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1、AStatisticalMeasureofComplexityR.L´opez-Ruiz1,‡,H.L.Mancini2andX.Calbet31LaboratoiredePhysiqueStatistique,ENS,24ruedeLhomond,Paris(France)2DepartamentodeF´ısica,UniversidadPrivadadeNavarra,Pamplona(Spain)3InstitutodeAstrof´ısicadeCanarias,LaLaguna,Tenerife(Spain)Abstrac
2、tAmeasureofcomplexitybasedonaprobabilisticdescriptionofphysicalsystemsisproposed.Thismeasureincorporatesthemainfeaturesoftheintuitivenotionofsuchamagnitude.Itcanbeappliedtomanyphysicalsituationsandtodifferentdescriptionsofagivensystem.Moreover,thecalculationofitsvaluedoe
3、snotrequireaconsiderablecomputationaleffortinmanycasesofphysicalinterest.PACSnumber(s):05.20.-y,02.50.+s,02.70.+darXiv:nlin/0205033v1[nlin.CD]15May20021Onthemostbasicgrounds,anobject,aprocedure,orsystemissaidtobe”complex”whenitdoesnotmatchpatternsregardedassimple.Thissou
4、ndsratherlikeanoxymoronbutcommonknowledgetellsuswhatissimpleandcomplex:simplifiedsystemsoridealizationsarealwaysastartingpointtosolvescientificproblems.Thenotionof”complexity”inphysics[1,2]startsbyconsideringtheperfectcrystalandtheisolatedidealgasasexamplesofsimplemodelsa
5、ndthereforeassystemswithzero”complexity”.Letusbrieflyrecalltheirmaincharacteristicswith”order”,”information”and”equilibrium”.Aperfectcrystaliscompletelyorderedandtheatomsarearrangedfollowingstringentrulesofsymmetry.Theprobabilitydistributionforthestatesaccessibletotheper
6、fectcrystaliscenteredaroundaprevailingstateofperfectsymmetry.Asmallpieceof”information”isenoughtodescribetheperfectcrystal:thedistancesandthesymmetriesthatdefinetheelementarycell.The”information”storedinthissystemcanbeconsideredminimal.Ontheotherhand,theisolatedidealgasi
7、scompletelydisordered.Thesystemcanbefoundinanyofitsaccessiblestateswiththesameprobability.Allofthemcontributeinequalmeasuretothe”information”storedintheidealgas.Ithasthereforeamaximum”information”.Thesetwosimplesystemsareextremainthescaleof”order”and”information”.Itfoll
8、owsthatthedefinitionof”complexity”mustnotbemadeintermsofjust”order”or”information”.Itmightseemreasonabletopropo
