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Non-equilibriumStatisticalMechanicsof
theStochasticNaVier1StokesEquations
andGeostrophicTurbulence
FreddyBouchet,CesareNardini,andTomźsTangarife
AbstractTwo-dimensionalandgeophysicalturbulentflowshaVethepropertyto
selforganizeandcreatelargescalecoherentjetsandVorticeslThisisforinstance
oneofthemajorprocessesforthedynamicsofEarth,satmospherelFollowingOn-
sagerinitialinsight,basedonconjugatedworksbymathematiciansandphysicists,
thisfundamentalphysicalprocesshasfoundsomeeXplanationsintheframework
ofstatisticalmechanicslAnimportantstep,initiatedtwentyyearsago,hasbeen
thestudyoftheequilibriumstatisticalmechanicsforthe2DEulerandtherelated
quasi-geostrophicmodels(theMiller-Robert-Sommeriatheory)l
RealgeophysicalandeXperimentalflowsarehoweVerdissipatiVeandmaintained
byeXternalforceslTheselecturesfocusonrecenttheoreticaldeVelopmentofthe
statisticalmechanicsofthosenon-equilibriumsituationslThoseprogresseshaVe
beenachieVedusingtoolsfromfieldtheory(pathintegralsandinstantons),non-
equilibriumstatisticalmechanics(largedeViations,stochasticaVeraging)lTheaim
oftheselecturesistobrieflyintroducethetheoreticalaspectsofthisprogramin
thesimplestconteXtothe2DstochasticEulerorNaVier-Stokesequationsandthe
quasi-geostrophicequationsl
WereViewpathintegralrepresentationsofstochasticprocesses,largedeViations
fortransitionprobabilities,actionminimization,instantontheory,forgeneralme-
chanicalsystemsforcedbyrandomforceslWewillapplythisframeworkinorder
topredictequilibriumandnon-equilibriumphasetransitionsforthe2DEuler,
NaVier-Stokes,andquasi-geostrophicdynamics,andtopredicttheratesofrare
transitionsbetweentwoattractorsinsituationsoffirstorderphasetransitionsl
Kinetictheoryofsystemswithlongrangeinteractions,bothwithandwithout
stochasticeXternalforces,areeXplainedlBasedonthiskinetictheory,wepredict
non-equilibriumphasetransitions,anddiscusstheirrecenteXperimentalobserVa-
tionsandnumericalsimulationsl
EVenifthemodelwehaVeconsideredsofararetoosimpleacademicmodels,the
FreddyBouchet
Laboratolredephyslque9EcoleNormaleSupfleuredeLyonetCNRS946allśed9Italle969007
Lyon9France
emall.FreddyºBouchet©ens-lyonºfr
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