Contents - Max-Planck-Institut für Physik komplexer Systeme
Contents - Max-Planck-Institut für Physik komplexer Systeme
Contents - Max-Planck-Institut für Physik komplexer Systeme
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WhennonlineartermsareaddedtotheHamiltonian,theAndersonmodesbecomecoupledbyhigher<br />
orderterms.ThisnonlinearcouplingallowsinprincipleenergytransferbetweentheAndersonmodesand<br />
itisthusnontrivialtopredictwhetherthereisenergydiffusionthroughtheAndersonmodesornot.<br />
Manyearlyworkssuggestedonthebasisofnumericalsimulationsinsuchsystems,thatanyinitialwavepacketwithfiniteenergyshouldspreadtozeroamplitude.<br />
Thisspreadingwascharacterizedbythe<br />
divergencyasafunctionoftime,ofthesecondmomentoftheenergydistributionofthewavepacket.It<br />
wasfoundthatthissecondmomentwasdivergingasapowerlawoftime,slowerthanstandarddiffusion<br />
(sub-diffusion).Thisnumericalobservationisexplainedwiththeassumptionthatmosttrajectoriesinan<br />
infinitenonlinearsystemarechaotic.Assumingthischaoticbehaviorisbroadband,theAndersonmodes<br />
farapartfromthecoreofthewavepacketshouldbeexcitedbythechaoticmotionoftheoscillatorof<br />
thecore.Thiseffectshouldmaintainenergyspreadingatalltimeifoneassumethatthechaosremains<br />
broadbandforever.<br />
OurinvestigationsduringmyGutzwillerfellowshipat mpipksin2009-2010concludethatthisexplanation<br />
forenergyspreadingisnotconsistentwithsomeexactanalyticalresults(thoughweonlyhaveincomplete<br />
answers)andalsowithnewnumericalinvestigations. Newquestionsariseandstillremainopened.<br />
Weessentiallyfocusedonthe1DrandomDNLSmodelwhichhasbeenintensivelystudiedearlierasa<br />
prototypemodelforthisproblem.Itobeysthedynamicalequation<br />
i ˙ ψn = (ǫn + χ|ψn| 2 )ψn − C(ψn+1 + ψn−1)<br />
where ψnisthecomplexcoordinateatsite n. ǫnarerandomuncorrelatedonsiteenergieswhichfor<br />
exampleareuniformlydistributedinaninterval [−V,+V ], Cisthecouplingconstantbetweennearest<br />
neighboursitesand χistheparameteroftheonsitenonlinearity. Besidetheenergy,thismodelhasa<br />
secondinvariantwhichisthetotalnormsquare <br />
n |ψn| 2 .Itisgenerallybelievedthatenergyspreading<br />
behavessimilarlyinotherrandomnonlinearmodelswithnoextrainvariantsuchasrandomnonlinear<br />
Klein-Gordonarrays.<br />
Wefoundnumerically[?,JKA10]hatafinitenormwave-packetinitiallylocalizedmaygeneratetwokinds<br />
oftrajectorieswhichbothareobtainedwithanonvanishingprobability.Thefirstkindofwave-packets<br />
consistsoftrajectorieswhicharerecurrentinthesenseofHaraldBohr. Thosetrajectoriesarealmost<br />
periodicandthuscannotspread.EmpiricalanalyticalargumentssuggestthatKAMtheorymaystillhold<br />
ininfinitesystemsundertwoconditionswhichare(1)thelinearizedspectrumispurelydiscreteand(2)<br />
theconsideredsolutionsaresquaresummablewithanormwhichisnottoolarge.Wechecknumerically<br />
thatindeedinappropriateregionsoftheparameterspace,manyinitialconditionscanbefoundwith<br />
finiteprobabilitywhichgenerate(nonspreading)infinitedimensiontori(almostperiodicsolutions)a<br />
fatCantorsetin(projected)phasespace. ManyarefoundinparameterdomainswheretheAnderson<br />
localizationlengthisrathershortbutthisdomainshrinkstozeromeasureveryfastwhenthelocalization<br />
lengthdiverges.Letusnotethatmostearlynumericalsimulationsweredoneindomainswithrelatively<br />
longlocalizationlengthwhereKAMtorisurviveonlyatverysmallamplitude. Otherwise,ourfindingis<br />
supportedbyfewrigorousresultsbutonlyonsomespecificmodelsofinfinitearraysofcouplednonlinear<br />
oscillatorswhereKAMtoridoexistwithnonvanishingprobability.<br />
Thesecondkindoftrajectorieslookinitiallychaoticandoftenspreadoverlongtimes.Wefirstrigorously<br />
provethatinitialchaosdoesnotnecessarilyimplycompletespreadingforexamplewhenthenormofthe<br />
wavepacketistoolarge.Otherwise,insomemodifiedmodels,nospreadingatallisproventobepossible<br />
despitethepresenceofinitialchaosincontradictionwithearlybeliefs. Thenatureoftheasymptotic<br />
stateisstillunknown.<br />
However,weattempttopresentempiricalargumentssuggestingthatifatrajectorystartswithchaotic<br />
spreading,therewillnecessarilyexist(generallylarge)criticalspreadingdistancesdependingonthedisorderrealizationwherethetrajectorywillnecessarilybecomestickingKAMtori.<br />
Thiseffectcouldbe<br />
viewedas”inverseArnoldDiffusion”sincethetrajectoryapproachesmoreandmoreKAMtoriinstead<br />
ofleavingthem.SinceKAMtoribecomemoreandmorerareaslocalizationlengthincreases,thiseffect<br />
couldonlymanifestafterlongerandlongertimesothatitcouldbeinvisibleattheavailabletimescale<br />
innumericalsimulation. Afteracertaintimescale,thewavepacketspreadingshouldslowdownmore<br />
andmore. Thisbehaviorshouldbeassociatedwithaself-organizationofthechaoticbehaviorofthe<br />
wavepacketbecomingnarrowband. Incases,wherecompletespreadingisimpossibleandmaybein<br />
allcaseswithinitialchaos,thelimitprofileshouldbeatrajectorywithmarginalchaos(withsingular<br />
Gutzwiller-Fellowship 109