Page - 130 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 of theparticlemoving inuniformstraightmotion at velocity v. Owing (27) and (28),we can determine thenewcomponentsof the torsor inX: p=mv, q=mx0, l= l0+q×v, e= e0+m2 ‖ v‖ 2 , (42) The third relation of (42) is the classical transport law of the angularmomentum. In fact, it is a particularcaseof thegeneral transformation laws(28)whenconsideringonlyaGalileanboost. The transformation lawreveals thephysicalmeaningof themomentumtensorcomponents: – Thequantity p,proportional to themassandto thevelocity, is the linearmomentum. – Thequantityq,proportional to themassandtothe initialposition,provides the trajectory equation. It is calledpassagebecause indicating theparticle ispassing through x0 at time t=0. – Thequantity l splits into twoterms. Thesecondone,q×v = x×mv = x×p, is theorbital angularmomentum. Thefirstone, l0= l− q×p/m, is the spinangularmomentum. Theirsum, l, is theangularmomentum. • Step3: parameterizing theorbit. If theparticlehasan internal structure, introducingthemomentof inertiamatrixJ andthespin ,wehave,accordingtoKönig’s theorem: l0=J , e0= 12 ·(J ) . Henceeachorbitdefinesaparticleofmassm, spin s0, inertiaJ andcanbeparameterizedby8 coordinates, the3componentsofq, the3componentsof pandthe2componentsof theunitvector ndefining the spindirection, thanks to themomentummapR3×R3×S2 → g∗ : (q,p,n) → μ=ψ(q,p,n) suchthat: l= 1 m q×p+s0n, e= 12m ‖ p‖ 2+ s20 2 n ·(J−1n) . The correspondingmeasure is dλ = d3qd3pd2n. For simplicity, we consider further only a singularorbitofdimension6representingaspinlessparticleofmassm,whichcorresponds to the particularcase l0=0 thenn=0. It canbeparameterizedby6coordinates, the3componentsofq andthe3componentsof p thanks to themap: ψ :R3×R3→g∗ : (q,p) →μ=ψ(q,p) , suchthat: l= 1 m q×p, e= 1 2m ‖ p‖2 . (43) • Step 4: modelling the deformation. Statisticalmechanics is essentially basedona set of discrete particles and, in essence, incompatiblewith continuummechanics. Thus, according tousual arguments, thepassage fromthestatisticalmechanics to continuummechanics isobtainedby equivalencebetweenthesetofNparticles(inhugenumber)andaboxoffinitevolumeVoccupied bythem, largewithrespect totheparticlesizebutsosmallwithrespect tothecontinuousmedium that it can be considered as infinitesimal. Let us consider N identical particles contained in V, largewith respect to the particles but representing the volume element of the continuum thermodynamics. Themotionof thematterbeingcharacterizedby(13), letusconsider thechange ofcoordinate t= t′, x=ϕ(t′,s′) . 130