Vol. 37 (2006) ACTA PHYSICA POLONICA B No 5
OVERDAMPED DETERMINISTIC RATCHETS
DRIVEN BY MULTIFREQUENCY FORCES∗
Da id Cube o, Jesús Casado-Pascual, Azucena Al a ez
Manuel Mo illo
Uni e sidad de Se illa, Facul ad de Física
Apdo. Co eos 1065, Se illa 41080, Spain
Pe e Hänggi
Ins i u ü Physik, Uni e si ä Augsbu g
Uni e si ä ss aße 1, 86135 Augsbu g, Ge many
(Recei ed Feb ua y 21, 2006)
We in es iga e a dissipa i e, de e minis ic a che model in he o e -
damped egime d i en by a ec angula o ce. Ex ensi e nume ical calcula-
ions a e p esen ed in a diag am depic ing he d i eloci y as a unc ion o
a wide ange o he d i ing pa ame e alues. We also p esen some heo e -
ical conside a ions which explain some ea u es o he men ioned diag am.
In pa icula , we p oo he exis ence o egions in he d i ing pa ame e
space wi h bounded pa icle mo ion possessing ze o cu en . Mo eo e ,
we p esen an explici analy ical exp ession o he d i eloci y in he
adiaba ic limi .
PACS numbe s: 05.60.–k, 05.45.Pq, 05.45.Ac, 05.45.X
1. In oduc ion and model se -up
Di ec ed cu en in a che sys ems ha e ecei ed much a en ion o e
ecen yea s [1–5]. One o he easons o s udy hese ype o sys ems is mo-
i a ed by he a emp o unde s and he physical mechanism o mo ion o
molecula mo o s in biological sys ems [2,6] and o in es iga e i s ole in he
design o new ma e ial p ope ies [3,4]. The dynamics o a pa icle mo ing
in a pe iodic po en ial unde he ac ion o an applied ime pe iodic e m is
a he complex and ich, displaying ypically e en chao ic beha io [7–10].
A ea u e o pa icula in e es is he eme gence o di ec ed cu en s in he
∗P esen ed a he XVIII Ma ian Smoluchowski Symposium on S a is ical Physics,
Zakopane, Poland, Sep embe 3–6, 2005.
(1467)
1468 D. Cube o e al.
sys em esponse o an ex e nal ime-pe iodic d i ing wi h ze o ime-a e aged
alue. Though mos o he wo ks abou a che sys ems conside he p es-
ence o noise, his phenomenon may also a ise in de e minis ic sys ems, bo h
in he o e damped [11–13] and unde damped [7, 8] egimes. Mo eo e , he
phenomenon o an icipa ed synch oniza ion occu ing in ine ial de e min-
is ic a che s ha e been s udied ecen ly [14].
In pa icula , in he o e damped egime, he one-dimensional pa icle
dynamics x( )usually conside ed is go e ned by a i s o de di e en ial
equa ion o he ype
˙x( ) = −U′[x( )] + F( ),(1)
whe e he do and he p ime deno e ime and spa ial de i a i es, espec-
i ely, U(x)is a pe iodic po en ial wi h spa ial pe iod λ[i.e.,U(x+λ)=U(x)],
and F( )is a ime-pe iodic d i ing o ce wi h pe iod T[i.e.,F( +T)=F( )].
