Page - 133 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 • Step6: boostmethod. For theboxat rest in thecoordinatesystemX, the temperature4-vector is givenby(46): W= ( β 0 ) . Anewcoordinatesystem X¯ inwhichtheboxhas thevelocityvcanbededucedfromX=PX¯+C byapplyingaboostu=−v (hencek=0,τ0=0andR=1R3). The transformation lawofvectors gives thenewcomponents W¯= ( β βv ) , and(9) leads to: z¯= z+ mβ 2 ‖v‖2= z+ m 2β ‖w‖2 . Taking intoaccount (47)andleavingout thebars: z= 1 2 ln(det(C))− 3 2 lnβ+ m 2β ‖w‖2+Cte . (48) It isclearfrom(11) thats isLegendreconjugateof−z, then, introducingtheinternalenergy(which isnothingother thantheGalilean invariant (40)): eint= e− 12m ‖ p‖ 2 , theentropyis: s= 3 2 lneint+ 1 2 ln(det(C))+Cte , and,byZ=∂s/∂M,wederive thecorrespondingmomenta: β= ∂s ∂e = 3 2eint , w=−gradps= 32eint p m . AsEquation(47),Equation(48)andtheexpressionsof s,βandwarenotaffectedbythearbitrary choiceofV0. • Step 7: link between z and ζ. As z is an extensivequantity, its value forN identical particles is zN=Nz. Planck’spotentialζbeingaspecificquantity,weclaimthat: ζ= zN Nm = z m = 1 2m ln(det(C))− 3 2m lnβ+ 1 2β ‖w‖2+Cte . By (16)and(17),weobtain the linear4-momentumΠ=(H,−pT)andCauchy’sstresses: H=ρ ( 3 2 kBT m + 1 2 ‖v‖2 ) , p=ρv, σ=−q1 R3 , where,by theexpressionof thepressure,werecover the idealgas law: q= ρ m kBT= N V kBT . Thefirstprincipleof thermodynamics (18) reads: ∂H ∂t +div (Hv−σv)=0, ρdv dt =−gradq, ∂ρ ∂t +div(ρv)=0 . 133