Page - 131 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 The jacobeanmatrixreads: ∂X ∂X′=P= ( 1 0 v F ) . (44) Fromthenon, themomentumisconsideredasanaffinetensor, i.e., its componentsaremodified bytheactionofanyaffinetransformation. Besides,wesupposethat theboxof initialvolumeV0 isatrest intheconsideredcoordinatesystem (v=0)andthedeformationgradientF isuniforminthebox, then: dx=Fds′ . Accordingto (3), the linearmomentumis transformedaccordingto: p=F−Tp′ . (45) Foraparticle initiallyatpositionx, thepassage isgivenby(42): q=mx . Themeasurebecomes dλ=m3d3xd3pd2n=m3d3s′d3p′d2n . For reasons thatwillbe justifiedatStep5,weconsider the infinitesimalgenerator: Z=(−W,0) . As thebox isat rest in theconsideredcoordinatesystem, thevelocity isnulland,owingto (14): W= βU= ( β 0 ) . (46) Hence thedualpairing(26) is reducedto: μZ= βe , and,owingto (43), (45)and(15), foraspinlessmassiveparticle: μZ= β 2m ‖ p‖2= β 2m ‖F−Tp′ ‖2= β 2m p′TC−1p′ . For reasonsof integrability as explained inSection6, it is usual to replace the orbit by the subset V0×R3×S2 orb(μ) . It isworthremarkingthat,unlike theorbit, this set isnotpreservedby theactionbut the integrals in (10)and(11)are invariant. Equation(10)gives foraparticle: z= ln(m3I0I1I2) , where: I0= ∫ V0 d3s′=V0 , I1= ∫ R3 e− β 2m p ′TC−1p′d3p′ , I2= ∫ S2 d2n=4π . 131