In his ype o sys ems, he cu en is de ined as he a e age eloci y
= lim
→∞
x( )−x(0)
.(2)
As shown in Re . [11] co-exis ing a ac o s can exis o la ge d i ing s en-
g hs, which, howe e , a e no cu en -ca ying. A ini e cu en , possessing
an unbounded x( )- ajec o y, is consequen ly independen o he ini ial con-
di ion x(0). I can also be shown ha a di ec ed cu en ( 6= 0) is only
possible i a leas one o he ollowing symme ies is b oken [2,4,5,15]:
∃x0∈ ℜ such ha U(x0−x) = U(x0+x),∀x∈ ℜ,(3)
F( +T/2) = −F( ),∀ ∈ ℜ.(4)
Ou main in e es in his pape is o gain a deepe insigh in o hese ype
o sys ems explo ing some quan i a i e and quali a i e aspec s o i s e y
ich dynamics. Speci ically, we will conside he same saw oo h po en ial as
in Re s. [11,16]
U(x) = −1
2πsin(2πx) + 1
4sin(4πx),(5)
which has a spa ial pe iod λ= 1. In addi ion, a he han using a sinusoidal
d i ing o ce F( ), in his wo k we will conside a mul i equency ime-
pe iodic o ce gi en by
F( ) = (A o 0≤ < T
2
−A o T
2≤ < T, (6)
wi h Abeing a cons an . Since he po en ial (5) b eaks he spa ial symme y
(3), a di ec ed cu en is possible, e en hough he ime symme y (4) is
ul illed.
O e damped De e minis ic Ra che s D i en by . . . 1469
The pape is o ganized as ollows. In Sec. 2, we pe o m a de ailed
nume ical s udy o Eq. (1) o a wide ange o he d i ing pa ame e alues A
and ω= 2π/T. In pa icula , we p esen a colo ed phase-diag am e sus A
and ω, which is inspi ed by he igu e p o ided by P o . Pe e Talkne and
collabo a o s in [16]. In o de o explain some ea u es o his diag am we
p opose in Sec. 3 some simple heo e ical conside a ions. Finally, in he las
sec ion, we summa ize ou indings.
2. Dynamical egimes and nume ical e alua ion o he cu en
Rega dless o he alue o he d i ing equency, he e exis wo c i ical
ampli udes A∗
1= 3/4and A∗
2= 3/2sepa a ing h ee di e en egimes. Fo
ampli udes A∈[0, A∗
1], bo h po en ial s a es, U(x) + Ax and U(x)−Ax,
possess a pe iodic a ay o equilib ium poin s (see Fig. 1). Since he ine ial
e m m¨x( )is absen , he pa icle canno c oss hese equilib ium poin s, and
consequen ly, i emains apped be ween hem, leading o a ze o cu en .
Fo A∈(A∗
1, A∗
2], he po en ial s a e U(x) + Ax e ains i s equilib ium
poin s, while U(x)−Ax does no ha e any, ha allowing non-bounding
mo ion in he posi i e di ec ion. Finally, when A > A∗
2, nei he o he
po en ial s a es possesses equilib ium poin s, and he pa icle mo ion is no
bounded in ei he di ec ion.
-3 -2 -1 0 1 2 3
x
-1
0
1
2
ins an aneous eloci y
Fig. 1. Plo s o he ins an aneous eloci ies −U′(x)+A(solid line) and −U′(x)−A
(dashed line) as a unc ion o x o a d i ing o ce wi h A= 0.5. (All quan i ies in
dimensionless uni s.)
In o de o go u he in ou analysis, we ha e eso ed o a nume ical
ea men o ou model using he eely a ailable in eg a o RKSUITE [17].
Wide egions o pa ame e space ha e been explo ed. In Fig. 2 we ha e
used a colo code o ep esen he d i eloci y o a ec angula d i ing
o ce o a ying ampli ude Aand equency ω. The egions in black co e-
spond o bounded, ime-pe iodic pa icle mo ion wi h ze o cu en . Mo e
p ecisely, he eloci y in hose egions is smalle han 10−6. The diag am has
a ich and in e es ing s uc u e. Regions o ze o d i eloci y a e in e min-
gled wi h egions wi h ini e alues, gi ing ise o a inge -shaped s uc u e.
1470 D. Cube o e al.
No ice ha o he pa ame e alues conside ed, he d i eloci y is always
posi i e o ze o, showing he la ges magni ude in he in e media e egion
A∈(A∗
1, A∗
2]. This las ea u e could be unde s ood by aking in o accoun
ha in his egime he mo ion in he posi i e di ec ion is ne e compensa ed
by mo ion in he opposi e di ec ion.
