Page - 31 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 to thekinetic energydue to thevelocityof its centreofmass, akinetic energydue to theparticle’s rotationarounditscentreofmass. Thereadercanconsult thebooksbySouriau[14]andMackey[18] where thekinetic theoryofpolyatomicgases isdiscussed. The pressure in the gas, denoted byΠ(b) because the notation P(b) is already used for the partition function, isdue to thechangeof linearmomentumof theparticleswhichoccursatacollision of theparticlewith thewallsof thevessel containing thegas (orwithaprobeusedtomeasure that pressure).Aclassicalargument in thekinetic theoryofgases (see forexample [52,53]) leads to Π(b)= 2 3 E(b) V = N Vb . This formula is thewellknown equationof stateofanidealmonoatomicgasrelatingthenumberof particlesbyunitofvolume, thepressureandthe temperature. Withb= 1 kT , theaboveexpressionsareexactly thoseused inclassical thermodynamics foran idealmonoatomicgas. 6.3.2.Classical IdealMonoatomicGas inaGravityField Letusnowassumethat thegas, contained inacylindricalvesselof sectionΣandlengthh,witha verticalaxis, is submittedto theverticalgravityfieldof intensitygdirecteddownwards.Wechoose Cartesiancoordinatesx,y,z, thezaxisbeingverticaldirectedupwards, thebottomof thevesselbeing in thehorizontal surface z= 0. TheHamiltonianof a freeparticle ofmassm, position and linear momentumvectors−→x (componentsx,y,z) and−→p (components px, py and pz) is 1 2m (p2x+p 2 y+p 2 z)+mgz . As in theprevioussectionweneglect thepartsof theHamiltonianof thegascorrespondingto collisionsbetweentheparticles,orbetweenaparticleandthewallsof thevessel. TheHamiltonianof thegas is therefore H= N ∑ i=1 ( 1 2mi (p2ix+p 2 iy+p 2 iz)+migzi ) . Calculationssimilar to thoseof theprevioussection leadto P(b)= N ∏ i=1 [ Σ ( 2πmi b )3/2 1−exp(−migbh) migb ] , ρb= 1 P(b) exp [ −b N ∑ i=1 (‖−→pi‖2 2mi +migzi )] . Theexpressionofρb showsthat the2N stochasticvectors −→xi and−→pi stillare independent,andthat foreach i∈{1,. . . ,N}, theprobability lawofeachstochasticvector−→pi is thesameas in theabsence ofgravity, for thesamevalueofb. Eachstochasticvector−→xi isnomoreuniformlydistributedinthe vessel containingthegas: itsprobabilitydensity ishigherat loweraltitudesz, andthisnonuniformity ismore important for theheavierparticles thanfor the lighterones. As in theprevioussection, the formulaegiven inProposition11allowthecalculationofE(b)and S(b).Weobserve thatE(b)nowincludes thepotentialenergyof thegas in thegravityfield, therefore shouldnomorebecalledthe internalenergyof thegas. 6.3.3. RelativisticMonoatomic IdealGas InaGalilean reference frame,weconsider a relativisticpointparticle of restmassm,moving at avelocity−→v . Wedenote by v themodulus of−→v andby c themodulus of thevelocity of light. 31