exact results for first-passage-time statistics in biased quenched trap models takuma akimoto资料.pdf

该【exact results for first-passage-time statistics in biased quenched trap models takuma akimoto资料 】是由【赖大文档】上传分享，文档一共【9】页，该文档可以免费在线阅读，需要了解更多关于【exact results for first-passage-time statistics in biased quenched trap models takuma akimoto资料 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。PHYSICALREVIEWE99,052127(2019)Exactresultsfor?rst-passage-timestatisticsinbiasedquenchedtrapmodelsTakumaAkimoto1,*andKeijiSaito21DepartmentofPhysics,TokyoUniversityofScience,Noda,Chiba278-8510,Japan2DepartmentofPhysics,KeioUniversity,Yokohama223-8522,Japan(Received7January2019;published20May2019)Weprovideexactresultsforthemeanandvarianceof?rst-passagetimes(FPTs)-Llimit,weobtainexactformulasfortheFPTstatistics,,we?ndthattheseformulasarestillusefulfornonperiodicheterogeneousenvironments;.,?:.,andhencetheFPTstatisticswiththeCTRWhavebeeninten-Encounteringareactivemoleculeor?ndingareactivesitesivelystudied[18–21].However,in1Dsystems,thediffu-byamoleculeisthe?,sioninquenchedpotentialscannotbeapproximatedwiththe?ndingaspeci?ctargetinstochasticprocessesisafundamen-CTRW,[1].Inparticular,thistarget-searchproblemattractsdiffusion[14,22].signi?cantinterestinbiomolecularreactionsincellssuchasHence,?llingthelacunaontheFPTstatisticsin1Dtranscriptionfactorssearchingforaspeci?cDNAsequencesystemsiscriticaltoupdatefundamentalunderstandingof[2–4].?cientlyreachthetargetsincellsInthispaper,weclarifyseveralpropertiesoftheFPT[5–8],binationofthree-dimensional(3D)freestatisticsinherenttothequenchedpotentialbylookingatdiffusionandone-dimensional(1D)?rstconsideraperiodicrandomplaysavitalroleinreducingthe?rst-passagetime(FPT)-[9].employedtosimplifythesetupinthispaper,stochasticdy-DNAsequencesexhibitanomalous?uctuations,suchaslongnamicsintheperiodicpotentialhavebeenintensivelystudiedcorrelationsand1/f?uctuations[10].Thus,a1Dslidinganalytically[23,24]andexperimentally[25–28].Moreover,,theelerationofdiffusivityinperiodicpotentialsdiffusioncoef?cientsofarepressorproteindiffusingonDNA[23,25],?eld-inducedsuperdiffusion[29–32],anddistinctareobtainedbysingle-particle-trackingmeasurementsandinitialensembledependenceofdiffusivityindisorderedmediashowlargetrajectory-to-trajectory?uctuations[11,12].ThesesuchastheCTRW[33,34].Ouranalysisalsounravelsseveral?uctuationsareevidenceoftheheterogeneityoftheenviron-,intrinsic?uctuationsofthediffusioncoef?(QTM),whichrepresentsarandomwalk(RW)onarandomenergylandscape,,whichpresentscontinuous-timeWeconsidereffectsofbiasontheFPTstatisticsinaRW(CTRW)[13,14].,weuseTheFPTstatisticsinheterogeneousenvironmentsarekeyabiasedRWona1Dquenchedrandomenergylandscape,quantitiesintarget-searchproblems[15,16]andplayanim-whichisperiodicallyarranged,.,abiasedQTMwithaportantroleinestimatingthediffusioncoef?cient[17].Inperiodiclandscape[(a)][22].Thetargetislo-higherdimensions,=1?p,=1/2,and*******@>1/-0045/2019/99(5)/052127(9)052127-1?2019AmericanPhysicalSocietyTAKUMAAKIMOTOANDKEIJISAITOPHYSICALREVIEWE99,052127(2019)δ≡?σ2(a)QTMwithperiodiclandscape(b)QTMwithnonperiodiclandscapewhereTTTLandListhesamplevariance:biastargetbiastargetL?1qpqp2122-L0L0Lσ=τ?μ.(4)i?∞iLLiLi=0σ2SamplevarianceLquanti?<δ2～σ2+μ2/?esTLL(LL)(p2μ2=σ2q)becauseLo(L),,=qbecausethemeanreturntimetotheorigindiverges.(a)withperiodL(L=8)and(b),theresultsincludethecasep=,itjumpstotherightissmall,p?qcanbeexpressedasp?q=～F/T,,andthetargetisanexternal?,-handsidesmallFdependenciesoftheFPTstatisticsarerepresentedas(notargetat0,?L,?2L,...).Inotherwords,weconsideratime23TL∝1/FandδTL∝1/(clockwiseorcounterclockwise)-potentialsare?at;.,thetopsarethesameheight,,theydependonμLandthatprobabilitypdoesnotaffecttheshapeoftherandomσL,whichare?niteanddependoneachrealizationofthedis-,theVFPTareconnectedtothediffusivityblikestructures[35],inthebiasedQTMonaring[23].Therefore,.,porousmedia[31]andneuronaldendrites[36,37],inasigni?cantroleinestimatingdiffusivity,?