Table Of ContentPRAMANA (cid:13)c IndianAcademyofSciences
—journalof
physics
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0 Thermalconductivityofnonlinearwavesindisorderedchains
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a SERGEJ FLACHa,∗, MIKHAILIVANCHENKOa,b,†, NIANBEILIa,‡
J aMax-Planck-Institutfu¨rPhysikkomplexerSysteme,No¨thnitzerStr.38,01187Dresden,Germany
4 bTheoryofOscillationsDepartment,UniversityofNizhniyNovgorod,Russia
2
Abstract. Wepresentcomputationaldataonthethermalconductivityofnonlinearwavesindis-
]
h orderedchains. DisorderinducesAndersonlocalizationforlinearwavesandresultsinavanishing
c conductivity. Cubic nonlinearity restores normal conductivity, but with a strongly temperature-
e dependentconductivityκ(T). Wefindindicationsforanasymptoticlow-temperatureκ∼T4and
m intermediatetemperature κ ∼ T2 laws. Thesefindings are inaccord withtheoretical studies of
- wavepacketspreading,wherearegimeofstrongchaosisfoundtobeintermediate,followedbyan
t
a asymptoticregimeofweakchaos(EPL91(2010)30001).
t
s
Keywords. disorder,nonlinearwaves,thermalconductivity
.
t
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m PACSNos. 05.45.-a,05.60.-k,05.60.Cd,63.20.Pw
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1. Introduction
o
c
[ Intheabsenceofnonlinearity(ormany-bodyinteractionsinquantumsystems)alleigen-
states-normalmodes(NM)-inone-dimensionalrandomlatticeswithdisorderarespa-
1
tiallylocalized. ThisisAndersonlocalization[1],whichhasbeendiscoveredfiftyyears
v
0 ago in disordered crystals as a localization of electronic wavefunctions. It can be in-
3 terpretedas an interference effectbetween multiple scattering eventsof the electron by
5 randomdefectsofthepotential. RecentexperimentsontheobservationofAndersonlo-
4
calizationwereperformedwithlightpropagationinspatiallyrandomopticalmedia[2,3],
.
1 with noninteracting Bose-Einstein condensates expanding in random optical potentials
0
[4,5],andwithwavelocalizationinamicrowavecavityfilledwithrandomlydistributed
1
scatterers[6].
1
: Inmanysituationsnonlineartermsinthewaveequations(respectively,manybodyin-
v
teractiontermsinquantumsystems)havetobeincluded. Thus,afundamentalquestion
i
X which has attracted the attention of many researchersis: what happensto an initial ex-
r citation of arbitrary shape in a nonlinear disordered lattice. Nonlinearity renormalizes
a
excitationfrequencies,and inducesinteractionsbetweenNMs. Numericalstudiesshow
thatwavepacketsspreadsubdiffusivelyandAndersonlocalizationisdestroyed[7–10].In
theregimeofstrongnonlinearity,farfromwhereitcanbetreatedperturbatively,newlo-
calizationeffectsofselftrappingoccur[11]. Atheoreticalexplanationofthesubdiffusive
∗[email protected]
†[email protected]
‡[email protected]
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SergejFlach,et. al.
spreadingwasofferedinRefs. [7,9,12]. Itisbasedonthefactthattheconsideredmod-
elsareingeneralnonintegrable.Thereforedeterministicchaoswillleadtoanincoherent
spreading. Estimatesoftheexcitationtransferrateacrossthepackettailareobtainedby
calculatingprobabilitiesofmode-moderesonancesinsidethepacket.Somepredictionsof
thisapproachincludetheeffectofdifferentdegreesofnonlinearityandweresuccessfully
tested in [13]. First experimentaldata were recently presented for repulsively interact-
ingBose-Einsteincondensatesinquasiperiodicpotentials[14,15]which,amongothers,
confirmthebasictheoreticalfindings.
While the above wave packets are spreading into an ’empty’ system, another set of
relatedproblemsconcernsthethermalconductivityatfinitetemperatureT (i.e. atfinite
energy density ε ∼ T) [16, 17]. In the absence of nonlinearity Anderson localization
of NMs leadsto a vanishingconductivity. Itwas observedthat evena small amountof
anharmonicityleadstoadiffusivetransportofenergy,andafinitethermalconductivityκ
[18]. Somenumericalstudiesindicatedthatforlowtemperaturesκ ∼ T1/2 [18]which
would indicate a singularity at zero temperatures. Other expectations claim that heat
conductivityvanishesstrictlyforweakenough(butstillfinite)anharmonicity[17].
In this report we use the theoretical results on wave packet spreading and derive a
connectionbetweenκ and T. We predictthatforlow temperaturesκ ∼ T4 whichcor-
respondsto the regimeofweak chaos. Athighertemperaturesan optionalintermediate
regimeofstrong chaosmay lead to κ ∼ T2. We performnumericalsimulationswhich
indicatethatourpredictionsareplausible.
