The Homogeneous Chaos.pdf

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1、TheHomogeneousChaosAuthor(s):NorbertWienerSource:AmericanJournalofMathematics,Vol.60,No.4(Oct.,1938),pp.897-936Publishedby:TheJohnsHopkinsUniversityPressStableURL:http://www.jstor.org/stable/2371268.Accessed:04/12/201308:21YouruseoftheJSTORarchiveindicatesyouracceptanceoftheTerms&Conditi

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4、esubjecttoJSTORTermsandConditionsTHEHOMOGENEOUSCHAOS.ByNoRBERTWIENER.1.Introduction.Physicalneedfortheory.2.Definition.Typesofchaos.3.Classicalergodictheorem.Lebesgueform.4.Dominatedergodictheorem.Multidimensionalergodictheorem.5.Metrictransitivity.Spaceandphaseaveragesinachaos.6.Pureone

5、-dimensionalchaos.7.Puremultidimensionalchaos.8.Phaseaveragesinapurechaos.9.Formsofchaosderivablefromapurechaos.10.Chaostheoryandspectra.11.Thediscretechaos.12.Theweakapproximationtheoremforthepolynomialchaos.13.Thephysicalproblem.Thetransformationsofachaos.1.Introduction.Physicalneedfor

6、theory.StatisticalmechanicsmaybedefinedastheapplicationoftheconceptsofLebesgueintegratiolntomechanics.Historically,thisisperhapsputtingthecartbeforethehorse.StatisticalmechanicsdevelopedthroughtheentirelatterhalfofthenineteenthcenturybeforetheLebesgueintegralwasdiscovered.Nevertheless,it

7、de-velopedwithoutanadequatearmoryofconceptsanldmathematicaltechnique,whichisonlynowintheprocessofdevelopmentatthehandsofthemodernschoolofstudentsofintegraltheory.Inthemoreprimitiveformsofstatisticalmechanics,theintegrationorsummationwastakenoverthemanifoldparticlesofasing