Hidden Symme ies, Ins abili ies, and Cu en Supp ession in B ownian Ra che s
Da id Cube o1,* and Fe uccio Renzoni2,†
1Depa amen o de Física Aplicada I, EUP, Uni e sidad de Se illa, Calle Vi gen de Á ica 7, 41011 Se illa, Spain
2Depa men o Physics and As onomy, Uni e si y College London, Gowe S ee , London WC1E 6BT, Uni ed Kingdom
(Recei ed 12 Sep embe 2015; published 7 Janua y 2016)
The ope a ion o B ownian mo o s is usually desc ibed in e ms o ou -o -equilib ium and symme y-
b eaking se ings, wi h he ele an spa io empo al symme ies iden i ied om he analysis o he equa ions
o mo ion o he sys em a hand. When he app op ia e condi ions a e sa is ied, symme y- ela ed
ajec o ies wi h opposi e cu en a e hough o balance each o he , yielding supp ession o anspo . The
di ec ion o he cu en can be p ecisely con olled a ound hese symme y poin s by inely uning
he d i ing pa ame e s. He e we demons a e, by s udying a p o o ypical B ownian a che sys em, he
exis ence o hidden symme ies, which escape iden i ica ion by he s anda d symme y analysis, and which
equi e di e en heo e ical ools o hei e ela ion. Fu he mo e, we show ha sys em ins abili ies may
lead o spon aneous symme y b eaking wi h unexpec ed gene a ion o di ec ed anspo .
DOI: 10.1103/PhysRe Le .116.010602
Mo ion a he nanoscale p esen s ea u es e y di e en
om hose encoun e ed in he mac oscopic wo ld. Noise is
a dominan p ocess a such a scale, and may con ibu e
cons uc i ely o he dynamics a he han playing he usual
ole o a dis u bance. New mechanisms o anspo eme ge
a he nanoscale; in pa icula , di ec ed mo ion may occu in
he absence o an applied bias o ce. B ownian a che s
[1–3], he a che ypal model sys em cap u ing he mecha-
nisms behind such a anspo p ocess, ep esen a key o
unde s anding se e al biological p ocesses [4,5]; hey also
ha e inspi ed a ple ho a o new nanode ices displaying
di ec ed mo ion [6–21]. All hese sys ems a e usually
desc ibed in e ms o ope a ion away om he mal equi-
lib ium, wi h di ec ed mo ion ollowing om he b eaking
o ce ain spa io empo al symme ies, which a e iden i ied
om he analysis o he equa ions o mo ion o he sys em
a hand. He e we p o e he exis ence o hidden symme ies,
which escape iden i ica ion by he s anda d symme y
analysis [2,3,22,23], and equi e di e en heo e ical ools
o hei e ela ion. The main assump ion o he s anda d
symme y analysis—i.e., ha wo ajec o ies connec ed by
a symme y ans o ma ion ca y he same s a is ical
weigh , a easoning ha can be aced back o
Loschmid ’s pa adox [24]—yields inco ec p edic ions
in hese dissipa i e sys ems, ailing o accoun o sys em
ins abili ies ha lead o spon aneous symme y b eaking.
Resul s.—A la ge class o B ownian mo o sys ems,
which includes pa icles in solu ion [8], o ices in supe -
conduc o s [14], and a oms in dissipa i e op ical la ices
[21], co esponds o a B ownian pa icle di using in a
pe iodic po en ial unde he ac ion o a d i ing o ce wi h
ze o a e age. The pa icle’s mo ion is desc ibed by he
ollowing Lange in equa ion:
m
x¼−γ_
xþFðx; Þþξð Þ;ð1Þ
whe e γis he ic ion coe icien , Fðx; Þis a gene ic
de e minis ic o ce, and ξð Þis a luc ua ing o ce, modeled
as a Gaussian whi e noise wi h au oco ela ion
hξð Þξð 0Þi ¼ 2Γδð − 0Þ, wi h he noise s eng h Γ ela ed
o he empe a u e To he en i onmen ia he luc ua ion-
dissipa ion ela ion Γ¼γkBT. The di ec ed cu en is
de ined as h i¼lim →∞hxð Þi= , whe e he angle b acke s
deno e he a e age o e noise ealiza ions. Fo ini e noise
s eng hs Γ>0, e godici y implies h i¼lim →∞xð Þ= .In
e y small sys ems, om he nanoscale o he mic oscale,
he B ownian dynamics o small pa icles is equen ly in
he o e damped egime, whe e ine ia e ec s— he e m m
x
in (1)—can be neglec ed. This is he egime o in e es he e.
