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1
ClassicalTransport,SteadyStatesandLarge
DeviationsinNon-equilibrium1dSystems
BernardDerrida
CollegedeFrance?11placeMarcelinBerthelot?75231ParisCedex05?France
andLaboratoiredePhysiqueStatistique?DépartementdePhysique?EcoleNormale
Supérieure24?rueLhomond?75231ParisCedex05?France
Summary.Thegoalofthesethreelectureswastoreviewaseriesofresultsobtained
overthelastfifteenyearsonnon-equilibriumdilusivesystems[7;8;17;24].
Theystartedbyanintroductiontothedilerenttypesofmodelswhicharestudied
todescribenon-equilibriumsystems(modelswithdeterministicthermostats[35;39;52]
anddeterministicdynamics?modelswithstochasticthermostatsanddeterministicinter-
naldynamics?modelswithstochasticthermostats[27;47;57;58]andstochasticinternal
dynamics[40;41;48;49;56]):intheparticularcaseofstochasticthermostatsitwas
shownhowtheideaofdetailedbalancedcanbeextendedtodescribesystemsincontact
withseveralheatbathsatunequaltemperaturesorseveralreservoirsofparticlesatun-
equalchemicalpotentials[24;54;59].ClosetoequilibriumthereisanEinsteinrelation
betweentheheatcurrentgeneratedbyasmalltemperaturedilerencebetweentwoheat
bathsandthevarianceoftheheatexchangesbetweenthetwothermostats[43].This
relationcanbeviewedasaconsequenceofthefluctuationtheorem[30Ź33;44;45;50]
whichwasdiscussedinthecaseofstochasticdynamics[45].Finallythelastpartofthe
firstlecturegaveseveralexampleswhichdoordonotsatisfyFourier)slaworFick)slaw
[27;47].
Thesecondlecturewasdevotedtothefluctuationsandthelargedeviationfunctions
ofthecurrentinnon-equilibriumdilusivesystems[1;5;6;11Ź13;25;26;36;42;51].
whenthedynamicsisdescribedbyaMarkovprocess?theLegendretransformofthe
largedeviationfunctionofthecurrentcanbecomputedasthelargesteigenvalueofa
matrixobtainedbydeformingtheMarkovmatrix[23;24].Forarbitrarydilusivesys-
temsoflargelinearsizethewholedistributionofthefluctuationsofthecurrentcanbe
obtainedusingthemacroscopicfluctuationtheory.Conditionnedonthecurrent?onecan
calculatethedensityprofilealongthesystem[11;29;38].Insomecases?thisoptimal
profileundergoesphasetransitions?forexamplebybecomingtimedependent[12;37].
Fordilusivesystemsatequilibriumonecanalsoshow?usingthemacroscopicfluctuation
theory?thatthedistributionofthefluctuationsofthecurrentisuniversal[2;46].
Thethirdlecturewasfocusedonanumberofresultsaboutthefluctuationsandthelarge
deviationfunctionofthedensityinnon-equilibriumsteadystates.Theseresultswere
obtainedeitherbyexactmicroscopiccalculationsbasedonthematrixansatz[10;18;19]
forsomespecificmodelssuchasthesymmetricexclusionprocess[20;21;28]orusing
fluctuatinghydrodynamicsandthemacroscopicfluctuationtheory[3;4;9].Thesedif-