-numberLoflatticesiteswithdifferentenergiesis?(L<∞).Ateachsite,depthE(>0)oftheenergytrapisdistributioninCTRWhasalreadybeenstudied[20],theex-,thedepthsareindependentplicitformsoftheFPTstatisticshaveneverbeenobtainedsoandidenticallydistributed(IID),ourresultsalsoleadtotheexactexpressionsρ=?1?/exponentialdistribution,(E)Tgexp(ETg),whereTgofthebiasedCTRW(seetheAppendices),T～,(5)whenaparticleescapesfromtheithtrapisarandomvariable,ctrwLp?qandthedistributionfollowstheexponentialdistributionwith{σ2?2+μ2}??τ/τ2L(pq)4pqmeanτ:ψ(i)(τ)=τ1ei[38].Meanwaitingtimeτfol-δT～,(6)iiictrwL(p?q)3lowstheArrheniuslaw,τi∝exp(Ei/T),-ThroughtheArrheniuslawandtheenergydistributionρ(E),<1andtheprobabilitydensityfunctionofτifollowsapowerlaw,α<2,respectively,whiletheQTMresultsare?niteforall?1?αψα(τ)∝τwithα≡T/Tg[39].-CTRWisnotgivenbybelowTg,α<1,themeanofτidiverges,inducinganomalousastraightforwardextensionobtainedfromthatofthebiasedfeaturessuchasanomalousdiffusionandaging[14,22,40]..?1L1μL=τi(1)-TO-SAMPLEFLUCTUATIONSL=i0Wepresentnumericalveri?cationsofexactresults(2)fora?xeddisorderintheQTMwithaperiodiclandscapeand(3)toseehowtheseformulasworkforeachdisorderneverdivergeswhenL<∞.,.,atimewhenaparticlestartingdisorderrealizations,wherethenumericalvaluesareplottedfromtheoriginreachesthetarget(siteL)forthe?,weshowthemeanFPT(MFPT)collapsedonthey=xline,whichshowsaperfectagree-andthevarianceoftheFPT(VFPT):thenumericalresultsareprovidedforparameterα=,wherethedisorderrealizationsshowlargesample-to-sampleLμT～L≡T,?-L?MFPT(2)pqto-sample?uctuationsbecausethedisorderstronglyaffectsLσ2(p?q)+,thetheoreticalvaluesfordifferentδT2～LL≡T,(3)LLL(p?q)-2EXACTRESULTSFORFIRST-PASSAGE-TIME…PHYSICALREVIEWE99,052127(2019)1024MFPTVFPT102210201018**********.(a)Ratioην,εasafunctionofL/Ndis(Ndis=10,ε=,Simulation10andp=)?xeddisorderrealization.(b)Correlationplot810fortheMFPTandVFPTinthein?nite1Dsystemswithnonperiodic36landscapes(α==10).(α==104).Symbolsrepre-[(b)].Fora?nitebiasandlargeL,aparticlewillex-sentresultsofnumericalsimulationsfor100differentdisorderperiencedeeppotentialsmainlyinthepositiveregimei>,theFPTmightbedominatedbywaitingtimesforregimei>,theexactresultsfortheperiodicQTMmaystillbeusefultounderstandtheFPTstatisticsinNext,wediscusssample-to-sample?uctuationsbyconsid-thein?,,whentheheterogeneityhefollowingtwoquantities:ofadisorderrealizationissuf?cientlyweak,physicalobserv-priseself-averaging(SA)properties[22].Hereweδ2quantifysample-to-sample?uctuationsbytheSAparametersTLTLTM≡andTV≡,(8)de?nedasTTMFPTVFPT22O?OLSA(L;O)≡Ldisdis,(7)whereTandTarethesameexpressionsasinEqs.(2)LO2MFPTVFPTLdisμσ2and(3),=0,...,L?1(inwhere·,wedonotusetheinformationofpotentialsforrealizationofSA,andhencetheseparameterssystematicallyi<0).Inaddition,toquantifyhowourpredictionworkswell,?nitionisweintroducethefollowingratio:analogoustothatofthediffusivityintheQTMdiscussedinRef.[41].NdisL1(j)Asμisarandomvariable,theSAparameterfortheην,ε≡I[1?ε,1+ε]Tν,ν=M,V,(9)LNNMFPT,O=T,canberewrittenasdisdisj=1μ2?μ2τ2?τ2=LdisLdis=disdis.(j)SA(L;TL)22whereTνisanumericalvalueofTνforthejthrealizationμLLτdisdisofdisorderandIA(x)isanindicatorfunction,.,IA(x)=1α<τ2ifx∈AandIA(x)=?esaratiothatFor2,theSAparameterisin?nitebecausedisdi-(j)verges,whileforα>2,itvanishesinthelarge-Llimit,Tνiswithinthecorrespondingtheorywithaε-dependentimplyingthattheSAissatis?edforα>2,(a),theratioapproaches1withforα<,thetransitionbetweenSAandnon-=T,thetransition(j)2νfromSAtonon-SAforquantityδTLcanbediscussedrealization,(b)wepresent(j)throughEq.(7).Fromasimilarcalculation,thecriticalvaluenumericaldataofTνasafunctionofTMFPTorTVFPTde-ν=canbeeasilyobtainedasαc=4;thatis,theVFPThasanSApendingonMorV,(b)showsthatpropertyforα>4,whileitisbrokenforα<δTLforalmostallrealizations,