Internal Mode Mechanism for Collective Energy Transport in Extended Systems

In e nal Mode Mechanism o Collec i e Ene gy T anspo in Ex ended Sys ems Luis Mo ales-Molina, 1,2, *Niu ka R. Quin e o, 3,4,† F anz G. Me ens, 1,‡ and Angel Sa ´nchez 2,x 1 Physikalisches Ins i u , Uni e si a ¨ Bay eu h, D-85440 Bay eu h, Ge many 2 G upo In e disciplina de Sis emas Complejos (GISC) and Depa amen o de Ma ema ´ icas, Uni e sidad Ca los III de Mad id, A enida de la Uni e sidad 30, 28911 Legane ´s, Mad id, Spain 3 Depa amen o de Fı ´sica Aplicada I, E.U.P., Uni e sidad de Se illa, Vi gen de A ´ ica 7, 41011 Se illa, Spain 4 Ins i u o Ca los I de Fı ´sica Teo ´ ica y Compu acional, Uni e sidad de G anada, 18071 G anada, Spain (Recei ed 9 May 2003; published 5 Decembe 2003) We s udy di ec ed ene gy anspo in homogeneous nonlinea ex ended sys ems in he p esence o homogeneous ac o ces and dissipa ion. We show ha he mechanism esponsible o unidi ec ional mo ion o opological exci a ions is he coupling o hei in e nal and ansla ion deg ees o eedom. Ou esul s lead o a selec ion ule o he exis ence o such mo ion based on esonances ha explain ea lie symme y analysis o his phenomenon. The di ec ion o mo ion is ound o depend bo h on he ini ial and he ela i e phases o he wo ha monic d i ings, e en in he p esence o noise. DOI: 10.1103/PhysRe Le .91.234102 PACS numbe s: 05.45.Y , 02.30.J , 05.60.Cd, 63.20.Pw One in iguing phenomenon ha is ecei ing much a en ion ecen ly is ne di ec ed mo ion induced by ze o a e age o ces. O iginally mo i a ed by s ochas ic models o biomolecula (b ownian) mo o s [1], de e min- is ic a che like sys ems [2,3] a e being in ensi ely s udied, chie ly because o hei many po en ial echno- logical applica ions [4]. Many such models consis o one o wo pa icles on a pe iodic, asymme ic po en ial and a pe iodic o ce ( ocking a che [1]). La e , he in es iga- ion was gene alized o sys ems wi h many in e ac ing pa icles, om noisy soli on-bea ing sys ems [5,6] o o he spa ially ex ended (s ochas ic and de e minis ic, o e damped and unde damped) sys ems, bo h heo e i- cally [7] and om a mo e applied [8] iewpoin . Among his class o p oblems, ne anspo in homo- geneous ex ended sys ems d i en by homogeneous ac o ces is pa icula ly in e es ing. A pa adigma ic example is he ac d i en, damped sine-Go don (sG) equa ion:  xx sin   :(1) A symme y analysis, p oposed o one-pa icle sys ems in [3] and ex ended o his p oblem [9,10], indica ed ha a di ec ed ene gy cu en appea ed i  b oke he sym- me y    T=2,Tbeing he pe iod o he ex e nal d i ing. One such choice is  1sin  02sinm 0([9,10] wi h 0=2), a case o which nume ical simula ions o he sG equa ion con i med he symme y analysis esul s. In wha ollows, we will e e o 0as he ini ial phase and o as he ela i e phase. T anspo equi ed a nonze o opological cha ge, implying he exis ence o sG soli ons (kinks) in he sys em. In his espec , we s ess ha kink-media ed anspo is impossible wi h only one ha monic o any alue o he damping coe icien [11]. I was a gued in [10] ha he obse ed ec i ica ion a ises om he non- adiaba ic exci a ion o in e nal kink modes and hei in e ac ion wi h he ansla ional kink mo ion. This con- jec u e had no igo ous suppo ; a he , i was based on plo s o sG soli on e olu ion and on he ailu e o a collec i e coo dina e (CC) app oach [12] wi h one deg ee o eedom, which assumed ha sG soli ons beha e simi- la o igid pa icles. An a emp o include he wid h deg ee o eedom has been ecen ly p esen ed in [13], whe e i was concluded ha he di ec ed ene gy cu en anishes unless he wid h o he kink, l , is a dynamical a iable. Howe e , his condi ion is only a necessa y one: l is a dynamical a iable in he one-ha monic case bu he kink eloci y is ze o o any alue o he damping as al eady men ioned [11]. Ano he poin no