Page - 125 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 whereψ → ψ′= a ·ψ is the induced action of the one ofG onN . It isworth observing it is just (6)with μ= ψ(η) andμ′= ψ′(η). In this sense, the values of themomentummap are just the components of the momentumG-tensorsdefined in thepreviousSection. Remark3. WesawatRemarkofSection3thatthemomentumG-tensorμ is identifiedtotheorbitμ= orb(μ, f) and,disregarding the frames for simplification,wecan identifyμ to the componentorbit orb(μ). 5. LieGroupStatisticalMechanics In order to discover the underlined geometric structure of the statisticalmechanics, we are interested in theaffinemapsΘon theaffine spaceofmomentumtensors. Inanaffine frame,Θ is representedbyanaffinefunctionΘ fromg∗ intoR: Θ(μ)=Θ(μ)= z+μZ , where z=Θ(0)=Θ(μ0)andZ= lin(Θ)∈ gare theaffinecomponentsofΘ. If thecomponentsof themomentumtensorsaremodifiedaccordingto (6), thechangeofaffinecomponentsofΘ isgivenby the inducedaction: z= z′−θ(a)Ad(a)Z′, Z=Ad(a)Z′ . (9) ThenΘ isaG-tensors. In [3,4],Souriauproposedastatisticalmechanicsmodelusinggeometric tools. Letdλbeameasureonμ= orb(μ)andaGibbsprobabilitymeasure pdλwith: p= e−Θ(μ) = e−(z+μZ) . Thenormalizationcondition ∫ orb(μ)pdλ=1 links thecomponentsofΘ, allowingtoexpressz in termsofZ: z(Z)= ln ∫ orb(μ) e−μZdλ . (10) Thecorrespondingentropyandmeanmomentaare: s(Z) = − ∫ orb(μ) p lnpdλ= z+MZ, M(Z) = ∫ orb(μ) μpdλ=−∂z ∂Z , (11) satisfyingthesametransformationlawastheone(6)ofμ.HenceMarethecomponentsofamomentum tensorMwhichcanbeidentifiedtotheorbitorb(M), thatdefinesamapμ →M, i.e., acorrespondance betweentwoorbits. Thisconstruction is formaland, forreasonsof integrability, the integralswillbe performedonlyonasubsetof theorbitaccordingtoanheuristicwayexplainedlatteron. Peoplegenerallyconsider that thedefinitionof theentropy is relevant forapplications insofaras thenumberofparticles in thesystemisveryhuge. For instance, thenumberofatomscontained inone mole isAvogadro’snumberequal to6×1023. It isworthnotingthatValléeandLerintiuproposeda generalizationof the idealgas lawbasedonconvexanalysisandadefinitionofentropywhichdoes not require theclassicalapproximations (Stirling’sFormula) [11]. 6.RelativisticThermodynamicsofContinua Independently of his statistical mechanics, Souriau proposed in [7,8] a thermodynamics of continua compatiblewith general relativity. Following in his footsteps, one can quote theworks by Iglesias [12]andVallée [13]. InhisPh.Dthesis,Vallée studied the invariant formofconstitutive laws in thecontextof special relativitywhere thegravitationeffectsareneglected. In [14], theauthor andValléeproposedaGalileanversionof this theoryofwhichwerecall thecornerstoneresults. For moredetails, thereader is referredto [2]. 125