymptotic power law ¯u ⬀ ␣ ⫺(1⫹2 ␤ ) that fits the simulation data of Fig. 2共b兲 for ␣ Ⰷ ␻ 20 /⍀ 共or ⍀ ␶ Ⰷ1). B. Damping constant dependence

Our analysis of the kink ratchet velocity dependence on the damping constant is summarized in Fig. 3 for a DP substrate and in Fig. 4 for a RP substrate, respectively. Curves of ¯u ␣ have been drawn at increasing values of the tilt frequency in order to highlight a few important properties of kink ratchets: 共i兲 The ¯u (t) peak. The curves ¯u ( ␣ ) peak for an optimal value ¯␣ of the damping constant. At variance with Ref. 关15兴兲, we relate this effect to the relativistic nature of the kink dynamics, as on lowering ␣ the nonrelativistic approximation p⯝M (t)u is no longer tenable. In the parameter regime F 0 Ⰶ ␻ 0 and ⍀→0, but with no restrictions on ␣ , LE 共12兲 yields immediately the relativistic version of Eq. 共14兲, ¯u 共 ␣ 兲 ⫽ c0

冓

aF 共 t 兲 / ␣ M 共 t 兲 c 0

冑1⫹ 关 aF 共 t 兲 / ␣ M 共 t 兲 c 0 兴 2

冔

,

共21兲

plotted in Fig. 3 for the reader’s convenience. In the overdamped regime ␣ Ⰷ ␻ min , Eq. 共21兲 tends to the Smoluchowski limit 共14兲; as a matter of fact, we know from Sec. 2 /⍀, that is III A that this law only applies for ␻ minⰆ␣Ⰶ␻min as long as the adiabatic approximation ⍀ ␶ Ⰶ1 holds. In the

FIG. 4. Kink ratchet velocity ¯u vs ␣ 共linear scale兲 on the RP substrate 共4兲 for different values of the ac frequency ⍀. Other simulation parameters: c 0 ⫽50 and F 0 ⫽0.3.

underdamped regime ␣ Ⰶ ␻ min , the kink velocity ¯u grows quadratically with ␣ , ¯u ( ␣ )/c 0 ⯝ ␣ 2 ␦ M (c 0 /aM 0 F 0 ) 2 ; hence, the appearance of a peak in the ¯u ( ␣ ) curve at an intermediate value ¯␣ of the damping constant. 共ii兲 The ¯u (␣ ) plateau. The quadratic branch of ¯u ( ␣ ) predicted by Eq. 共21兲 is actually observable only at fairly small values of the tilt frequency ⍀; instead, the curves of Figs. 3 and 4 approach a plateau as ␣ is decreased below a characteristic value of the order of ⍀. The explanation of this property touches upon the mechanism of phonon damping 关19,20兴, viz. the interplay of discreteness and relativistic effects. At extremely low ␣ values, the kink responds to an external tilt by raising the modulus of its speed up close to c 0 ; as a result its effective size, see Eq. 共2兲, contracts until it becomes of the order of the discretization constant ⌬x introduced in Eq. 共16兲, no matter how large the static kink size c 0 / ␻ min . At this point the kink starts radiating phonons, thus dissipating the excess kinetic energy being pumped into it by the external forcing term; as a consequence, the kink speed levels off, growing insensitive to any further decrease of ␣ ; this means that for ␣ →0 the effective damping ␣ ph acting on the moving kink is caused mostly by phonon radiation, that is ␣ ph Ⰷ ␣ 关17兴. As phonon radiation is essentially a resonant process taking place at the bottom of the substrate valleys, we expect the underdamped limit of the phononless law 共21兲 to fail us for ␣ ⱗ⍀ 关17兴, in agreement with our numerics. One might wonder why in Figs. 3 and 4 the ¯u ( ␣ ) plateaus increase with ⍀. In the limit ␣ →0, the effective kink damping constant boils down to the phonon damping constant ␣ ph ; accordingly, each plateau value ¯u (0) is expected to rest close to the corresponding ideal value ¯u ( ␣ ph ), obtained from Eq. 共21兲 by replacing ␣ with ␣ ph in the absence of phonon

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radiation. Moreover, ␣ ph increases with ⍀ 共stimulated phonon radiation兲 and so does the relevant kink velocity plateau ¯u (0), as long as ␣ ph ⱗ ¯␣ . One might also argue that the properties of ¯u ( ␣ ) due to discreteness are a mere numerical artifact, as we actually integrated Eq. 共1兲 instead of Eq. 共16兲 共namely, the dynamics of a chain instead of a string 关16兴兲. Although more sophisticated simulation algorithms may be especially devised to eliminate such a difficulty, most of the physical systems we can model by means of an elastic string, are indeed discrete on the microscopic scale 关17兴, such as like dislocations in crystals, magnetic flux lines through layered type II superconducting films, etc. Therefore, the discreteness effects reported here have a physical interest of their own, beyond the moot question of their numerical explanation. 共iii兲 ¯u (␣ ) inversions. At higher ⍀ values instability effects start playing an important role. In Fig. 3 for the DP substrate we notice that an abrupt dynamics change occurs around ⍀ ⯝0.2: When on increasing ⍀ the corresponding plateau ¯u (0) attains the maximum of the ideal ¯u ( ␣ ph ) curve 共21兲, all of a sudden the actual ¯u ( ␣ ) peak shifts to higher ␣ values. The same behavior was detected for the RP substrate, too 关see Fig. 4共a兲兴. In the case of a DP kink we could not increase ⍀ any further 共over the entire ␣ domain兲, as a fast kinkantikink nucleation phenomenon sets in, thus making us unable to monitor the time evolution of the tagged kink; in other words, kink and antikink become unstable, thus signaling a route to the chaotic string dynamics. The RP kink is more stable towards periodic tilting 共see Sec. III A兲 and, therefore, we did manage to see what happens on increasing ⍀: In the limit ␣ →0, ¯u ( ␣ ) drops fast but continuously towards negative values, namely, a kink current inversion takes place when ␣ ph ⱗ ¯␣ 共not an unusual occurrence in the underdamped particle rocked ratchets 关1,2,21兴兲. The ⍀ dependence of ¯u ( ␣ ⯝0) in the adiabatic regime is shown in the inset of Fig. 5. C. Frequency dependence