accoun ed o in [13] is he connec ion be ween he in e nal mode mechanism and he symme y analysis, which also p o- hibi s mo ion in o he cases whe e l is a dynamical a iable. The e o e, he easons o he phenomena ob- se ed in [9,10] emained la gely obscu e. In his Le e , a di e en CC app oach allows us o iden i y he mechanism h ough which he wid h oscil- la ion d i es he kink and i s ela ion wi h he symme y condi ions. Fu he mo e, ou heo y p edic s, and nu- me ical simula ions o Eq. (1) con i m, ha he di ec ion o mo ion depends on he ini ial phase o he d i ing, e en in he p esence o addi i e noise. Ou CC heo y is based on an ansa z, p oposed in [14], o he pe u bed kink depending on wo CC, X and l ( espec i ely, posi ion and wid h o he kink). I is no di icul o show [14–16] ha he dynamics o hese wo CC is gi en by dP d P q  ;(2) _ ll22l ll 2l _ ll 2 Rl21P2 M2 01 ;(3) whe e he momen um P M0l0_ XX=l ,R 1=  pl0wi h 2=12 is he so-called Rice’s e- quency, and M08,q2,andl01a e, espec- i ely, he dimensionless kink mass, opological cha ge, and unpe u bed wid h. Equa ion (2) can be sol ed PHYSICAL REVIEW LETTERS week ending 5 DECEMBER 2003 VOLUME 91, NUMBER 23 234102-1 0031-9007=03=91(23)=234102(4)$20.00 2003 The Ame ican Physical Socie y 234102-1 exac ly, and in he la ge ime limi ( 1) yields P    pa1sin 01 a2sinm 02; whe e is me ely a escaling pa ame e in he pe - u ba ion expansion, o be de e mined la e ; 1 a c an=,2a c anm=,a1q1= 22 p, and a2q2= 2m22 p:As we a e in e es ed in he damped (0) case and Eq. (3) canno be sol ed in ha case [15,16], we will s udy i by a pe u ba i e expansion, l l0l1 2l2 . A o de O, we ob ain  ll 1 _ ll1 2 Rl1 2 RP2 l0=2M2 0:(4) The key poin is ha , by subs i u ing he exp ession o P in o (4), we see ha he equa ion o l1 con ains ha monics o equencies 2,2m,andm1;i.e.,  ll1 _ ll1 2 Rl1 A1A2cos2 2021A3cos2m 20222 A4cosm1 21A4cosm1 2021; whe e A1A2A3,A2Ra2 1=4  pM2 0,A3Ra2 2=4  pM2 0,andA4Ra1a2=2  pM2 0. A e ansien s elapse, we ind l1 A1 2 RA2sin2 2021~ 2  2 R422422 qA3sin2m 20222~ 2m  2 R4m2224m222 q A4sinm1 21~ m1  2 Rm12222m122 qA4sinm1 2021~ m1  2 Rm12222m122 q;(5) whe e ~ ma c an2 Rm22=m. A cumbe some bu o he wise i ial calcula ion yields he ha monics con- ained in l2 , collec ed in Table I. Nex , we need o compu e he a e age eloci y o e one pe iod T2=: In he CC app oach, we use he de ini ion o he momen um and ind h_ XX i  1 TZT 0 P l  M0l0 d : (6) A O0, he a e ages hP i and h_ XX0 i anish i ially; he e o e, ne kink mo ion can a ise only in nex o de . By s aigh o wa d calcula ions om Eqs. (5) and (6), we ind o m2 ha , o la ge enough imes, h_ XX1i q32 R2 12 8M3 022 242 p2 cos0221~ 1  2 R2222 qcos0221~ 2  2 R422422 q:(7) F om Eq. (7), we see ha o  o be small he p e ac o on he igh -hand side has o be much smalle han 1. A de ini e, e i iable p edic ion om his asymp o ic ex- p ession is he exis ence o a nonze o eloci y o m 2, wi h a sinusoidal dependence on 0and . This means ha he eloci y depends on bo h he ini ial and he ela i e phases; indeed, by le ing 0 0in Eq. (1) and changing a iables o 0  0, i can be immedi- a ely seen ha an ini ial phase 0is equi alen o a ela i e phase 0m10 o a kink wi h i s cen e shi ed o x0V 0.The dependence o he eloci y on ag ees wi h (and explains) [9,10], whe eas he de- pendence on 0is a o ally new esul . Ne e heless, hese analy ical esul s as well as he nume ical simula ions we p esen below s ongly suppo he p esen conclusion. Fo he case m3, he a e age eloci y is ze o a all o de s, a esul con i med by di ec nume ical simula ion o he ull sG Eq. (1) as we will see below. The eason can be unde s ood by looking a Table I: Fo m3, he equencies o he ac o ce (o he momen um) a e odd ha monics (and 3), whe eas he wid h o he kink oscilla es only wi h e en ha monics (2n,n2N). This leads us o ou main