Lyapunov instability of fluids composed of rigid d

arXiv:chem-ph/9602001v25Feb1996LyapunovinstabilityoffluidscomposedofrigiddiatomicmoleculesIstv´anBorzs´akLaboratoryofTheoreticalChemistry,E¨otv¨osUniversity,H-1518Budapest112,Pf.32,HungaryH.A.PoschInstitutf¨urExperimentalphysik,Universit¨atWien,Boltzmanngasse5,A-1090Wien,AustriaAndr´asBaranyaiLaboratoryofTheoreticalChemistry,E¨otv¨osUniversity,H-1518Budapest112,Pf.32,Hungary(February5,2008)WestudytheLyapunovinstabilityofatwo-dimensionalfluidcomposedofrigiddiatomicmolecules,withtwointeractionsiteseach,andinteractingwithaWCAsite-sitepotential.WecomputefullspectraofLyapunovex-ponentsforsuchamolecularsystem.Theseexponentscharacterizetherateatwhichneighboringtrajectoriesdivergeorconvergeexponentiallyinphasespace.Qualitativedifferentdegreesoffreedom–suchasrotationandtrans-lation–affecttheLyapunovspectrumdifferently.Westudythisphenomenonbysystematicallyvaryingthemolecularshapeandthedensity.Wedefineandevaluate“rotationnumbers”measuringthetimeaveragedmodulusofthean-gularvelocitiesforvectorsconnectingperturbedsatellitetrajectorieswithanunperturbedreferencetrajectoryinphasespace.Forreasonsofcomparison,varioustimecorrelationfunctionsfortranslationandrotationarecomputed.Therelativedynamicsofperturbedtrajectoriesisalsostudiedincertainsub-spacesofthephasespaceassociatedwithcenter-of-massandorientationalmolecularmotion.PACSnumbers:05.45.+b,02.70.Ns,05.20.-y,66.90.+rI.INTRODUCTIONThechaoticmolecularmotioninfluidsand(nonlinear)solidshasbeenmainlystudiedinthepastintermsofcorrelationfunctionsorrelatedpowerspectraofassorteddynamicalvariables.Thereis,however,amorefundamentalpointofview:Thebasicunderlyingdynamicalprocessesarecollisions(interactions)betweenparticleswithconvexpotentialsurfaces.Asaconsequence,thephasespacetrajectoryishighlyunstablewhichisreflectedinaverysensitivedependenceoninitialconditions.ThisphenomenonischaracterizedintermsofthesetofLyapunovexponents{λl},l=1,···,L,usuallyorderedfromthelargesttothesmallest.Thelargestexponentλ1describesthetime-averagedlogarithmicrateatwhichnearbyphase-spacetrajectoriesseparate.ThesumsofexponentsPli=1λidescribetheexpansionorcontractionratesofl−dimensionalphase-spaceobjects.Thus,theLyapunov1exponentsrepresentthetime-averagedlocaldeformationratesintheneighborhoodofaphase-spacetrajectoryspecifiedbythetimeevolutionofspecially-selectedperturbationvectorsδl(t).Fromapracticalpointofviewthepreciseorientationofthesetofinitialvectors{δl(0)}isnotknown,norisitneededforthedeterminationoftheλl.ThisisdiscussedinmoredetailinSectionII.ThetotalnumberLofexponentsisequaltothedimensionalityofthephasespace,andthewholesetofexponentsisreferredtoastheLyapunovspectrum.FortheevaluationofallLexponentsthesimultaneousintegrationofL×(L+1)first-orderdifferentialequationsisrequired.Themethodhasthereforebeenrestrictedtoratherlow-dimensionaldynamicalsystemsinthepast.Thefeasibilityofsuchstudiesformany-bodysystemshasbeendemonstratedbyHooverandPosch[1,2]whoinvestigatedfluidsystemswithupto100atomsintwodimensions.LyapunovspectraofHamiltoniansystemsinthermalequilibriumexhibitapronouncedsymmetrywhichhelpstoreducethenumberofdifferentialequationstoL(L+2)/2:foreachpositiveexponentthereexistsanothernegativeexponentwithequalabsolutemagnitude.ThisisreferredtoasSmalepairing[1]orconjugatepairing[3,4].Itisaconsequenceofthesymplecticnatureoftheequationsofmotion[5],whichmeansthatthephaseflow–viewedasacanonicaltransformationofthephasespaceontoitself–leavesthedifferentialtwo-formPL/2i=1dpi∧dqiinvariant.Herethesumisoveralldegreesoffreedom,andpiandqidenoteallcomponentsofparticlemomentaandpositions,respectively.Duetothispairingsymmetrythecalculationmayberestrictedtothepositiveexponents,thuspermittingthesimulationofmorecomplexandlargersystems.Withtheavailablecomputerhardwaresystemswithupto400degreesoffreedommaybesimulatedatpresent[6]byfarexceedingthecomplexityoneusuallyencounterswhenstudyingdynamicalsystemswithalow-dimensionalphasespace[7].ForatomicfluidsonefindsthattheshapeoftheLyapunovspectrumchangesqualita-tivelyifthedensityisisothermallyincreasedfromthatofadensegastosoliddensities[8],andthatthelargestLyapunovexponentλ1exhibitsamaximumatthesolid-fluidphasetransitiondensity[8],[9].Fromasimulationofthermostattedbutanemoleculesonemayfurtherdeducethatthelargestcontributiontoλ1-andthereforethelargestsourceforchaos-isduetothetorsionalmotionaroundthecentralCC-bond.Theotherdegreesoffreedomcontributemainlytothesmallerexponents[10].Thisconclusionhasbeenreachedbyob-servingthevariationsintheLyapunovspectrumofaNos´e–Hoover-thermostattedmolecule,ifvariouscontributionstothemolecularHamiltonianspecifyingdifferentdegreesoffreedomareselectivelyswitchedoff.Consideringtheseexampleswemayexpectthatalsoformolecu-larfluidsthestretchingandcontractionpropertiesareverydifferentlyaffectedbytranslationandrotation,andthat-asaconsequence-theshapeoftheLyapunovspectrawillstronglydependonthestateofthesystem.ToclarifythispointwereportinthispaperfirstresultsofamoleculardynamicssimulationstudyofasimpleplanarmolecularfluidinequilibriumconsistingofNrigidhomonucleardiatomicmoleculeswithanisotropyd/σ.