The s anda d symme y analysis [2,3,22,23] elies on he
iden i ica ion o ans o ma ions ha lea e he equa ion o
mo ion (1) unchanged and e e se he sign o he pa icle
momen um. T ajec o ies wi h opposi e momen um a e
equi alen , wi h a ne null con ibu ion o he di ec ed
cu en , which hus u ns ou o be ze o. We will show ha
his pic u e does no ully cap u e he basic p inciples
behind he ope a ion o B ownian a che s. To do his, we
conside a mo e gene al app oach [25], and ega d he
di ec ed cu en h ias a gene ic unc ional o he d i ing
o ce F, hus using he no a ion ½Fðx; Þ. Se e al p ope -
ies ollow om symme y conside a ions.
Fi s , due o he ec o ial na u e o bo h he o ce Fand
he cu en , he ans o ma ion x→−xyields he ollowing
p ope y:
½−Fð−x; Þ ¼ − ½Fðx; Þ:ð2Þ
Second, an a bi a y ansla ion along he xo axis does
no al e he cu en , i.e.,
½Fðx; Þ ¼ ½Fðxþx0; Þ ¼ ½Fðx; þ 0Þ:ð3Þ
PRL 116, 010602 (2016) PHYSICAL REVIEW LETTERS week ending
8 JANUARY 2016
0031-9007=16=116(1)=010602(6) 010602-1 © 2016 Ame ican Physical Socie y
Le us conside now a o ced a che , i.e., Fðx; Þ¼
ðxÞþFð Þ, whe e ðxÞ¼−∂UðxÞ=∂xis a conse a i e
o ce and Fð Þis a d i ing o ce.
I he sys em is spa ially symme ic wi h espec o a
ce ain poin x0, hen he po en ial sa is ies Uðxþx0Þ¼
Uð−xþx0Þ. Wi hou loss o gene ali y, we choose he
coo dina e’s o igin such ha x0¼0. Then − ð−xÞ¼ ðxÞ,
which oge he wi h (2) yields a cha ac e is ic p ope y o
spa ially symme ic sys ems,
½ ðxÞ−Fð Þ ¼ − ½ ðxÞþFð Þ:ð4Þ
A shi -symme ic o ce is de ined as Fð þ 0Þ¼−Fð Þ—
o pe iodic d i es 0¼τ=2, whe e τis he pe iod,
Fð þτÞ¼Fð Þ. The di ec applica ion o p ope ies (4)
and (3) yields no cu en o shi -symme ic o ces in
spa ially symme ic sys ems,
½ ðxÞ−Fð Þ ¼ ½ ðxÞþFð þ 0Þ ¼ ½ ðxÞþFð Þ
¼− ½ ðxÞþFð Þ:ð5Þ
This is a well-known esul o spa ially symme ic sys ems,
al eady cap u ed by he s anda d symme y analysis
[2,3,22,23]. Howe e , ou cu en app oach e eals wo
addi ional symme ies o o e damped one-dimensional
sys ems, which a e no cap u ed by he s anda d app oach.
