Table Of ContentTwo-dimensional particlemotioninarandom potential under acbias
Maxim A. Makeev,1,∗ Imre Dere´nyi,2,† and Albert-La´szlo´ Baraba´si1,‡
1 Dept. of Physics, University of Notre Dame, Notre Dame, IN 46566, USA
2 Dept. ofBiological Physics, Eo¨tvo¨sUniversity, Pa´zma´nyP.stny. 1A,H-1117Budapest, Hungary
(February2,2008)
4
0 WestudytheBrownianmotionofasingleparticlecoupledtoanexternalacfieldinatwo-dimensionalrandom
0 potential. Wefindthatforsmallfieldsalarge-scalevorticitypatternofthesteady-statenetcurrentsemerges,
2 a consequence of local symmetry breaking. In this regime the net currents are highly correlated, the spatial
n correlationfunctionfollowsalogarithmicdependence,andthecorrelationlengthisoftheorderofthesystem
a size. For large external fields correlations disappear and only random net currents are observed. Numerical
J analysis indicates that the correlation length scales as a power law with both the size of the system and the
7 amplitudeoftheacfield.
] PACSnumbers:79.20.Rf,64.60.Ht,68.35.Rh
h
c
e Drivendiffusioninrandommediaisamuchstudiedprob- potential,sincesuchapotentialobeysinversionsymmetry(in
m
lemimpactingonanumberofvariousfields.Bynow,itiswell astatisticalsense). Inparticular,theratchetvelocityinasys-
- established that the interplay between the annealed random- temofidenticalenergywellsseparatedbyhighenoughenergy
t
a nessofthediffusionprocessandthequenchedrandomnessof barriersis alwayszero, irrespectiveof the distribution of the
t
s the media gives raise to unexpected scaling phenomena that barrierheights.
.
t havebeenextensivelystudiedinrecentdecades[1,2,3]. The On the other hand, in two-dimensional (2D) systems any
a
rangeof applicabilityoftheproblemofdrivendiffusivemo- finite region of a random potential exhibits some degree of
m
tion in random media covers various fields, with relaxation broken symmetry, and the particle can follow different tra-
-
d phenomena in spin glasses [4], dislocation motion in disor- jectoriesduringthetwohalfperiodsoftheacdriving. Inthis
n deredcrystals[5],transportinporousmedia[6],andturbulent Letter,weshowthattheinterplaybetweenthelocalsymmetry
o diffusion[7]beingbutafewtypicalexamples.Inthepresence breakinginarandom2Dpotentiallandscapeandthemotion
c
ofanexternalacfield,themotionofaparticleistheresultof of the particle biased by an externalac field leads to the ap-
[
thecombinedeffectofthermallyactivateddiffusivityandpe- pearanceofa highlycorrelatedsteady-statenetcurrentfield,
2 riodicmotion,inducedbythecouplingtotheappliedfield.At whichischaracterizedbyalarge-scalevorticity. We alsoin-
v
time scales muchlarger than the period of the drivingforce, vestigatethescalingpropertiesofthesefields.
8
one would expectthat the influence of the field is negligible Asamodelsystem, weinvestigatethedrivendynamicsof
6
0 andthedynamicsofthesystemcanbedescribedbytheclas- a single particle moving in a random uncorrelated potential
1 sicalBrownianmotion.Indeed,whiletheparticlemovesback ona2DsquareEuclideanlattices(i,j)withvariednumberof
0 and forth along the direction of the ac field, a stroboscopic identical sites N = L×L, where each site is connected to
4
viewofthesystem,obtainedbytakingpicturesonlyattimes itsnearestneighborsbybondsofunitlength. Randompoten-
0
that are integer multiples of the external field period, would tialbarriersareassignedtoeachbondandperiodicboundary
/
t stillshowarandomlydiffusingparticle,almostasiftheexter- conditionsare assumed. A schematic illustration of the sys-
a
m nalfieldwasabsentinthesystem. Incontrasttothisintuitive
picture, we demonstrate that the presence of an external ac
-
d field can fundamentally change the nature of the dynamics,
n
whenquenchedrandomnessispresentinthesystem.
o
c Anindication,thatanacfieldbiasmaydrasticallyinfluence
: the nature of diffusive motion in a random potential comes
v
i fromtherecentadvancesinthefieldofthermalratchetsdeal-
X
ing with one-dimensional(1D) systems, where the x → −x
r symmetryisbroken[8]. Inequilibrium,aparticlemovingina
a
periodicasymmetricpotentialdisplaysasimplediffusivebe-
havior. However,if the particleis also drivenbyanac field,
it driftsin the directiondefinedby the asymmetryof the po-
tential, on time scales exceeding the period of the ac field.
