Page - 62 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Φ(β) =−log M e−〈β,U(ξ)〉dλ [99]. Jean-Marie Souriau thengeneralizes theGibbs equilibriumstate to all symplecticmanifolds that have adynamical group. To ensure that all integrals thatwill be definedcouldconverge, the canonicalGibbs ensemble is the largest openproper subset (inLie algebra)where these integrals are convergent. This canonical Gibbs ensemble is convex. Thederivative ofΦ,Q = ∂Φ∂β (thermodynamic heat) is equal to themean value of the energyU. Theminus derivative of this generalizedheatQ,K=−∂Q∂β is symmetricandpositive (this isageometricheatcapacity). Entropy s is thendefinedbyLegendre transformofΦ, s= 〈β,Q〉−Φ. If thisapproach isappliedfor thegroup of timetranslation, this is theclassical thermodynamics theory.However,Souriau [10]hasobserved that ifweapply this theory fornon-commutativegroup (GalileoorPoincarégroups), the symmetryhasbeenbroken. ClassicalGibbs equilibriumstatesareno longer invariantby thisgroup. Thissymmetrybreakingprovides newequations,discoveredbySouriau[10]. We can read in his paper this prophetical sentence “This Lie group thermodynamics could be also of first interest formathematics (Peut-être cetteThermodynamiquedes groupsdeLie a-t-elle un intérêt mathématique)” [30]. Heexplains that for thedynamicGalileogroupwithonlyoneaxeof rotation, this thermodynamic theory is the theoryof centrifugewhere the temperaturevectordimension is equal to2 (sub-groupof invarianceof size2),usedtomake“uranium235”and“ribonucleicacid” [30]. Thephysicalmeaningof these twodimensions forvector-valuedtemperature is“thermicconduction” and“viscosity”. Souriausaidthat themodelunifies“heatconduction”and“viscosity” (Fourierand Navierequations) in thesametheoryof irreversibleprocess. Souriauhasappliedthis theory indetail for relativistic idealgaswith thePoincarégroupfor thedynamicalgroup. Before introducing theSouriauModelofLiegroupthermodynamics,wewillfirst remindreaders of theclassicalnotationofLiegrouptheory in theirapplicationtoLiegroupthermodynamics: • Thecoadjoint representationofG is thecontragredientof theadjoint representation. Itassociates toeachg∈G the linear isomorphismAd∗g∈GL(g∗),whichsatisfies, foreachξ∈ g∗ andX∈ g:〈 Ad∗g−1(ξ),X 〉 = 〈 ξ,Adg−1(X) 〉 (23) • Theadjoint representationof theLiealgebrag is the linear representationofg into itselfwhich associates, to eachX∈ g, the linearmap adX ∈ gl(g). adTangent applicationofAdatneutral element eofG: ad=TeAd :TeG→End(TeG) X,Y∈TeG → adX(Y)= [X,Y] (24) • Thecoadjointrepresentationof theLiealgebrag is thecontragredientof theadjointrepresentation. Itassociates, toeachX∈ g, the linearmap ad∗X∈ gl(g∗)whichsatisfies, foreachξ∈ g∗ andX∈ g:〈 ad∗−X(ξ),Y 〉 = 〈ξ,Ad−X(Y)〉 (25) Wecanillustrate forgroupofmatrices forG=GLn(K)withK=RorC. TeG=Mn(K), X∈Mn(K),g∈G Adg(X)= gXg−1 (26) X,Y∈Mn(K) adX(Y)=(TeAd)X(Y)=XY−YX=[X,Y] (27) Then, thecurve from e= Id= c(0) tangent toX= c(1) isgivenby c(t)= exp(tX)andtransform byAd:γ(t)=Adexp(tX) adX(Y)=(TeAd)X(Y)= d dt γ(t)Y ∣∣∣∣ t=0 = d dt exp(tX)Yexp(tX)−1 ∣∣∣∣ t=0 =XY−YX (28) 62