They a e
½ ð−xÞþFð Þ ¼ ½ ðxÞþFð Þ;ð6Þ
½ ðxÞþFð− Þ ¼ ½ ðxÞþFð Þ:ð7Þ
A p oo o (6) and (7) based on he Smoluchowski equa ion
is gi en in he Supplemen al Ma e ial [26]. The symme ies
(6) and (2), oge he wi h (3), yield he ollowing p ope y
o shi -symme ic po en ials:
½ ðxÞþFð Þ¼ ½ ð−xÞþFð Þ¼− ½− ðxÞ−Fð Þ
¼− ½ ðxþL=2Þ−Fð Þ¼− ½ ðxÞ−Fð Þ:
ð8Þ
This is he same p ope y as Eq. (4) and, p oceeding as
be o e, i implies cu en supp ession when combined wi h
a shi -symme ic d i ing o ce. The e o e, qui e coun e -
in ui i ely, in one-dimensional o e damped sys ems, he
condi ion o cu en supp ession o sys ems wi h shi -
symme ic po en ials—like he one shown in Fig. 1(b)—is,
despi e being spa ially asymme ic, he same as o spa ially
symme ic po en ials. Figu e 1(c) con i ms his unexpec ed
beha io o la ge enough ic ions. In he unde damped
egime his p ope y is no sa is ied exac ly. Ne e heless,
e en in his egime he o e damped symme y (6) iden i ied
he e has a las ing e ec : The ze o-cu en poin de e mined
by he o e damped symme y is displaced o a lowe alue
o he symme y pa ame e ain he unde damped egime.
Thus, he o e damped symme y (6) de e mines a cu en
e e sal in he unde damped egime.
The disco e y o hidden symme ies epo ed abo e does
no ep esen he only depa u e om he conclusions ha
can be d awn om he s anda d symme y analysis. The
p esence o ins abili ies may also al e he pic u e, as
ajec o ies ha a e solu ions o he equa ions o mo ion
wi h opposi e momen a may ha e e y di e en s abili y
p ope ies, and, hus, esul in o a o al nonze o con ibu ion
o he sys em cu en . Such a scena io o spon aneous
symme y b eaking is bes illus a ed ia a speci ic
case s udy.
In he o e damped egime, om e e y solu ion xð Þ, he
ajec o y ~
xð Þ¼xð− ÞþL=2is also a solu ion o (1)
p o ided he po en ial is shi symme ic. I co esponds o
a ans o med andom o ce ~
ξð Þ¼−ξð− Þ, which is
s a is ically equi alen o ξð Þ, and a d i ing o ce
~
Fð Þ¼−Fð− Þ. Following he s anda d symme y analy-
sis, no cu en is expec ed when an isymme ic d i ing
o ces Fð þ 0Þ¼−Fð− þ 0Þa e applied [3,22,23,41]—
an app op ia e choice o he ime o igin yields 0¼0. This
p edic ion is co ec in one-dimensional sys ems, as eadily
e i ied by nume ical simula ions. Howe e , he same
easoning also p edic s no cu en in he case o highe
dimensions, a esul ha is con adic ed by ou nume ical
0.6
0 0.2 0.4 0.8 1
a
-0.005
0
0.005
0.01
γ=30
γ=20
γ=10
γ=5
U(x) o U(x) o
a=0 a=1
(a) (b)
(c)
FIG. 1. Shi -symme ic po en ials ac like spa ially symme ic
ones in one-dimensional o e damped sys ems. (a) Ra che po-
en ial U a ðxÞ¼−U0½sinðkxÞþð1=4Þsinð2kxÞ wi h pe iod
L¼2π=k. (b) Shi -symme ic po en ial de ined om U a ðxÞ
as UssðxÞ¼U a ðxÞin he i s hal -pe iod, and UssðxÞ¼
−Ussðx−L=2Þin he second hal -pe iod. (c) Di ec ed cu en
o a B ownian pa icle subjec o he mixed po en ial UðxÞ¼
U a ðxÞð1−aÞþUssðxÞaand o a shi -symme ic o ce de ined
by Fð Þ¼gð Þ≡A½sinðω Þþð1=4Þsinð2ω Þ in he i s hal -
pe iod, and Fð Þ¼−gð −τ=2Þin he second hal , whe e
τ¼2π=ω. Reduced uni s a e de ined such as m¼L¼10ω¼1.