The appearanceof this non-equilibriumsteady-state net cur-
rentiscalledtheratcheteffect,andtheaveragedriftvelocity FIG. 1: Schematic illustrating the system under consideration. A
oftheparticleisoftenreferredtoastheratchetvelocity[8]. It particleonthesite(i,j)hopstoitsneighboringsites,withhopping
mustbe noted, however,thatthe ratchetvelocityis expected rates,kγ ,definedbyboththepotentialbarriersbetweenthesesites
i,j
andbytheacfield.
to vanishwhentheparticleis movingina quenchedrandom
2
tem’s geometry is shown in Fig. 1. The particle, diffusing τ ≈ L2max(1/kγ ). In this case, for each half period
relax i,j
in the system, is also driven by an external ac field, applied oftheacdrivingtheprobabilitydistributionP andcurrents
i,j
alongthex-direction.Wedefinethelocalprobabilitycurrents J~ relax to their steady state values: P (+F), J~ (+F),
i,j i,j i,j
Jix,j [Jiy,j] on site (i,j) as the currents flowing over the po- andPi,j(−F),J~i,j(−F)),whichcanbedeterminedfromthe
tential barriers between the lattice sites (i,j) and (i + 1,j) stationary solution of the master equation for φ(t) = +F
[(i,j + 1)]. As shown in Fig. 1, a particle, located at site and φ(t) = −F, respectively. From directanalogy with the
(i,j)atanarbitrarymomentoftimet, canhoptoanyofthe ratchet velocities, we can then define the net currents in the
fournearestneighborlatticesitesbyovercomingthepotential
systemas
barriers, Eγ , assigned to each bond. To simplify the nota-
i,j
tions,wedenotebyEir,j,Eil,j,Eiu,j andEid,j thelocalpoten- J~ = 1[J~ (+F)+J~ (−F)]. (4)
tial barriers associated with the right Er = E , i,j 2 i,j i,j
i,j (i,j)→(i+1,j)
left El = E , up Eu = E , and
i,j (i,j)→(i−1,j) i,j (i,j)→(i,j+1) Theappearanceofnon-zeronetcurrentsisacharacteristicsof
downEid,j = E(i,j)→(i,j−1) jumps, such thatEir,j ≡ Eil+1,j systemswith locallybrokensymmetry,andtheirmagnitudes
and Eu ≡ Ed . The barrier heights are chosen to be aregivenbyhigherthanlineartermsintheresponsefunctions
i,j i,j+1
random and quenched, with the spatial distribution given by (i.e.,byF2andhigherorderterms)[9].
Eiγ,j = E0 +Uηiγ,j, where γ stands for r, l, u, and d; E0 is WesolvedthesystemofEqs.(1)-(3)numericallybyusing
a constant; and ηr ≡ ηl and ηu ≡ ηd are uncor- the conjugate gradient method, modified for sparse matrices
i,j i+1,j i,j i,j+1
relatedrandomnumbersuniformlydistributedbetween0and [10]. InFigs.2(a)-(d),weshowthesteadystatelocalnetcur-
1. Throughoutthepaperwe measuretheenergiesinunitsof rent patternsobtained for a system with linear size L = 50,
k T, where k stands for the Boltzmann constant, and T is randomnessparameterU =0.5,andexternalfieldamplitudes
B B
theabsolutetemperature. F = 0.01, 0.05, 0.1, and 0.9. These figures offer a visual
Inadditiontothethermallyactivateddiffusion,theparticle prooffortheemergenceofhighlynon-trivialsteady-statenet
isalsodrivenbyanexternalacforce,2φ(t),appliedalongthe current fields, characterized by long-range correlations and
x-direction(seeFig.1). Sincetheenergybarriersareplaced large-scalevorticity. Thevorticitystructuresaremostappar-
halfway betweenadjacentsites, the ac force modulatestheir ent for small external field amplitudes (F ≪ U). As F in-
hight by −φ(t) in the positive x-direction and by +φ(t) in creases,however,thecorrelationsgraduallydisappearandthe
thenegativeone.Thus,forlargeenoughbarriers,thehopping netcurrentfieldconvergestoanearlyrandomstructure.