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 “Talkne ”-cu en phase diag am. The d i eloci y as a unc ion o he
d i ing pa ame e s Aand ωis ep esen ed using a colo densi y plo . Black colo has
been used o eloci ies less han 10−6. (All quan i ies in dimensionless uni s.)
0 2 4 6 8
ω
0
0.1
0.2
0.3
0.4
0.5
Fig. 3. D i eloci y (solid line) as a unc ion o he d i ing equency ω o A=
1.4<A∗
2. The dashed line shows he adiaba ic alue. The a ows depic s he ze o-
eloci y bands gi en by he heo y in he ex . (All quan i ies in dimensionless uni s.)
O e damped De e minis ic Ra che s D i en by . . . 1471
In Fig. 3 we p esen a sec ion o he diag am o a d i ing ampli ude
in he in e media e egime A= 1.4. A se ies o peaks co esponding o
he inge s in Fig. 2 a e obse ed. Fig. 4 shows a ep esen a i e sec ion o
A > A∗
2. By con as wi h Fig. 3, i p esen s an in e media e gap o pa icle
localiza ion. I we u he inc ease he alue o A, mo e in e media e gaps
appea , as can be seen in he phase diag am.
0 2 4 6 8
ω
0
0.1
0.2
0.3
Fig. 4. The same as in Fig. 3 bu o A= 2 > A∗
2.
3. Some heo e ical esul s
E en hough he nonlinea i y o he sys em p ecludes a comple e and
de ailed analy ical solu ion o he p oblem, i is possible o explain some
ea u es o he phase diag am by simple conside a ions.
3.1. P oo o exis ence o egions displaying pa icle localiza ion
Due o he unca ion implici in any nume ical calcula ion, he simula-
ions epo ed abo e a e no able o dis inguish be ween a si ua ion o exac
pa icle localiza ion o a e y small d i , lowe han he ole ance we chose
o de e mine he black egions in Fig. 2. Fu he mo e, a simple explana ion
o he exis ence o such egions in he non- i ial egime A > A∗
1would be
desi able.
Le us assume ha he pa icle s a s a x0a he beginning o a d i ing
pe iod. I hen mo es unde a o ce −U′(x) + A o hal a pe iod un il i
eaches he posi ion x1> x0. Then he d i ing o ce swi ches sign so he
o al o ce on he pa icle is now −U′(x)−A, which makes i (in gene al)
mo e backwa ds up o a posi ion x′
0a e ano he T/2. I x′
0=x0, he
pa icle has e u ned o he ini ial posi ion, and consequen ly, he p ocess is
epea ed successi ely, leading o a d i eloci y s ic ly equal o ze o. Fo
his si ua ion o happen he ollowing equa ions mus hold

1472 D. Cube o e al.
T
2=
x1
Z
x0
dx
−U′(x) + A(7)
T
2=
x0
Z
x1
dx
−U′(x)−A.(8)
Sub ac ing bo h equa ions we a i e a he condi ion
G(x0) = G(x1),(9)
whe e
G(x) =
x
Z
0
d˜xU′(˜x)
A2−U′(˜x)2.(10)
The e o e, we can de e mine he se o pai s (x0, x1)wi h ze o cu en o a
gi en d i ing s eng h Aby plo ing he unc ion G(x) e sus x(see Fig 5).
The in e sec ion o a ho izon al line wi h G(x)in his plo p o ides he
possible alues o he pai (x0, x1). The pe iod Tassocia ed wi h he pai is
hen gi en by Eq. (7) o (8). Since he d i eloci y is independen o he
ini ial condi ions [15], i would be exac ly ze o o hose d i ing pa ame e
alues Aand T.
In Fig. 5, we plo G(x) o he same alue o A= 2(> A∗
2)as in Fig. 4.