The dependence of the kink ratchet velocity ¯u on the modulation frequency ⍀ is also interesting. In Fig. 5 we display three curves ¯u (⍀) for the RP kink in the low damping regime. The ␣ values were chosen in order to explore the ⍀ dependence of the ¯u ( ␣ ) peak of Fig. 4共a兲. All curves in Fig. 5 exhibit a main 共positive兲 resonance peak at ⍀⫽ ␻ b with ␻ b ⯝0.8␻ min ; the peak height increases with decreasing ␣ . At the lowest ␣ value, a secondary 共negative兲 peak shows up at half the fundamental resonance frequency, i.e., for ⍀ ⬃ ␻ b /2. The onset of such a negative peak is clearly related to the ¯u inversion shown in Fig. 4共b兲. The fundamental frequency ␻ b lends itself to a simple interpretation in terms of the perturbation Eq. 共20兲. By means of standard numerical integration 共see also Ref. 关15兴兲, one can prove that the RP kink admits of one internal mode 共or bound state兲 whose eigenfrequency coincides right with ␻ b . In the presence of the periodic kink mass modulation M (t), Eq. 共17兲, such an internal mode gets easily excited, hence the

FIG. 5. Kink ratchet velocity ¯u ⍀ on the RP substrate 共4兲 for different values of the damping constant ␣ . Note that ¯u has been rescaled by the relevant factor ␣ . Inset: the extremely underdamped case ␣ ⫽10⫺3 in the adiabatic regime of small ⍀. Other simulation parameters: c 0 ⫽50 and F 0 ⫽0.3.

main resonance peak of Fig. 5. When accounting for higher order corrections, Eq. 共19兲, one soon recognizes that the time-dependent kink profile sustains n⍀ harmonics of the drive frequency 共with amplitude decreasing with increasing n); therefore, for sufficiently small ␣ values, the kink deformation ␦ ␾ 共viz. the kink mass兲 may appear to resonate under the parametric condition 2⍀⬃ ␻ b 关22兴. In the limit of vanishingly small ac frequencies the ratchet current turns positive again as explained in item 共iii兲 of Sec. III B. The argument above, while locating correctly the position of the resonance peaks of ¯u (⍀), fails to explain the current inversion corresponding to the onset of the parametric resonance 2⍀⬃ ␻ b . This question would require a sophisticated perturbation analysis 关23兴 capable of reaching beyond the leading order approximation adopted throughout the present work. On the other hand, current inversions in particle rocked ratchet at low damping and/or high frequency 关1,2,21兴 have never been related to a resonant dynamics. IV. CONCLUSIONS

In the present work we addressed the problem of ac driven asymmetric kinks. The kink ratchet current has been shown to exhibit nontrivial dependence on both the string damping constant and the drive frequency. More importantly, we have analyzed two classes of asymmetric kinks: 共I兲 The RP kinks, whose asymmetry is built-in due to the anisotropy of the substrate. In such a case the kink shape 共and mass兲 gets modulated in second order and, consequently, the relevant kink ratchet velocity decays like ␣ ⫺5 in the overdamped regime. The existence of a ratchet mechanism for this class of kinks is largely expected in view of the rocked ratchet theory for pointlike Brownian particles; 共II兲 The DP kinks, supported by a deformable symmetric substrate, whose valleys and barriers change continuously in time; substrate modulation and external tilt are arbitrarily phase locked. The ensuing kink mass modulation is of the first

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order in the substrate perturbation, as proved by the asymptotic ␣ ⫺3 decay of the corresponding kink ratchet velocity. The ratchet mechanism for a string on a symmetric DP substrate has no counterpart in the theory of particle ratchets. As stated in the Introduction, we neglected the presence of noise sources, otherwise required for the string to thermalize. The role of thermal fluctuations as the trigger of kinkantikink nucleation has been mimicked by simulating a onekink bearing string. In our strategy a full understanding of the dynamics of an individual ac driven kink ought to allow us to

reconstruct a posteriori the corresponding time evolution of the entire string. Noise, however, is capable of influencing the diffusion of a single kink, as well, a possibility ignored altogether in the present study. It is believed, though, that at low temperatures 共i.e., in the presence of low intensity fluctuation sources兲, the ratchet efficiency is only marginally affected by noise. The diffusion of an ac driven asymmetric kink in equilibrium at finite temperature and, eventually, the ab initio simulation of a thermalized string diffusing on a rocked asymmetric substrate are certainly topics that deserve more accurate investigation.

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