2. Models
WestudytwodifferentHamiltonianmodels. Thefirstmodeldescribesinteractinganhar-
monicoscillatorsinaquarticKlein-Gordon(KG)chain,givenas
p2 ǫ˜ 1 1
H = l + lu2+ u4+ (u −u )2 (1)
K X 2 2 l 4 l 2W l+1 l
l
whereu andp arerespectivelythegeneralizedcoordinate/momentumonthesitel,and
l l
ǫ˜ arethedisorderedpotentialstrengthschosenuniformlyintheinterval[1/2,3/2].Like-
l
wise,∂2u =−∂H /∂u generatestheequationsofmotion
t l K l
1
u¨l =−ǫ˜lul−u3l + W(ul+1+ul−1−2ul). (2)
By neglecting the nonlinear terms, the KG model (1) reduces to a linear eigenvalue
problem.ThisleadstoasetofNMamplitudes,A ,withNMsquaredfrequenciesω2 ∈
ν,l ν
[1/2,3/2+4/W] and ∆ = 1+4/W being the width of the squared eigenfrequency
spectrum. The NMasymptoticspatialdecayis givenbyAν,l ∼ e−l/ξ(ων2) whereξ(ων2)
isthelocalizationlength. Itisapproximated[19]inthelimitofweakdisorder(W ≪ 1)
asξ(ω2) ≤ 96W−2. TheNM participationnumberP = 1/ A4 characterizesthe
ν ν Pl ν,l
NMspatialextent.AnaveragemeasureofthisextentisthelocalizationvolumeV,which
is on the order of 3.3ξ(0) for weak disorder and unity in the limit of strong disorder
[20]. TheaveragesquaredfrequencyspacingofNMswithinalocalizationvolumeisthen
d ≈ ∆/V. Thetwosquaredfrequencyscalesdand∆withd < ∆arethusexpectedto
determinethedynamicsdetailsinthepresenceofnonlinearity.
NonlinearityinducesaninteractionbetweenNMs. SinceallNMsareexponentiallylo-
calizedinspace,eachofthemiseffectivelycoupledtoafinitenumberofneighbormodes,
2
Thermalconductivityofnonlinearwavesindisorderedchains
i.e. theinteractionrangeisfinite. However,thestrengthofthiscouplingisproportional
toacharacteristicenergydensity,ε. Thesquaredfrequencyshiftduetothenonlinearity
isthenδ ∼ε.
TheKGmodelissuitableforstudiesofthedynamicsforweakandintermediatedisor-
derW < 20,sinceotherwisethecouplingstrengthbetweennearestneighboroscillators
becomestooweakandrequiredcomputationtimestoolong. Inthelimitofstrongdisor-
dertheNMsbecomelocalizedonasinglesite,andwillbecoupledbynonlinearitytotheir
nearestneighbors.Thereforethislimitcanbemodeledbyachainofharmonicoscillators
withrandomfrequenciesandpurelyanharmonicnearestneighborcoupling:
p2 ˜ǫ 1
H = l + lu2+ (u −u )4 (3)
FSW X 2 2 l 4 l+1 l
l
whichisanexamplefromaclassofmodelsstudiedbyFro¨hlich,SpencerandWayne[21]
and therefore called here the FSW model. It follows qualitatively from the KG model
byfirsttransformingintoNMspace,droppingsecularterms[12],andfinallythroughthe
limits of strong disorder, and of high temperatures (energies). Its quartic potential part
howeverisinvariantunderhomogeneouscoordinateshifts,contrarytotheKGcase.
3. Thermalconductivity
Thethermalconductivityismeasuredbyastandardapproach,whenthethermalbathsat-
tachedtotheendsgenerateatemperaturegradientandheatcurrentalongthechain[16].
FortheKGmodelweimplementLangevinthermostatswithtwonext-to-endatomscou-
pledtoLangevinheatbathswithu¨ =−∂H/∂u +ξ −λu˙ withwhitenoise
1,N 1,N 1,N 1,N
hξ i = 0 and hξ (t)ξ (0)i = 2λT δ(t) where h·i denotesensemble averages.
1,N 1,N 1,N 1,N
T isthetemperature,andλdenotesthecouplingstrengthbetweensystemandheatbath.
FortheFSW modelweuse Nose´-Hooverthermostatsaddingtheterms−ζ±u˙1,N to the
respectiveequationsofmotion,whereζ˙± =u˙21,N/T1,N −1.
