uniform line fillings evangelos marakis资料.pdf

该【uniform line fillings evangelos marakis资料 】是由【宝钗文档】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【uniform line fillings evangelos marakis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。PHYSICALREVIEWE99,043309(2019)Uniformline?llingsEvangelosMarakis,*,,RavitejUppu,?(COPS),MESA+InstituteforNanotechnology,UniversityofTwente,,7500AEEnschede,herlands(Received5December2018;revisedmanuscriptreceived4March2019;published25April2019)Deterministicfabricationofrandommetamaterialsrequires?llingofaspacewithrandomlyorientedandrandomlypositionedchordswithanon-averagehomogenousdensityandorientation,?llingswithsuchchords,linesthatrunfromedgetoedgeofthespace,-averagehomogeneousandrotationallyinvariant?llingsofcircles,balls,andarbitrary-’sparadoxandJaynes’?llings,mappingoutthedensitypro?leandtheline-,?-laser-writingfabricationdesignofopticalmultiple-scatteringsamplesasthree-:.?llathree-dimensionalvolumeofuptomm3[13].Inthisestablished[1,2].Whilerandommultiple-scatteringmediapaper,weaddresstheproblemofuniform?llingofaspaceareabundantinnature,theirmicroscopicstructureoriginateswithchordswithminimallong--Theproblemof?llingspacewithrandomlyorientedscatteringmediawithdesignermicroscopicfeaturesaswellchordsismoreinvolvedthanitseemsat?[14],who??llingsinacircle:rystals[3],quasicrystalsanddeterministicSupposeone?llsacirclewithrandomlinescrossingthecircleaperiodicstructures[4],andhyperuniformmedia[5].An(suchlinesarealsocalled“chords”)andgiventhelargestemergingapplicationofrandommultiple-scatteringmediaisenclosedequilateraltriangle,whatistheprobabilityPthatatheiruseasopticalphysicalunclonablekeysincryptographyrandomchordislongerthanthesideofthetriangle?Perhaps[6–8].Thesecurityofthesecryptographyprotocolsreliessurprisingly,-?-,whichresultinthreedifferentursinthickisotropicmultiple-scatteringmedialack-probabilitiesP,concludingthatspeci?edas“random?lling”inganysymmetryorlong-rangeorder[9],whichmakethemonly,theproblemisillde?-averageuniformline?llingsofaboundedtocreatethebestpossiblereplicasofisotropicmultiple-domaininann-dimensionalspace(n2).-expressionsforthechordlengthdistributionsandvalidateicalscattererseasethedesignofisotropicscatteringmedia,conventionalmethodsinhibittheprecisepositioningofthethechordgenerationmethodusingMonteCarlosimulationsscatterersinathree-dimensionalvolume[10].Signi?,weanalyzethesizeadvancesindirectlaserwriting(DLW)methodshaveover-distributionofvoidsformedinuniformline?llings,whichcomethislimitationandenableprecisepositioning(uptocouldprovideinsightintothestructuralcorrelations,?ndingfewnanometers)ofsubmicron-sizedfeatures[11,12].-scatteringmediathatcanbeimmediatelyimplementedusing*e.******@?Presentaddress:works(Hy-Q),forotherproblemswhererandombutstraightpathwaysareNielsBohrInstitute,UniversityofCopenhagen,Blegdamsvej17,found,suchas,.,dosimetryorsamplingofnonstationaryDK-2100Copenhagen,-0045/2019/99(4)/043309(9)043309-1?,043309(2019)-(1)(2,2*),rischosenfromauniformdistribution[0,1],whileformethod2*,r2ischosenuniformlyfrom[0,1].In(3),bothmethods1and2*giveanon-’sparadox:Thetoprowillustratesthreewaysof?llingacirclewithrandomlines:(1)connecttworandompointsonthecircumferenceofthecircle,(2)choosearandomradiusarerandomlyinterceptedbystraightlinessuchasinapplica-andanglefromthecenterandthendrawthelineperpendiculartionsinacoustics,microscopy,textureanalysis,anddosimetrytothat,and(3)choosearandompointinsidethecircleand?nd[20,21].Inthesestudiestheon-averagehomogeneous?