O he pa ame e s a e A¼4,U0¼10=2π, and Γ¼10. The
di ec ed cu en anishes in he o e damped limi (la ge ic ions
γ) o he shi -symme ic po en ial (a¼1) because o hidden
symme ies.
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simula ions, as shown in Fig. 2 o a wo-dimensional
po en ial and an applied spli biha monic d i e, as well as
by independen esul s by Reimann’s g oup (see Re . [43],
p. 16). The p esence o ins abili ies is he key o unde -
s anding such an unexpec ed, spon aneous symme y-
b eaking beha io . The s anda d analysis ails o accoun
o he ac ual ins abili y o he ans o med solu ions ~
xð Þ,
which makes hem e y unlikely. E en in he noiseless
limi , he abo e ans o ma ion maps s able oscilla ions
abou he po en ial minima in o highly uns able oscilla ions
abou po en ial maxima [44]. We ha e e i ied ia nume i-
cal simula ions ha , gi en a s able solu ion xð Þ, he
ans o med solu ion ~
xð Þis uns able and hus quickly
collapses on o xð Þ. This occu s bo h in one dimension as
well as in highe -dimensional sys ems [26]. Gi en ha
ins abili ies des oy he mechanisms o cu en supp ession
due o he con ibu ions o a ajec o y and he ans o med
one, he obse ed supp ession o di ec ed anspo in one-
dimensional sys ems mus be associa ed o a di e en
mechanism. This supp ession unde an isymme ic o ces
is ac ually a consequence o he symme y (7), which yields
no cu en o sys ems—which include spa ially symme ic
as well as spa ially shi -symme ic sys ems o in e es
he e—sa is ying he p ope y (4)
½ ðxÞþFð Þ ¼ − ½ ðxÞ−Fð Þ ¼ − ½ ðxÞþFð− Þ
¼− ½ ðxÞþFð Þ:ð9Þ
A consequence o his analysis is ha uly spa ially
symme ic sys ems should also exhibi no cu en in
one-dimensional o e damped sys ems when an isymme ic
o ces a e d i ing he sys em. This phenomenon is illus-
a ed in Fig. 3. I was al eady expe imen ally obse ed in
Re . [45], bu i emained unexplained un il he p esen
Le e . These esul s a e a con i ma ion o he alidi y o he
app oach based on a mo e gene al symme y analysis ha
does no ely on he di ec analysis o he solu ions o he
equa ion o mo ion.
I is wo h s essing ha in he p esen discussion he
dimensionali y o he sys em co esponds o he numbe o
spa ial deg ees o eedom aking pa in o he ec i ica ion
mechanism, and no necessa ily o he dimensionali y o he
po en ial landscape. The iola ion o he symme ies (6),
(7) in he abo e 2D o e damped se up is due o a
ec i ica ion mechanism aking place in he wo
pe pendicula di ec ions. Howe e , he symme ies (6),
(7) a e no es ic ed o s ic ly one-dimensional sys ems;
hey a e s ill p esen in highe -dimensional o e damped
sys ems p o ided ha he ec i ica ion mechanism in ol es
one spa ial dimension only. Fo example, he dashed line in
Fig. 3(c) shows he supp ession o cu en o an isym-
me ic d i ing o he same 2D sys em shown in Fig. 2
when he biha monic d i ing o ce is applied in he y
di ec ion only. Addi ional examples a e shown in [26].
