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
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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
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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
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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
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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.
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5 DECEMBER 2003
VOLUME 91, NUMBER 23
234102-4 234102-4