FortheKGmodeltheheatfluxalongthechainisdefinedasthetimeaverageofj =
KG
− 1 (u˙ +u˙ )(u −u )[16]. FortheFSWmodeltheheatfluxfollowsfromthe
2W P l+1 l l+1 l
l
timeaverageofj =−1 (u˙ +u˙ )(u −u )3 [16]. Thethermalconductivity
FSW 2P l+1 l l+1 l
l
coefficientreadsκ = jN/(T −T )then[16]. We makeuseof themeantemperature
N 1
T = (T + T )/2 as a parameter corresponding to the energy density hεi = T, and
N 1
(T −T )/T =0.5furtheron.
N 1
4. Predictions
Thenumericalevolutionofawavepacketspreadingintoanemptysystem[7–10]yields
subdiffusivegrowthofthesecondmoment.Thisimpliesthatthediffusionrateisdepend-
ingontheenergydensity(temperature). Atheoreticalapproach[7,9,12]postulatesthat
thewavepacketevolveschaotically,andestimatesthediffusionratefortheKGmodelto
be[12]
D ∼ε2P2(ε) (4)
withtheresonanceprobability[20]
P(ε)≈1−e−aε/d (5)
3
SergejFlach,et. al.
1
0.1
T=0.005
T=0.01
T=0.02
0.01 T=0.05
T=0.1
T=0.2
100 1000
N
Figure1. KGchain: thesize-dependentthermalconductivityκN versusN fordiffer-
enttemperaturesandW =2.
andabeinganunknownconstantoforderone.
Forsmalltemperaturesε ∼ T. Therefore,andassumingthatthethermalconductivity
isproportionaltothediffusionrate[10],weconcludethatfortheKGmodel
2
κ∼T2 1−e−bT/d (6)
(cid:16) (cid:17)
withbanotherconstantoforderone. Inparticularwefindthat
κ∼T4 , T ≪d(weakchaos) (7)
κ∼T2 , T ≫d(strongchaos) (8)
Note that for larger energydensities (temperatures)the KG model enters the regime of
selftrapping, and the abovepredictionsdo not apply anymore. The selftrappingregime
isexpectedwhenε ∼ T ≈ 2/(3W)[10]. Itistheasymptotichightemperaturelimitof
theKGmodel,wherethethermalconductivityshoulddropwithincreasingtemperature
[17]. Thus,dependingonthestrengthofdisorder,weexpectthatthethermalconductivity
growswithtemperatureasT4,then(fornottoostrongdisorder)itgrowsasT2,andfinally
goesthroughamaximumandstartstodecreasewithfurtherincreaseoftemperature.The
threephasesaresummarizedintheinsetofFig.3inRef. [10].
The application of the above approach to the FSW model leads to d ∼ 1 and to the
absence of a selftrapped regime. Instead, for large temperatures, the leading order an-
harmonicpotentialtermsoftheFSWmodelbringitclosetothemomentumconserving
β-Fermi-Pasta-Ulamchain (FPU). The FPU chainhas a thermalconductivitywhich di-
vergeswithincreasingsystemsize[16]. Thereforeκwillincreasewithsystemsizealso
for the FSW model for large temperatures, until the harmonic oscillator potential parts
can not be neglected anymore, leading to a final saturation of the thermal conductivity
4
Thermalconductivityofnonlinearwavesindisorderedchains
withfurthersystemsize increase. Indeed,the harmonicoscillatorpotentialpartsarevi-
olatingtotalmomentumconservation,andthereforethethermalconductivitystaysfinite
forlargeenoughsystemsize[10,17].SincethemomentumconservingtermsoftheFSW
modelbecomemoredominantatlargertemperatures,itfollowsthatthehigherthetem-
perature,thelargerthesystemsizesareneeded,andthelargerthesaturatedconductivity
value.Thecrossovertothispseudo-FPUregimecanbeexpectedwhenthetemperatureis
oforderunity(sinceallrelevantFSWmodelparametersareoforderunity).Thereforewe
conclude,thatwewillnotseeapronouncedregimeofstrongchaos,insteadtheregimeof
weakchaosatlowtemperaturesshouldgraduallytransformintothepseudo-FPUregime.
5. Numericalresults
The thermal conductivity is computed along the above lines, with additional averaging
overdisorderrealizations.
5.1 KGchain
We first present results for the disorder strength W = 2. In that case the localization
length ξ ≈ 25 and the localization volume V ≈ 75. The thermal conductivity κ is
N
plotted for various temperaturesversus system size N in Fig.1. For large temperatures
T ≥0.1theconductivityreachesitssaturatedlevelκatsystemsizeN ≈ξ. Forthelinear
waveequation,exponentiallylocalizedNMsinducea dependenceκ ∼ e−N/ξ. In
linear
thepresenceofnonlinearitytheexponentialdecaywithN willstopatthesaturatedvalue
of κ. Assuming that κ ∼ Tα it follows that the system size for saturated conductivity
valuesscalesasN ∼ −αξlnT. Thusforlowertemperaturesthesystemsize neededto
reachsaturatedvaluesofκ increases. Thisisduetothedroppingofthesaturatedvalue
ofκwithT. WewereabletoreachtemperaturesaslowasT = 0.005andconductivity
valuesκ = 0.008. AccordingtoRef. [10](insetinFig.3)theregimeofstrongchaosis
realizedfor0.01<T <0.3. Thereforeweexpecttoobserveκ ∼T2for0.01<T <1,
followed by a maximum and a further dropping with increasing temperatures. These
predictionsareindeedcompatiblewiththenumericalresultsinFig.2. We alsocompute
theconductivityforareferenceorderedchainwithǫ˜ =1,whichcoincideswiththehigh
l
temperature part of κ(T) for the disordered case. Therefore the observed maximum is
due to disorder, as well as the decrease of κ with further lowering of the temperature.
