Page - 127 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 Combiningthis resultwith thegeometricversionof thefirstprincipleof thermodynamics: DivT=0, Div −→ N =0, , (18) In [7,8],Souriauclaimedthat the4-fluxofentropyisgivenby: −→ S =T −→ W+ζ −→ N , (19) andprovedit isdivergence free.Moreover thespecificentropy s isan integralof themotion[2]. Letus introducenowthe5-temperature −ˆ→ W representedbythe5-column: Wˆ= ( W ζ ) , (20) andthe tensor Tˆ representedbythe4×5matrix Tˆ= ( T N ) (21) whichallowsgatheringEquation(18) in themorecompact form Div Tˆ=0 andrepresenting(19) in themorecompact form: S= TˆWˆ , localexpressionof thecontractedproductof Tˆand ˆW: S= Tˆ · ˆW , (22) It is thecornerstoneequationofSouriau’s theory. In this form, it canbeseenasageometrization ofClausius’definitionof theentropyasstate functionofasystem: S=Q θ , (23) whereQ isa theamountofheatabsorbedinanisothermalprocess. Scalarquantitiesarereplacedby analogoustensorialones:S byits4-flux S,Qby Tˆandβ=1/θbyits5-flux ˆW. Replacing(19)by(22) isnotapurelyformalmanipulationbut it takesastrongmeaningwhenconsideringBargmann’sgroup B [15],acentralextensionofGalileo’sone[16], setof theaffinetransformationsdXˆ′ → dXˆ= PˆdXˆ′+Cˆ ofR5 suchthat. Pˆ= ⎛⎜⎝ 1 0 0u R 0 1 2 ‖u‖2 uTR 1 ⎞⎟⎠ . (24) TheB-tensorsarecalledBargmanniantensors. Fromthisviewpoint, the5-column(20) represents aBargmannianvector ˆW of transformation law: Wˆ= PˆWˆ′ , (25) and the 4×5matrix (21) represents a Bargmannian 1-covariant and 1-contravariant tensor Tˆ of transformation law: Tˆ=P Tˆ′ Pˆ−1 . 127