ratesoftheparticleoccupyingsite(i,j)canbewrittenas ItmustbeemphasizedthatnumericalsolutionofEqs.(1)-
(3)forarbitrarilysmallvaluesofF and/orforarbitrarilylarge
kr =νexp{−[Uηr −φ(t)]},
i,j i,j system sizes is notpossible. Thisis due to the slow conver-
kl =νexp{−[Uηl +φ(t)]}, gence of the conjugate gradient method in these limits and,
i,j i,j
ku =νexp{−Uηu }, most importantly,due to the growingimpactof the random-
i,j i,j ization,inducedbythenumericalerrors[9].
kid,j=νexp{−Uηid,j}, (1) To understand the nature of the observed correlationsand
investigatethe scaling propertiesof the system, we compute
where ν = ν exp{−E }, and the attempt frequency ν is
0 0 0 the ensemble-averaged current-current correlation function,
alsoaconstant.
definedintherealspaceas
Next, we turn to an ensemble description, in which the
probabilityofthe particleoccupyingsite (i,j)is denotedby
1 2
Pdei,sjc,riabneddtbhyetthimeefoelvloowluitnigonseotfotfhmisapsrtoerbaebqiuliattyiodniss:tributionis C(r)=* (i,j)J~i2,jNr Xi,j Xi′,j′(cid:16)J~(ri,j)−J~(ri′,j′)(cid:17) +.
(5)
∂P P
i,j =−Jx +Jx −Jy +Jy , (2) Here summation over all the lattice site indexes (i,j) is im-
∂t i,j i−1,j i,j i,j−1 plied,whilesummationover(i′,j′)indexesisperformedonly
where forlatticesitepairssuchthat|ri,j −ri′,j′|= r,Nr beingthe
numberof such pairs. The averaging < ... > was taken for
Jix,j=kir,jPi,j −kil+1,jPi+1,j, variousdifferentrealizationsofthedisorder{ηiγ,j}. Thecor-
Jy =ku P −kd P (3) relationfunctiondefinedthiswayisboundedfrombelowby0
i,j i,j i,j i,j+1 i,j+1
(incaseofperfectcorrelation),fromaboveby4(perfectanti-
are the x and y componentsof the local probabilitycurrents correlation),andtakesthevalueof2forvanishingcorrelation.
J~ ,asdefinedabove. In Fig. 3, we show the normalized correlation function,
i,j
Forsimplicity,inthefollowingwerestrictourselvestothe C(r)/C(1), computed for different system sizes, and fixed
consideration of symmetric square wave external fields, i.e, external field amplitude, F = 0.01. As Fig. 3 demonstrates,
when the field φ(t) alternates between +F and −F at con- the correlationfunctionfollowscloselya logarithmicdepen-
stanttimeintervalsτ . Wealsoassumethatτ ismuchlarger denceonr, forsmallradialdistances,anddeviatesfromthat
F F
than the relaxation time of the entire system, estimated as asrgrowsbeyondacertainvalue,withasaturationthreshold
3
determinedbythesystemsize,L. Thisallowsustointroduce
acorrelationlengthξinthesystem,whichwedefineasachar-
acteristiclengthatwhichthecorrelationvanishes: C(r) = 2,
thatis,1−0.5C(r)∼<J~(ri,j)J~(ri′,j′)>turnszero.
