CONVERGENCE TO EQUILIBRIUM: ENTROPY PRODUCTION AND HYPOCOERCIVITY

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Niveau: Supérieur, Doctorat, Bac+8
CONVERGENCE TO EQUILIBRIUM: ENTROPY PRODUCTION AND HYPOCOERCIVITY CEDRIC VILLANI Abstract. These is the text for my Harold Grad lecture, delivered in the 24th Rarefied Gas Dynamics conference, Bari (July 2004). They describe recent developments about convergence to equilibrium in kinetic theory, related to Boltzmann's H Theorem and entropy production. The style is intentionally informal, at the price of rigor and precision; full details can be found in the research papers quoted within the text. The text is partially based on my earlier contri- bution to the proceedings of the 14th International Congress of Mathematical Physics, Lisbon (July 2003). 1. 1872: Boltzmann's H Theorem Boltzmann's equation models the dynamics of a rarefied gas via the (time-dependent) position- velocity density f(x, v); here x is the position variable, varying in a box ? ? R3, while v is the velocity variable, varying in R3. In the 1870's, Boltzmann discovered that, under ad hoc boundary condition, the entropy S(f) = ?H(f) := ? ∫ ??R3v f(x, v) log f(x, v) dv dx, is nondecreasing as time increases if f evolves according to the Boltzmann equation. More than 100 years later, this theorem is still striking and beautiful: - Boltzmann managed to recover irreversibility from a model based on reversible mechanics and statistics; - the H Theorem is a manifestation of the second law of thermodynamics, but it is a theorem, as opposed

  • transport collisions

  • boltzmann's theory

  • full boltzmann equation

  • theorem implies

  • entropy principle

  • famous theorem

  • maxwellian collision

  • very high

  • boltzmann equation


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ENTROPCYONPVREORDGUECNTICOENTOANEDQUHIYLIPBORCIOUEMR:CIVITYCEDRICVILLANIAbstract.TheseisthetextformyHaroldGradlecture,deliveredinthe24thRare edGasDynamicsconference,Bari(July2004).Theydescriberecentdevelopmentsaboutconvergencetoequilibriuminkinetictheory,relatedtoBoltzmann’sHTheoremandentropyproduction.Thestyleisintentionallyinformal,atthepriceofrigorandprecision;fulldetailscanbefoundintheresearchpapersquotedwithinthetext.Thetextispartiallybasedonmyearliercontri-butiontotheproceedingsofthe14thInternationalCongressofMathematicalPhysics,Lisbon(July2003).1.1872:Boltzmann’sHTheoremBoltzmann’sequationmodelsthedynamicsofarare edgasviathe(time-dependent)position-velocitydensityf(x,v);herexisthepositionvariable,varyinginabox R3,whilevisthevelocityvariable,varyinginR3.Inthe1870’s,Boltzmanndiscoveredthat,underadhocboundarycondition,theentropyZS(f)=H(f):=f(x,v)logf(x,v)dvdx,3R visnondecreasingastimeincreasesiffevolvesaccordingtotheBoltzmannequation.Morethan100yearslater,thistheoremisstillstrikingandbeautiful:-Boltzmannmanagedtorecoverirreversibilityfromamodelbasedonreversiblemechanicsandstatistics;-theHTheoremisamanifestationofthesecondlawofthermodynamics,butitisatheorem,asopposedtoapostulate;-althoughnotperfectlyrigorous,theproofisbeautiful;-thistheoremappliesinallgenerality,evenfarfromequilibrium.Inviewoftheseconsiderations,itisinsomesenseashameforusmathematiciansthatwearestillunabletorigorouslyproveBoltzmann’stheoreminfullgenerality.Thedicultyliesina“slightanalyticaldiculty”(asEulersaidabouthisownequation):theexistenceof“smooth”solutions.TheregularitytheoryfortheBoltzmannequationisahugeworkinprogress;itis70yearsold(Carlemanmaybeconsideredasitsfounder),andhasneverbeensoactiveasnow.Yetitisstillfarfromcompletion,anditisstillariddlewhethertheBoltzmannequation,startingfromasmoothinitialdatum,admitssmoothsolutions.1