ThelowtemperaturedataarealsoconsistentwiththeT2 law. Howeverthetemperature
windowistoonarrowtoconcludeaboutaconfirmation.Alsoweexpectthatatevenlower
temperatures(sofarnotreachablewithourcomputationalmeans)theconductivityshould
crossoverintotheweakchaosregimeandfollowtheT4law.
Inordertofindevidencefortheasymptoticlow-temperaturebehaviorκ∼T4wechose
astrongerdisorderW = 6. Inthatcaseξ ≈ 3andV ≈ 10. Theconductivityisreduced
bytwo ordersofmagnitude. We are thereforequitelimited in the temperaturewindow,
and do not obtain conclusive data (see Fig.2). We again find a maximum in the κ(T)
dependence, and a clear indication that the small temperature conductivity drops faster
thanT2 withtemperature. Yetwedonothaveenoughlowtemperaturedatatoconclude
abouttheexponent.
5.2 FSWchain
Toobservetheasymptoticlowtemperaturebehavioroftheconductivity,weexplorethe
strongdisorderlimit and study the FSW chain. Since the localizationlength ξ = 0 for
5
SergejFlach,et. al.
110000
κ ~ T 2
11
κ
00..0011
00..00000011
00..000011 00..0011 00..11 11 1100
T
Figure2. KGchain: κ(T)forW = 2(filledsquares). Forcomparisonwealsoshow
the data for ǫ˜l ≡ 1 (filledcircles). Thin solid lines guide the eye. The dashed line
correspondstothepowerlawT2. ThestrongerdisordercaseW = 6correspondsto
theopendiamonddatapoints.
theFSWchain,wecanusesmallsystemsizesforsmalltemperatures,while(asdiscussed
above) we will need larger system sizes to obtain the saturated conductivity values for
largertemperatures. InFig.3weplottheconductivityκversustemperaturefordifferent
system sizes N = 16,64,256. Indeed,we observea significantdifferencebetween the
datafortemperaturesT >1,whilethedataagreewellforlowertemperatures.Therefore
wecanreachlowervaluesoftheconductivityascomparedtotheKGchaincase. Yetfor
κ<10−4(whichcorrespondstoT ≈0.01)weagainreachourcomputationallimitations,
sincethe numberofrealizationsandtheintegrationtimes, whichareneededin orderto
reduce the impact of fluctuations, increase drastically. The numerical results in Fig.3
indicatethatforlowtemperaturestheconductivitybehavesasκ ∼ T4. Alsoweobserve
atlargertemperaturesa crossoverto the predictedpseudo-FPUregime. Againwe need
more data in the low temperature regime to be more conclusive about the conductivity
data.
6. Discussionandoutlook
We computedthe thermalconductivityfornonlineardisorderedchainsandobtainedre-
sultswhichindicatethattheoreticalpredictionsbasedonwavepacketspreadingareplau-
sible. In particular we find indicationsfor an asymptotic low-temperatureκ ∼ T4 and
intermediate temperature κ ∼ T2 laws. To be more conclusive, we need data at even
lowertemperaturesthanthose presentedhere. The neededCPU timesare howevercur-
rentlyexceedingourcapabilities.Possiblesolutionsincludetheusageofmassiveparallel
computingequipment,orthesearchforothermodelswhichwillallowtomaketheneeded
observationswithstandardcomputationalequipment.
6
Thermalconductivityofnonlinearwavesindisorderedchains
110000
κ ~ T4
11
κκ
00..0011
00..00000011
00..0011 11 110000
T
Figure3. FSWchain: κ(T)forsystemsizesN = 16(filledcircles),N = 64(filled
squares),andN =256(filleddiamonds).ThedashedlinecorrespondstothelawT4.
Acknowledgments
MIacknowledgesthesupportoftheFederalProgram‘Scientificandscientific-diductional
brainpowerofinnovativeRussia’for2009-2013(contractsΠ2308,02.740.11.0839)and
RFBR 10-02-00865. We thank J. Bodyfelt, G. Gligoric, T. Lapteva, D. Krimer and H.
Skokosforstimulatingdiscussions.
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