A ho izon al line c osses G(x)a he poin s A, B, C, and D, p o iding he
-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. De e mina ion o ze o- eloci y bands wi h x( +T) = x( ). Solid line depic s
G(x), de ined in Eq. (10), as a unc ion o x o he same alue o Aas in Fig. 4.
O e damped De e minis ic Ra che s D i en by . . . 1473
coo dina es xA, xB, xC, and xD. I we choose x0=xA, and x1as any o he
o he poin s, we ob ain h ee pai s o poin s wi h d i ing pe iods TAB, TAC,
and TAD. Since he in eg and in Eq. (7) is always posi i e, he u he away
x1is om x0, he la ge he pe iod, which implies TAB < TAC < TAD.
Be o e we p oceed in analyzing he plo in g ea e de ail, le us s udy
he symme ies o G(x). Since i is an in eg al o a space-pe iodic unc ion
[see Eq. (10)] we ha e
G(x+ 1) = G(x) + φ , (11)
whe e φ=G(1). In addi ion, aking in o accoun ha U′(x)is an e en
unc ion, necessa ily
G(−x) = −G(x).(12)
P ope y (11) leads o he ac ha we only need o a y x0wi hin a spa ial
in e al o leng h uni y, whe eas p ope y (11) combined wi h (12) implies
ha G(x)has an in e sion cen e a x=n, wi h nany in ege . The d i ing
pe iod T, when iewed as a unc ion o x1o x0[see Eq. (7)], also obeys
hese p ope ies.
Because o hese symme y p ope ies, choosing he poin A as x0in
Fig. 5 is equi alen o choosing any o he poin s A′, A′ ′ and A′′′, in he sense
ha hey lead o he same d i ing pe iods. Le us conside a ho izon al line
be ween poin s A and A′. As his line ge s close o A, wo o he in e sec ion
poin s app oach each o he , and consequen ly, he d i ing pe iod associa ed
o hem ends o ze o. The e o e, T= 0 (ω=∞) co esponds o pa icle
localiza ion. Mo ing up he line con inuously we ob ain a whole band o
d i ing pe iods s a ing om ze o up o a maximum alue gi en by he
pai (xB′, xA′)when he line c osses A′. This pai is equi alen o (xA, xB).
Two mo e in e media e pe iods in he band can be ob ained om he pai s
(xB, xC)and (xC, xD)in he igu e.
The nex alue o he d i ing pe iod ob ained om Fig. 5 is he one
gi en by he pai (xA, xC). This pai ma ks he s a o a second band ha
ends a he maximum alue gi en by he pai (xA, xD). Mo ing up he line
up o A′ ′ gi es all he in e media e alues in he band.
Clea ly, he i s band is associa ed wi h mo emen wi hin a spa ial in-
e al less han uni y (i.e.,0≤x1−x0<1), whe eas he second band is
ela ed o displacemen s wo imes ha dis ance (1≤x1−x0<2). In
Fig. 4 we ha e indica ed wi h a ows he calcula ed equency bands. I can
be seen ha hey do no co e he en i e egion wi h nume ically e alua ed
ze o eloci y. This is due o he ac ha he abo e discussed mechanism
is jus he simples one leading o ze o cu en . The d i eloci y can also
anish because he pa icle e u ns o i s ini ial posi ion a e wo o mo e
d i ing pe iods, ins ead o a e he i s one.
1474 D. Cube o e al.
The same analysis can be ca ied ou o a sub h eshold d i ing A∈
(A∗
1, A∗
2]. In his case G(x)p esen s singula i ies a he equilib ium poin s,
which p e en s he pa icle om c ossing hose poin s when F( ) = −A.
This leads o a single band which is nea he ze o-cu en egion obse ed
in he simula ions, as shown in Fig. 3.