0246810 12 14 16 18
ky/ kx
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
〈 y〉
x / L
y / L
kykx
=4
FIG. 2. B eaking o symme ies (6),(7) in a 2D o e damped
sys em. The d i ing o ce is Fð Þ¼A½cosðω Þexþ
cosð2ω þπ=2Þey, i.e., a biha monic d i e spli [42] in wo
pe pendicula di ec ions. The po en ial is Uðx; yÞ¼
U0cosðkxxÞ½1þcosðkyyÞ, which is spa ially symme ic in bo h
di ec ions, and shi symme ic along he xdi ec ion. The cu en
is p oduced in he ydi ec ion only—due o he symme y (5) in
he xdi ec ion— h ough he coupling wi h he dynamics in he x
di ec ion. Though he d i ing o ce is an isymme ic, a nonze o
cu en is obse ed when kxand kya e compa able. Reduced
uni s a e de ined such ha m¼kx¼ω¼1. O he pa ame e s
a e U0¼γ¼50,A¼2γ, and Γ¼0.1γ2. The inse illus a es he
po en ial landscape o ky¼4kx, wi h L¼2π=kx.
0π/2 π
φ
-1.5
-1
-0.5
0
0.5
1
1.5
〈 〉
γ=20
γ=15
γ=10
γ=5
U(x) F( ) (φ=π/2)
(a) (b)
(c)
FIG. 3. Cu en supp ession in one-dimensional o e damped
sys ems wi h a spa ially symme ic po en ial and applied an i-
symme ic o ces. The d i ing o ce has a biha monic shape,
Fð Þ¼A½cosðω Þþcosð2ω þϕÞ. (a) Spa ially symme ic po-
en ial UðxÞ¼U0½cosðkxÞþcosð2kxÞ. (b) The d i ing o ce
Fð Þis an isymme ic when ϕ¼π=2. (c) Di ec ed cu en as a
unc ion o he d i ing phase ϕ, o di e en le els o damping.
Reduced uni s a e de ined such ha m¼k¼ω¼1. O he
pa ame e s a e U0¼20 and A¼Γ¼40. The dashed line shows,
o compa ison, he cu en (h yi) o he same d i ing o ce
applied in he ydi ec ion and a wo-dimensional po en ial
Uðx; yÞ¼U0cosðkxÞ½1þcosð4kyÞ in he o e damped egime
(γ¼U0¼50,A¼2γ,Γ¼0.1γ2).
PRL 116, 010602 (2016) PHYSICAL REVIEW LETTERS week ending
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Discussion.—The hidden symme ies iden i ied in he
p esen Le e a e o ele ance o cu en expe imen s, and
hey also allow us o ecas known esul s wi hin a mo e
gene al heo e ical amewo k. This is well exempli ied by
he wo speci ic case s udies ha a e p esen ed below.
The i s case s udy co esponds o he sys em o
ac-d i en o ices apped in a supe conduc o ha was
expe imen ally s udied in Re . [14]. He e, in e pa icle
in e ac ions p o ide an addi ional pa h o escape om
he symme ies (6),(7). Ou esul s o Fig. 4 e e p ecisely
o he one-dimensional sys em o in e ac ing B ownian
pa icles ha was success ully used in Re . [14] o explain
he mul iple cu en e e sals obse ed on ac-d i en o -
ices apped in a supe conduc o . Despi e no being s ic ly
sa is ied, he in luence o he symme ies (6),(7) is qui e
no iceable, canceling he a che e ec and mos o he
cu en e e sals in egions o he pa ame e space whe e
he appea ance o a cu en is no di ec ly ela ed o pa icle
in e ac ions.
Fo a second case s udy, we e e o he celeb a ed
lashing a che model [8–11,46,47], whe e he a che
po en ial is pe iodically swi ched on and o in he
absence o any addi ional addi i e d i ing Fð Þ—i.e., he e
Fðx; Þ¼−∂Uðx; Þ=∂x—and mo e speci ically o he
known esul [48] ha a lashing shi -symme ic po en ial
canno p oduce di ec ed mo ion. The heo e ical amewo k
and he ela ed new symme ies ha we in oduce he e
allow o a simple explana ion o such a esul .