conclusion, namely, he mechanism o he appea ance o ne mo ion and he co esponding selec ion ules. Equa ions (2) and (3) show ha he o ce ac s on he kink wid h h ough P2 , whe eas P i sel is in u n in e sely p opo ional o l . This coupling is he TABLE I. Ha monic con en o he i s con ibu ions o he pe u ba i e expansion o l . Ha monic l1l2 m2,2m,m12,4,2m,4m,m1, 2m1,m3,3m1 2,2,3,4,2,3,4,5,6,7,8 32,4,62,4,6,8,10,12 PHYSICAL REVIEW LETTERS week ending 5 DECEMBER 2003 VOLUME 91, NUMBER 23 234102-2 234102-2 esponsible o he ne kink mo ion bu , o i o be ac ually possible, he ha monic con en o he e ec i e o ce P2 ac ing on he wid h deg ee o eedom mus be able o esona e wi h i . This is e iden om Eq. (6), in which he in eg al is nonze o only i l con ains a leas one o he ha monics o P . I is impo an o ealize ha his condi ion is much mo e es ic i e han ha ound in [13], whe e only he necessi y o l being a dynamic a iable was poin ed ou . We ha e jus seen ha his is indeed necessa y, bu ha addi ional, c ucial esonance condi ions ha e o be ul illed. In e es ingly, ou heo y shows also ha dissipa ion can change o e en e e he kink eloci y [see Eq. (7)] in ag eemen wi h he nume i- cal esul s in [9,10]. A mo e de ailed discussion o his poin is o hcoming [17]. These p edic ions om he CC app oxima ion mus be con i med by a nume ical solu ion o he ull pa ial di e en ial Eq. (1). We do his by using he S auss- Va ´zquez scheme [18], on sys ems o leng h L100, 1000, wi h s eps  0:01,x0:1, ee bounda y condi ions, and a kink a es as an ini ial condi ion. Ins ead o he pe u ba i e exp essions (which a e only quali a i ely co ec unless 1), o assess he alidi y o ou heo y we nume ically in eg a e Eq. (3) and he equa ion o he eloci y ob ained om he exp ession o P [Eq. (2)] wi h a ou h-o de Runge-Ku a me hod. Ou main esul s a e shown in Figs. 1–3; hey ully con i m he accu acy, e en quan i a i e, o ou app oach. Figu e 1 exhibi s clea ly he sinusoidal dependence o he eloci y as a unc ion o he ini ial phase. The dependence on is also seen as a simple shi when changing om 0 o =2. The ag eemen wi h he CC esul s is pe ec . As a u he check o he obus ness o his de- pendence, ollowing [9] we ha e simula ed Eq. (1) wi h an addi ional addi i e Gaussian whi e noise e m wi h a iance D. While one could, in p inciple, hink ha his noise would supp ess he ini ial phase dependence, Fig. 2 shows ha he opposi e is he case: The noise enhances he dependence on he ini ial phase, inc easing he maximum alues o he eloci y while keeping he same gene al sinusoidal dependence and he loca ion o he ze os. I is emp ing o conclude om his plo ha he noise, a leas i i is no e y la ge (D1), assis s he p ocess o ene gy ans e be ween he wid h and he ansla ion deg ees o eedom, ac i a ing i . Finally, Fig. 3 makes i clea ha ou main esul , namely he in e p e a ion o he physics o he p oblem, is indeed ue, by showing he ha monic con en o l  o m2 and 3. In his case, he ag eemen be ween ou CC heo y and he ull nume ical simula ion o Eq. (1) is indeed imp essi e, and alida es i mly ou esonance c i e ion o ne kink mo ion. I is impo an o s ess ha he p esen heo y does no apply o he ne mo ion ound o m3in [10]. We ha e con i med hei esul in ou simula ions, which allowed us o ealize ha his is an al oge he di e en phenomenon: Fi s , i appea s only abo e a (mode a ely la ge, i*0:4) h eshold ampli ude, and, second, i is induced by he kink wings, which a e highly dis o ed in he p ocess yielding he CC pic u e inapp op ia e (e en kink-an ikink pai s a e c ea ed). In conclusion, we ha e ound ha he symme y con- di ions se o h in [9,10] ha e hei physical o igin in he mechanism o he di ec ed mo ion: he indi ec ac ion o he o ce h ough he coupling o he ansla ional and wid h deg ees o eedom. To make ne mo ion possible, his indi