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1、ThelargedeviationapproachtostatisticalmechanicsHugoTouchetteSchoolofMathematicalSciences,QueenMaryUniversityofLondon,LondonE14NS,UK(Dated:August20,2009)Thetheoryoflargedeviationsisconcernedwiththeexponentialdecayofprobabilitiesoflarge?uctuationsinrandomsystems.Theseprobabili
2、tiesareimportantinmany?eldsofstudy,includingstatistics,?nance,andengineering,astheyoftenyieldvaluableinformationaboutthelarge?uctuationsofarandomsystemarounditsmostprobablestateortrajectory.Inthecontextofequilibriumstatisticalmechanics,thetheoryoflargedeviationsprovidesexpon
3、ential-orderestimatesofprobabilitiesthatre?neandgeneralizeEinstein’stheoryof?uctuations.Thisreviewexploresthisandotherconnectionsbetweenlargedeviationtheoryandstatisticalmechanics,inanefforttoshowthatthemathematicallanguageofstatisticalmechanicsisthelanguageoflargedeviationt
4、heory.The?rstpartofthereviewpresentsthebasicsoflargedeviationtheory,andworksoutmanyofitsclassicalapplicationsrelatedtosumsofrandomvariablesandMarkovprocesses.Thesecondpartgoesthroughmanyproblemsandresultsofstatisticalmechanics,andshowshowthesecanbeformulatedandderivedwithint
5、hecontextoflargedeviationtheory.Theproblemsandresultstreatedcoverawiderangeofphysicalsystems,includingequilibriummany-particlesystems,noise-perturbeddynamics,nonequilibriumsystems,aswellasmultifractals,disorderedsystems,andchaoticsystems.Thisreviewalsocoversmanyfundamentalas
6、pectsofstatisticalmechanics,suchasthederivationofvariationalprinciplescharacterizingequilibriumandnonequilibriumstates,thebreakingoftheLegendretransformfornonconcaveentropies,andthecharacterizationofnonequilibrium?uctuationsthrough?uctuationrelations.PACSnumbers:05.20.-y,65.
7、40.Gr,02.50.-r,05.40.-aarXiv:0804.0327v2[cond-mat.stat-mech]20Aug2009TypesetbyREVTEX2ContentsI.Introduction4II.Examplesoflargedeviationresults6III.Largedeviationtheory10A.Thelargedeviationprinciple10B.Moreonthelargedeviationprinciple11C.Calculatingratefunctions131.TheGartner
8、-EllisTheorem¨132.PlausibilityargumentfortheGartner-EllisTheorem¨14D.Cramer