Onthebasisoftheaboveobservations,weconcludethat,in
themostgeneralform,thecurrent-currentcorrelationfunction
isdescribedwellbytherelation
r
C(r)∼log(r)f , (6)
ξ
(cid:18) (cid:19)
where f(x ≤ 1) ∼ const. Note that the correlation length
in Eq. (6) is dependent on both the amplitude of the exter-
nal field and the system size; i.e., ξ = ξ(F,L). Moreover,
as our results demonstrate, the correlation length, ξ, scales
withbothofthesequantitiesfollowingpowerlaws. Thus,for
small amplitudes of the external field, the correlation length
dependsstronglyonthesystemsize,approximatelyfollowing
ξ(L) ∼ Lα relation,wheretheexponentα = 0.983±0.008
(i.e.,ξ growsnearlylinearlywithL). Ontheotherhand,for
intermediateexternalfieldamplitudes,ξisfoundtobeweakly
dependent on L and to monotonically decrease with F, fol-
lowing closely the inverse power law, ξ ∼ F−β, where the
exponentisβ ≃0.78±0.01. Thetransitionbetweenthetwo
regimesoccursatacertainfieldamplitude,Fc(L),dependent
1
on L. The abovescaling lawscan be collapsedinto a single
scalingrelation:
ξ ∼F−βg(FLα/β), (7)
where g(x) ∼ x for F ≤ Fc(L) and g(x) ∼ const for
1
Fc(L)≤F ≤Fc(L). ThesignificanceofFc(L)willbedis-
1 2 2
cussed later. In Fig. 4 we show the numericaldata collapse,
performedaccordingto Eq. (7). As one can observe, it pro-
vides satisfactory results for small and intermediate FLα/β.
In general, three regimes of the scaling behavior can be dis-
FIG.2: Snapshotsofthesteady-statenetcurrentfieldsobtainedfor
modelsystemswithL = 50, U = 0.5, anddifferentexternalfield
FIG.3: Normal-logplotofthecurrent-currentcorrelationfunction
amplitudes:(a)F=0.01;(b)F=0.05;(c)F=0.10;(d)F=0.90.
C(r)/C(1), computed for U = 0.5 and F = 0.01 for different
systemsizes.
4
current fields are observed to be largely independent of the
particular realization of the disorder. The correlation length
has been found to scale nearly linearly with the system size
and display an inverse power law behaviorwith the external
fieldamplitude.
Anintuitiveexplanationfortheappearanceofsystem-wide
structures is that in 2D the particle has the opportunity to
“choose”globallydifferentpathswhendriveninthetwoop-
posingdirections. Thesuperpositionofthesegloballydiffer-
entflowfieldsresultsinlarge-scaleflowpatterns.Thispicture
isalso consistentwith aslow (logarithmic)decayofthecor-
relationsandacorrelationlengthbeingintheorderofthesys-
tem size. It is importantto note that the abovephenomenon
cannotoccurin1D,wherealltheenergybarriersareinseries,
andnoalternativetrajectoriesarepossible.
ResearchatNotreDamewassupportedbyNSF-PHYS.
FIG.4: Scalingplotofthecorrelationlengthaccording toEq.(7)
fordifferentsystemsizes.
criminated. For small F, the behavioris defined by the sys-
∗ Electronicaddress:[email protected]
tem size scaling and extendsover the region of F values up
† Electronicaddress:[email protected]
toFc(L). Inthesecondregime,thescalingisdominatedby
1 ‡ Electronicaddress:[email protected]
the externalfield amplitude. This regime correspondsto the
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(7)areobservedforlargevaluesofF (i.e.,forF ≥ Fc(L)). [3] J.-P.BouchaudandA.Georges,Phys.Rep.195,127(1990).
2
Thisbehaviorisanartifactofthediscretenatureofthemodel [4] T.NattermanandJ.Villain,PhaseTransit.11,5(1988).
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[6] J. Nittmann, G. Daccord, and H. E. Stanley, Nature 314, 141
oflargeF,thecorrelationlengthdoesnotfollowEq.(7)down
(1985).
toarbitrarysmalldistances,butrathersaturatesatξ ≃1. [7] L. F. Richardson, Proc. Roy. Soc. (London) Ser. A 110, 709
Insummary,wehavestudiedtheBrownianmotionofasin- (1926).
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following a logarithmic dependence. The properties of the 1992).
