Diffusion Effects on the Breakdown of a Linear Amp

arXiv:physics/0011010v1[physics.plasm-ph]3Nov2000DiffusioneffectsonthebreakdownofalinearamplifiermodeldrivenbythesquareofaGaussianfieldA.Asselaha,P.DaiPrab,J.L.Lebowitzc,andPh.MounaixdaLATP,UMR6632duCNRS,CentredeMath´ematiqueetInformatique,Universit´edeProvence,39rueF.Joliot-Curie,13453MarseilleCedex13,France.bDipartimentodiMatematica,PolitecnicodiMilanoPiazzaLeonardodaVinci32,I-20133Milano,Italy.cDepartmentsofMathematicsandPhysics,Rutgers,TheStateUniversityofNewJersey,Piscataway,NewJersey08854-8019.dCentredePhysiqueTh´eorique,UMR7644duCNRS,EcolePolytechnique,91128PalaiseauCedex,France.(February2,2008)AbstractWeinvestigatesolutionstotheequation∂tE−DΔE=λS2E,whereS(x,t)isaGaussianstochasticfieldwithcovarianceC(x−x′,t,t′),andx∈Rd.ItisshownthatthecouplingλcN(t)atwhichtheN-thmomenthEN(x,t)idivergesattimet,isalwayslessorequalforD0thanforD=0.EqualityholdsundersomereasonableassumptionsonCand,inthiscase,λcN(t)=Nλc(t)whereλc(t)isthevalueofλatwhichhexp[λRt0S2(0,s)ds]idiverges.TheD=0caseissolvedforaclassofS.ThedependenceofλcN(t)ondisanalyzed.Similarbehaviorisconjecturedwhendiffusionisreplacedbydiffraction,D→iD,thecaseofinterestforbackscatteringinstabilitiesinlaser-plasmainteraction.PACSnumbers:05.10.Gg,02.50.Ey,52.40.NkTypesetusingREVTEX1I.INTRODUCTIONWeinvestigatethedevelopmentofalinearamplificationinasystemdrivenbythesquareofaGaussiannoise.Thisproblemaroseandcontinuestobeofinterestinmodelingthebackscatteringofanincoherenthighintensitylaserlightbyaplasma.Thereisalargelitteratureonthistopic,andwerefertheinterestedreadertoRef.[1]forbackground.OurstartingpointhereistheworkbyRoseandDubois[2]whoinvestigatedthefollowingequationforthecomplexamplitudeE(x,z)ofthescatteredelectricfield∂zE(x,z)−iDΔE(x,z)=λ|S(x,z)|2E(x,z),z∈[0,L],x∈Λ⊂R2,andE(x,0)=E0(x).(1)InEq.(1),zandxcorrespondtotheaxialandtransversedirectionsinaplasmaoflengthLandcross-sectionaldomainΛ.Theinputatz=0,E0(x),isagivenfunctionofxandΛwillbegenerallytakentobeatorus(e.g.innumericalsolutionsofEq.(1)usingspectralmethods).Thecouplingconstantλ0isproportionaltotheaveragelaserintensityandDisaconstantparameterintroducedforconvenience.ThecomplexamplitudeofthelaserelectricfieldS(x,z)isahomogeneousGaussianstochasticfielddefinedbyhS(x,z)i=hS(x,z)S(x′,z′)i=0,hS(x,z)S(x′,z′)∗i=C(x−x′,z−z′),wherethecorrelationfunctionC(x,z)isthesolutionto∂zC(x,z)+i2ΔC(x,z)=0,z∈[0,L],x∈Λ,andC(x,0)=C(x),(2)withC(x)agivenfunctionofx[3],normalizedsothatC(0)≡h|S(x,z)|2i=1.Usingheuristicargumentsandnumericalsimulations,RoseandDuBoisfoundthattheexpectedvalueoftheenergydensityofthescatteredfieldh|E(x,L)|2idivergedforeveryL0asλincreasedtosomecriticalvalueλc(L).Theaverageh|E|2iisovertherealizationsoftheGaussianfieldS.Thisdivergenceindicatesabreakdownintheassumptionsmadein2derivingEq.(1),whichneglectsbothnonlinearsaturationandtransienttimeevolution[2,4].Physically,itcanbeinterpretedasindicatingachangeinthenatureoftheamplificationcausedbytheplasma.Toseetheoriginofthisdivergenceinitssimplestform,considerthecasewhereDissetequaltozeroinEq.(1),andneglectalldependenceofSonxandz.WearethenledtotheequationdE(z)dz=λS2E(z),(3)whichyieldsE(z)=E(0)eλS2z,z0.HereS2=S21+S22andS1,S2aretwoindependentrealGaussianrandomvariableswithzeromeanandunitvariance.ItiseasilyseenthattheprobabilitydistributionofE(z),settingE(0)=1,hasthedensityW(E,z)=(2λz)−1E−[1+(2λz)−1]forE≥1,z0.(4)IfwenowtakemomentsofEatsomevalueLofz,wefindthathEN(L)iwilldivergewhenever2NλL≥1.AtthecriticalcouplingλcN(L)=(2NL)−1,thereisaqualitativetransitionoftheamplificationofhEN(L)ifromaregimewhereitisdominatedbythebulkoftheorder-one-fluctuationsofStoaregimewhereitisdominatedbythelargefluctuationsofS.Thistoymodelcanbethoughtofasanidealizationinwhichthesizeoftheplasmaisverysmallcomparedtothecorrelationlengthofthelaserfield.ThisiscertainlynotareasonablephysicalapproximationandweshalllaterconsidersituationsinwhichSinEq.(3)isz-dependentwithacovarianceC(z,z′).Theequationisthenstillsolvablemoreorlessexplicitly,dependingontheformofC,atleastasfarasthedependenceofthedivergenceofthemomentsofEonλandLisconcerned.ThemaindifferencefromEq.(4)isthatforsmallenoughvaluesofλ,thefirstfewmomentsneednotdivergeforanyL.Inthispaper,weextendtheseresultstothex-dependentcasewhereiDinEq.(1)isreplacedbyD,i.e.weconsideradiffusiveprocessinxratherthanadiffractiveone.Some-whatsurprisinglythediffusiondoesnotsuppresstheonsetofdivergencesinmomentsofthe3field.Thissuggestsasimilarbehaviorforthediffractivecase–inaccordwiththenumericalresultsof[2]–butweareunabletoprovethisatthepresenttime.Beforegoingontotheformulationandpresentationofresultsforthediffusivecase,wemakesomeremarksabouttherelationbetweenexpectationsoverdifferentrealizationsoftheGaussiandrivingterm|S|2andtheoutcomeofagivenexperiment.AcceptingtheidealizationsinherentinassumingGaussianstatisticsandneglectofnonlinearterms,thephysicallyrelevantquestionrelatingtothesolutionofthestochasticPDE(1)appearstobethefollowing:WhatistheprobabilitythatforgivenΛandLtherewillbesmallregionsinΛthroughwhichasignificantfractionofthetotalincomingpowerisbackscattered,(here”total”means