3.2. Adiaba ic limi
In his sec ion we will ob ain an analy ical exp ession o he cu en in
he adiaba ic limi ω→0
ad(A) = lim
ω→0 (A, ω).(13)
Speci ically, as p o ed la e on, ad(A)is gi en by he a e age alue
ad(A) = 1
2[ +(A) + −(A)] ,(14)
whe e +(A)and −(A)a e, espec i ely, he eloci ies in he p esence o
he s a ic o ces −U′(x) + Aand −U′(x)−A. Ob iously, +(A) = 0 o
A∈[0, A∗
1]and −(A) = 0 o A∈[0, A∗
2], as he pa icle ends up being
apped by an equilib ium poin . Fo A > A∗
1,
+(A) = 1
τ+(A),(15)
whe e
τ+(A) =
1
Z
0
dx
A−U′(x)=2q2A+p−3 + 4A(1 + A)
p(−1 + 2A)(3 + 2A)(−3 + 4A)(16)
is he ime aken o he pa icle o a el a dis ance equal o 1( he spa ial
pe iod) in he p esence o he s a ic o ce −U′(x) + A. Analogously, o
A > A∗
2,
−(A) = −1
τ−(A),(17)
whe e
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 de ined as τ+(A)bu eplacing −U′(x) + Aby −U′(x)−A.
O e damped De e minis ic Ra che s D i en by . . . 1475
In o de o p o e Eq. (14), le us conside sepa a ely he h ee egimes
A∈[0, A∗
1],A∈(A∗
1, A∗
2], and A∈(A∗
2,∞). Fo A∈[0, A∗
1] he esul in
Eq. (14) is i ial, since ad(A) = −(A) = +(A) = 0. In he in e media e
egime A∈(A∗
1, A∗
2], le us assume ha he pa icle is ini ially loca ed a
a minimum o he po en ial U(x) + Ax (as we ha e men ioned be o e, he
d i eloci y does no depend on his pa icula ini ial condi ion). I we
choose he ime pe iod o F( )as T= 2Nτ+(A), wi h N= 1,2,3,...,
hen, du ing he i s hal -pe iod, Nτ+(A), he pa icle s ays apped a
he ini ial loca ion. A e he second hal -pe iod, 2Nτ+(A), he pa icle
a i es a a new minimum o U(x)+Ax sepa a ed om he ini ial loca ion
by a dis ance N. Du ing he hi d hal -pe iod, 3Nτ+(A), he pa icle s ays
apped a ha minimum, and so on. Thus, o a gi en A∈(A∗
1, A∗
2], all
he equencies
ωN(A) = π
Nτ+(A)(19)
lead o he same alue o he d i eloci y
[A, ωN(A)] = N
2Nτ+(A)=1
2τ+(A)= +(A)
2.(20)
Consequen ly, aking in o accoun ha limN→∞ ωN(A) = 0, i ollows ha
ad(A) = lim
N→∞
[A, ωN(A)] = +(A)
2.(21)
This p o es Eq. (14) o A∈(A∗
1, A∗
2], since in his egime −(A) = 0.
Finally, o A∈(A∗
2,∞), i is easy o p o e ha τ+(A)/τ−(A)is a con in-
uous s ic ly inc easing unc ion o Awhich akes alues in he in e al (0,1).
Le us assume i s ha o a gi en A∈(A∗
2,∞) he a io τ+(A)/τ−(A)is
a a ional numbe p/q, wi h p < q. Then, i we choose he ime pe iod o
F( )as T= 2Nqτ+(A) = 2Npτ−(A), wi h N= 1,2,3,..., he pa icle
a els a dis ance equal o N(q−p)e e y ime pe iod. Consequen ly, all he
equencies
ωN(A) = π
Nqτ+(A)=π
Npτ−(A)(22)
lead o he same alue o he d i eloci y
[A, ωN(A)] = N(q−p)
2Nqτ+(A)=1
2τ+(A)−1
2τ−(A)= +(A)+ −(A)
2.(23)
Taking in o accoun once again ha limN→∞ ωN(A) = 0, i ollows ha
ad(A) = lim
N→∞
[A, ωN(A)] = +(A) + −(A)
2.(24)