In one-dimensional o e damped sys ems, he ollowing
symme y is gene ally sa is ied [26]:
½Fð−x; − Þ ¼ ½Fðx; Þ:ð10Þ
In wo-s a e sys ems ha a e pe iodically swi ched, e e sing
he di ec ion o ime has no e ec , ½Fðx; − Þ ¼ ½Fðx; Þ;
his ac , oge he wi h (2),(10), and (3), yields no cu en o
shi -symme ic po en ials,
½Fðx; Þ ¼ − ½−Fð−x; Þ ¼ − ½−Fðx; − Þ
¼− ½−Fðx; Þ ¼ − ½−FðxþL=2; Þ
¼− ½Fðx; Þ:ð11Þ
The e o e,a lashing a che wi hashi -symme icpo en ial,
ega dless o whe he i is spa ially asymme ic, canno
p oduce di ec ed mo ion; his hus shows ha in o e damped
sys ems shi -symme ic po en ials beha e like spa ially
symme ic ones.
Conclusions.—The p esen Le e add esses he ou -
s anding issue o p o iding a gene al heo e ical amewo k
o he iden i ica ion o symme ies no cap u ed by he
s anda d symme y analysis, examples o which we e
al eady gi en in p e ious wo ks [44,48] wi h ad hoc
ea men s. We ha e p o en he exis ence in a p o o ypical
1D o e damped sys em o hidden symme ies, which
escape iden i ica ion by he s anda d symme y analysis
and equi e di e en heo e ical ools o hei e ela ion.
Though no igo ously sa is ied in highe -dimensional
sys ems, he e ec s o hidden symme ies ha e been shown
o be s ill no iceable in hem. Ou esul s pa e he way o
new mechanisms o manipula ing anspo . In ac , he
hidden symme ies de e mine cu en e e sals, which can
be used o p ecisely con ol anspo and implemen
mechanisms o pa icle sepa a ion. Speci ic ealiza ions
o op ical weeze s and cold a om se ups a e discussed in
he Supplemen al Ma e ial [26].
Financial suppo om he Royal Socie y (G an
No. IE130734) (D. C. and F. R.), and he Le e hulme
T us (G an No. RPG 2012 809) (F. R.) is acknowledged.
*[email p o ec ed]
†[email p o ec ed]k
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nian mo o , Science 276, 917 (2000).
[2] P. Reimann, B ownian mo o s: Noisy anspo a om
equilib ium, Phys. Rep. 361, 57 (2002).
[3] P. Hänggi and F. Ma chesoni, A i icial B ownian mo o s:
Con olling anspo on he nanoscale, Re . Mod. Phys. 81,
387 (2009).
[4] L. Mahade an and P. Ma sudai a, Mo ili y powe ed by
sup amolecula sp ings and a che s, Science 288,95
(2000).
4 3 2 1 0
0
1
2
3
4
U /E
n
-10
-5
0
5
10
p1 0
4 3 2 1 0
0
1
2
3
4
U /E
p1 0
0
E /L
10 -3
( )
U(x) U(x)-U(x+L/2)
~U p1
〈 〉
FIG. 4. Cancella ion o anspo , ia he use o shi -symme ic
po en ials, o he one-dimensional o e damped sys em o
in e ac ing pa icles om Re . [14]. The bo om panels show
he ne chain cu en as a unc ion o he numbe o pa icles pe
pe iod, n, and he po en ial dep h Up1=E0, wi h he le panels
e e ing o he o iginal one-pa icle po en ial UðxÞ(depic ed in
he uppe panel), and he igh panels o a shi -symme ic
po en ial buil om he o me as UssðxÞ¼UðxÞ−UðxþL=2Þ.
The in e ac ion be ween he pa icles is accoun ed o by he pai
po en ial Vin ð Þ¼−E0lnð Þ, wi h he pa icle sepa a ion. The
sys em is d i en by a single-ha monic o ce ac ing on each
pa icle, which is bo h shi symme ic and an isymme ic. The
pa ame e s a e he same as in Fig. 2 o [14].
PRL 116, 010602 (2016) PHYSICAL REVIEW LETTERS week ending
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