Seite - 46 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 This formuladescribes thebehaviour of a gasmadeofpoint particles of variousmasses in a centrifugerotatingataconstantangularvelocity ω ε : theheavierparticlesconcentrate farther from therotationaxis thanthe lighterones. 7.3.6.OtherApplicationsofGeneralizedGibbsStates ApplicationsofgeneralizedGibbsstates in thermodynamicsofcontinua,with theuseofaffine tensors,arepresented in thepapersbydeSaxcé [64,65]. Several applications of generalized Gibbs states of subgroups of the Poincaré group were consideredbySouriau. Forexample,hepresents inhisbook[14],Chapter IV,p. 308,ageneralized Gibbswhichdescribes thebehaviourofagas inarelativistic centrifuge,andinhispapers [15,16],very niceapplicationsofsuchgeneralizedGibbsstates inCosmology. Acknowledgments: I addressmy thanks toAlainChenciner for his interest andhis help to study theworks ofClaudeShannon, toRogerBalian forhis commentsandhis explanationsabout thermodynamicpotentials, andtoFrédéricBarbarescoforhiskindinvitationtoparticipate intheGSI2015conferenceandhisencouragements. Mywarmest thanks to theanonymousrefereeswhoseverycarefulandbenevolent readingofmyworkallowed metocorrect severalmistakesandto improvethispaper. Conflictsof Interest:Theauthordeclaresnoconflictof interest. References 1. Abraham,R.;Marsden, J.E.FoundationsofMechanics, 2nded.;AmericanChemicalSociety:Washington,DC, USA,1978. 2. Arnold,V.I.MathematicalMethods ofClassicalMechanics, 2nded.; Springer: Berlin/Heidelberg,Germany, 1978. 3. CannasdaSilva,A.LecturesonSymplecticGeometry; Springer: Berlin/Heidelberg,Germany,2001. 4. Guillemin, V.; Sternberg, S. Symplectic Techniques in Physics; Cambridge University Press: Cambridge, UK,1984. 5. Holm,D.GeometricMechanics,Part I:DynamicsansSymmetry;WorldScientific: Singapore,2008. 6. Holm,D.GeometricMechanics,Part II:Rotating,TranslatingandRolling;WorldScientific: Singapore,2008. 7. Iglesias,P.Symétries etMoment;ÉditionsHermann: Paris,France,2000. (InFrench) 8. Laurent-Gengoux, C.; Pichereau, A.; Vanhaecke, P. Poisson Structures; Springer: Berlin/Heidelberg, Germany,2013. 9. Libermann, P.;Marle, C.-M.SymplecticGeometry andAnalyticalMechanics; Springer: Berlin/Heidelberg, Germany,1987. 10. Ortega, J.-P.;Ratiu,T.-S.MomentumMapsandHamiltonianReduction;Birkhäuser: Boston,MA,USA;Basel, Switzerland;Berlin,Germany,2004. 11. Vaisman, I.Lectureson theGeometryofPoissonManifolds; Springer: Berlin/Heidelberg,Germany,1994. 12. Marle,C.-M.Symmetriesofhamiltoniansystemsonsymplecticandpoissonmanifolds. InSimilarityand SymmetryMethods,Applications inElasticityandMechanicsofMaterials;Ganghoffer, J.-F.,Mladenov, I.,Eds.; Springer: Berlin/Heidelberg,Germany,2014;pp.183–269. 13. Souriau, J.-M.Définitioncovariantedeséquilibres thermodynamiques.SupplementoalNuovoCimento1966,4, 203–216. (InFrench) 14. Souriau, J.-M.StructuredesSystèmesDynamiques;Dunod:Malakoff,France,1969. (InFrench) 15. Souriau, J.-M.MécaniqueStatistique,GroupesdeLieetCosmologie. InGéométrieSymplectiqueetPhysique Mathématique;CNRSÉditions: Paris,France,1974;pp.59–113. (InFrench) 16. Souriau, J.-M.Géométrie symplectiqueetPhysiquemathématique. InDeuxConférencesde Jean-Marie Souriau,ColloquiumdelaSociétéMathématiquedeFrance,Paris,France,19February–12November1975. (InFrench) 17. Souriau, J.-M.MécaniqueClassique etGéométrieSymplectique;Dunod:Malakoff,France,1984. (InFrench) 18. Mackey,G.W.TheMathematical Foundations ofQuantumMechanics;W.A.Benjamin, Inc.: NewYork,NY, USA,1963. 46