Page - 134 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 Werecognize thebalanceofenergy, linearmomentumandmass. Remark4. TheHessianmatrix I of−z, consideredas functionofW throughZ, ispositivedefinite [3]. It is Fishermetric of the InformationGeometry. For the expression (48), it is easy toverify it: −δMδZ= 1 β ( eint(δβ)2+m ‖ δw− δβm p‖ 2 ) >0 , for anynonvanishingδZtaking into accountβ> 0,eint> 0andm> 0. On this basis,we canconstruct a thermodynamic lengthof apath t →X(t) [21]: L= ∫ t1 t0 √ (δW(t))TI(t)δW(t)dt , whereδW(t) is theperturbationof the temperaturevector, tangent to the space-timeatX(t).Wecanalsodefine arelatedquantity, Jensen–Shannondivergenceof thepath: J=(t1− t0) ∫ t1 t0 (δW(t))TI(t)δW(t)dt . 8.Conclusions Theaboveapproach isnot limited to classicalmechanics but canbeusedasguiding ideas to tackle therelativisticmechanics. Beyondthestrictapplicationtophysics, it canbe takenassourceof inspiration tobroachother topicssuchas thescienceof informationfromtheviewpointofdifferential geometryandLiegroups.Wehopetohavemodestlycontributedto thisaim. Conflictsof Interest:Theauthorsdeclarenoconflictof interest. References 1. Souriau, J.-M. Milieux continus de dimension 1, 2 ou 3 : statique et dynamique. Available online: http://jmsouriau.com/Publications/JMSouriau-MilContDim1991.pdf (accessedon9July2016). (InFrench) 2. DeSaxcé,G.;Vallée,C.GalileanMechanicsandThermodynamicsofContinua;Wiley-ISTE:London,UK,2016. 3. Souriau, J.-M.StructuredesSystèmesDynamiques;Dunod: Paris,France,1970. (InFrench) 4. Souriau, J.-M. Structure of Dynamical Systems: A Symplectic View of Physics; Birkhäuser: NewYork, NY, USA,1997. 5. DeSaxcé,G.;Vallée,C.AffineTensors inMechanicsofFreelyFallingParticlesandRigidBodies.Math.Mech. Solid J.2011,17, 413–430. 6. Guillemin,V.;Sternberg,S.SymplecticTechniques inPhysics;CambridgeUniversityPress:Cambridge,MA, USA,1984. 7. Souriau, J.-M.ThermodynamiqueetGéométrie. InDifferentialGeometricalMethods inMathematicalPhysics II; Springer: Berlin/Heidelberg,Germany,1976;pp. 369–397. (InFrench) 8. Souriau, J.-M.ThermodynamiqueRelativistedesFluides;CentredePhsyiqueThéorique:Marseille,France,1977; Volume35,pp.21–34. (InFrench) 9. Tulczyjew,W.;Urbañski,P.;Grabowski, J.Apseudocategoryofprincipalbundles.AttidellaRealeAccademia delleScienzediTorino1988,122, 66–72. (In Italian) 10. Libermann, P.; Marle, C.-M. Symplectic Geometry and Analytical Mechanics; Springer: Dordrecht, TheNetherlands,1987. 11. Vallée,C.;Lerintiu,C.Convexanalysisandentropycalculation instatisticalmechanics.Proc.A.Razmadze Math. Inst. 2005,137, 111–129. 12. Iglesias,P.Essaide“thermodynamiquerationnelle”desmilieuxcontinus.Annalesde l’IHPPhysiqueThéorique 1981,34, 1–24. (InFrench) 13. Vallée,C.Relativistic thermodynamicsofcontinua. Int. J.Eng. Sci. 1981,19, 589–601. 134