“μ-randomness,”incontrasttootherrandomordingtothemethoddistributionssuchas“ν-randomness”(?ndingachordlongerthanarandompointinasphere),“λ-randomness”(straightlinestheedgelengthoftheenclosedequilateraltriangleisdifferentforthroughtworandompointsinasphere),and“i-randomness”“radiusmethod”(rayoriginatinginrandompoint)[20].(middlecolumn)resultsinensemble-averagedrandomline?llingBertrandandJaynesdiscussedline?llingsofcircles,butthatistranslationallyinvariant,.,??,’?rstmethod,twopointsarechosenatran-esnontrivialanddependentonthegeometrydomonthecircumferenceofthecircleandachordiscon-oftheobject[22,23].=-?,thebuton-(3D),“radiusmethod,”?llingrandanangle0φ,in3Dthismethod1producesanon-averageuniform=1theradiusofthecircle,,arandommethod,arandompointinsidethecircleischosenandthedirectionaldistancer<’sbasethislinesegmentischosentouniformly?=1,[01],thenthemethodBertrand’sparadoxisthatthereisnotonerandom?llingdoesnotleadtoahomogeneously?,however,forordingtotheprobabilitya“random?lling.”JaynesarguedthatthisisactuallynotdensityfunctionfR(r)=2r,thenweobtainanon-averagenecessarybecauseoftheprincipleofmaximumignoranceuniform?*.Method3in3D[15],dictatingauniformrandom?lling:Ifnothinghasbeenselectsarandompointintheball,?ndsthediskofthoselinesspeci?edonthespeci?clocationoftheobjectinspace,thenthathavethispointasmidpointandrandomlypicksalineonehastoassumethatitdoesnotmatter,,whereasin2Donlyonemethodthecaseforanon-averagehomogeneous?(2)producesauniformdistribution,in3Dwe?ndtherearebementionedthatthereisdisputeintheliteratureonhowtwomethods,1and2*.In3DwealsofoundanadditionalcompleteJayne’ssolutionis[16–18].eneratingrandomline?llings[24].attempttoresolvethisdisputebutmerelyremarkthatJayne’sTocheckthethreedifferentmethodsbyBertrandforuni-solutionseemstobethemostrelevantforphysicalproblemsformityinarigorousway,weuseaMonteCarlomethodim-suchasours[19].Thisissupportedbymore-recentliteratureplementedinMatlabtogeneratelinesandtesttheconvergenceonchord-(3D),thecircle043309-2UNIFORMLINEFILLINGSPHYSICALREVIEWE99,043309(2019)(a)2Dand(b),Bertrandsmethod2showsconvergence,whilebothmethods1and2*showconvergencefor3D.(sphere)isdividedinto100shellsofequalarea(volume).Thelinedensityinthatshellisthende?,dividedbytheshell’,andthesolidredlineistheanalyticresult√Inorderto?llother2Dshapeswithlines,=,wewillproducealargenumberof√,butneverlesswecutoutasquarethatis?lledwithrandomlinesformingthan,:??llingofthesquareofwhichasamplecanbeseenintheinsertwithθ≈π/4,3π/-horizontalornear-verticallines(≈1),(a)is??at,representingthefactthatinahomogeneously?-averagedhomogeneousdistribution,?rsttest,wechecktheaveragetheAppendixandhasasingularityatl=1,(a).θ=π/πbythemanypossiblelinesthatrunalmostparalleltooneofWeseethatanglescloseto0,2,(horizontalorthesidesandthereforehavealengthofslightlymore,butvertical)arelesslikely,whereaslinesorientedalongtheθ=π/,π/neverlessthan,,togetherdiagonaldirections(434)-,exactlyhalfofthelinesingparallelareshorter(orlonger),thenweobtaintheangulardistribution√(b).Sincetheaveragelinelength(≈12)forchords2Usingmethod2*tocreateauniformline?llingofacube,:Thedistributionhasasingularityatl=,thestraightlinefor0<l<1isnot?atasinthecaseofthe2Dsquarebutinclined:Shorterlengths√√maximumpossiblelengthisnow3,andthereisakink√at2expressing√thefactthatlineswithale