ec d i ing has o esona e wi h he a ailable equencies o he wid h. This in e p e a ion does no con adic he nonexis ence o in e nal modes in sG kinks, shown in [16], because ex e nal o ces can induce, ia exci a ion o ce ain phonons, beha io simila o he one expec ed om an in insic in e nal mode [19,20]. The ac ha a o ce wi h only one ha monic would no d i e he damped sG kink [11], and ha wo ha monics a e needed o simul aneously exci e he wid h oscilla ions and induce ne mo ion, i s nicely in his pic u e. On he o he hand, his poin aises he ques ion as o he gene al- i y o ou esul s, in iew o he ac ha mos kink- bea ing sys ems do ha e in e nal modes. To answe his ques ion, we ha e s udied he same p oblem in he ame- wo k o he 4model, eaching he same conclusions [17]: Indeed, he in insic in e nal mode o 4kinks -4.0 -2.0 0.0 2.0 4.0 δ0 -0.20 -0.10 0.00 0.10 0.20 <V> FIG. 1. Dependence o he kink eloci y on he ini ial phase. Pa ame e s a e 120:2,0:05,0:1. Rela i e phase =2: solid line, CC heo y; illed ci cles, simula ion esul s. Rela i e phase 0: dashed line, CC heo y; squa es, simula ion esul s. -4.0 -2.0 0.0 2.0 4.0 δ0 -0.20 -0.10 0.00 0.10 0.20 <V> FIG. 2. Dependence o he kink eloci y on he ini ial phase o ela i e phase =2in he de e minis ic (D0, emp y ci cles) and he s ochas ic (D0:03, diamonds) cases. O he pa ame e s a e as in Fig. 1. PHYSICAL REVIEW LETTERS week ending 5 DECEMBER 2003 VOLUME 91, NUMBER 23 234102-3 234102-3 makes he phenomenon e en mo e no iceable, making us con iden on he wide applicabili y o his wo k. Ano he impo an conclusion is he dependence o he eloci y on he ini ial phase 0, no men ioned in ea lie wo k [9,10]. We no e ha his dependence allows much mo e lexibili y in con olling he kink eloci y, p o id- ing an al e na i e o he use o he ela i e phase sug- ges ed ea lie . On he o he hand, his may ha e impo an consequences o applica ions as a way o sepa a ing, e.g., luxons in long Josephson junc ions [8]. In e es ingly, such supe conduc ing de ices p o ide he bes possible labo a o y o e i y ou esul s. This expe imen al con- i ma ion is c ucial in o de o asce ain hei applicabil- i y. Gi en he accu acy wi h which he sG equa ion desc ibes long Josephson junc ions, and he ac ha an ex e nal o ce such as he one p oposed in his and ea lie wo ks [9,10] is easy o implemen , we hope ha he co esponding measu emen s will soon be ca ied ou . A conclusi e, posi i e e i ica ion o ou heo y would yield he pic u e we p o ide he e e y use ul in ha and ela ed con ex s. This wo k has been suppo ed by he Minis e io de Ciencia y Tecnologı ´a o Spain h ough G an s No. BFM2001-3878-C02 (N. R. Q.), No. BFM2000- 0006, and No. BFM2003-07749-C05-01 (A. S.), by he Jun a de Andalucı ´a unde P ojec No. FQM-0207, by DAAD (Ge many) A0231253/Re . 314, and by he In e - na ional Resea ch T aining G oup ‘‘Nonequilib ium Phenomena and Phase T ansi ions in Complex Sys ems.’’ *Elec onic add ess: Luis.Mo ales-Molina@uni- bay eu h.de † Elec onic add ess: niu ka@eule .us.es ‡ Elec onic add ess: F anz.Me ens@uni-bay eu h.de x URL: h p://gisc.uc3m.es/~anxo [1] P. Reimann, Phys. Rep. 361, 57 (2002). [2] P. 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E 65, 065601 (2002); Chaos, Soli ons & F ac als ( o be published). 0.00 0.03 0.06 0.09 0.12 0.15 equency/(2π) 0.00 0.05 0.10 0.15 0.20 Ampli ude o DFT 2δ 4δ 6δ 8δ 0.00 0.03 0.06 0.09 0.12 0.15 equency/(2π) 0.00 0.05 0.10 0.15 0.20 Ampli ude o DFT δ2δ 3δ 5δ 6δ 8δ 9δ FIG. 3. Disc e e Fou ie ans o m o he kink wid h. Uppe panel: m2; lowe panel: m3. Solid line: ampli ude mea- su ed in simula ions. Dashed line: nume ical in eg a ion o he CC equa ions. Pa ame e s a e as in Fig. 1 o ela i e phase  =2and ini ial phase 02:5. PHYSICAL REVIEW LETTERS week ending 5 DECEMBER 2003 VOLUME 91, NUMBER 23 234102-4 234102-4