Page - 126 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 Galileo’s group GAL is a subgroup of the affine group Af f(4), collecting the Galilean transformations, that is theaffinetransformationsdX′ → dX=PdX′+CofR4 suchthat: C= ( τ0 k ) , P= ( 1 0 u R ) , (12) whereu∈R3 isaGalileanboost,R∈SO(3) isa rotation,k∈R3 isaspatial translationandτ0∈R is aclockchange.Hence,Galileo’sgroupisaLiegroupofdimension10. TheGAL-tensorswillalsobe calledGalilean tensors. M is thespace-timeequippedwithasymmetricGAL-connection∇ representing thegravitation, thematterand its evolution is characterizedbya linebundleπ0 :M →M0. The trajectoryof the particleX0∈M0 is thecorrespondingfiberπ−10 (X0). In local charts,X0 is representedby s′ ∈R3 and itspositionxat time t isgivenbyamap: x=ϕ(t,s′) . (13) The4-velocity: −→ U = −→ dX dt , is the tangentvector to thefiberparameterizedbythe time. Ina local chart, it is representedby: U= ( 1 v ) , (14) wherev is theusualvelocity.Conversely,ϕcanbeobtainedas theflowof the4-velocity. βbeing thereciprocal temperature, that is1/kBTwhere kB isBoltzmann’sconstantandT the absolute temperature, therearefivebasic tensorfieldsdefinedonthespace-timeM: • the4-fluxofmass−→N = ρ−→U whereρ is thedensity, • the4-fluxofentropy−→S = ρs−→U = s−→N where s is thespecificentropy, • Planck’s temperaturevector−→W= β−→U , • itsgradient f=∇−→W calledfrictiontensor, • themomentumtensorofacontinuumT, a linearmapfromTXM into itself. In localcharts, theyarerespectivelyrepresentedbytwo4-columnsN,W andtwo4×4matrices f andT. Thenweprovedin[14] the followingresult characterizingthereversibleprocesses: Theorem1. IfPlanck’spotentialζ smoothlydependsons′,WandF= ∂x/∂s′ throughrightCauchystrains: C=FTF , (15) then: T=UΠ+ ( 0 0 −σv σ ) (16) with Π=−ρ ∂ζ ∂W , σ=−2ρ β F ∂ζ ∂C F T , (17) represents themomentumtensorof the continuumand is such that: (∇ζ)N=−Tr (T f) , 126