Page - 30 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Interactionsof theparticleswith thewallsof thevessel are essential for allowing themotions ofparticles to remainconfined. Interactionsbetweenparticlesareessential toallowtheexchanges between themofenergyandmomentum,whichplayan importantpart in theevolutionwith time of thestatistical stateof thesystem. However it appears thatwhile these termsarevery important todeterminethesystem’sevolutionwith time, theycanbeneglected,whenthegas isdiluteenough, ifweonlywanttodeterminethefinalstatisticalstateof thesystem,onceathermodynamicequilibrium isestablished. TheHamiltonianusedwill thereforebe H= N ∑ i=1 1 2mi ‖−→pi‖2 . Thepartitionfunction is P(b)= ∫ M exp(−bH)dλω= ∫ D exp ( −b N ∑ i=1 1 2mi ‖−→p i‖2 ) N ∏ i=1 (d−→xid−→pi) , whereD is thedomainof the6N-dimensional spacespannedbythepositionvectors−→xi and linear momentumvectors−→pi of theparticles inwhich all the−→xi liewithin thevessel containing thegas. Aneasycalculation leads to P(b)=VN ( 2π b )3N/2 N ∏ i=1 (mi3/2)= N ∏ i=1 [ V ( 2πmi b )3/2] , whereV is thevolumeof thevesselwhichcontains thegas. Theprobabilitydensityof theGibbsstate associatedtob,withrespect to theLiouvillemeasure, therefore is ρb= N ∏ i=1 [ 1 V ( b 2πmi )3/2 exp (−b‖−→pi‖2 2mi )] . Weobserve thatρb is theproductof theprobabilitydensitiesρi,b for the i-thparticle ρi,b= 1 V ( b 2πmi )3/2 exp (−b‖−→pi‖2 2mi ) . The2N stochasticvectors−→xi and−→pi , i=1, . . . , Nare therefore independent. Theposition−→xi of the i-thparticle isuniformlydistributed in thevolumeof thevessel,while theprobabilitymeasureof its linearmomentum−→pi is theclassicalMaxwell–Boltzmannprobabilitydistributionof linearmomentum foran idealgasofparticlesofmassmi,firstobtainedbyMaxwell in1860.Moreoverwesee that the threecomponents pi1, pi2 and pi3 of the linearmomentum −→pi inanorhonormalbasisof thephysical spaceare independentstochasticvariables. Byusingtheformulaegiven inProposition11the internalenergyE(b)andtheentropyS(b)of thegascanbeeasilydeducedfromthepartitionfunctionP(b). Theirexpressionsare E(b)= 3N 2b , S(b)= 3 2 N ∑ i=1 logmi+ ( 3 2 ( 1+ log(2π) ) + logV ) N− 3N 2 logb . Wesee thateachof theNparticlespresent in thegashas thesamecontribution 3 2b to the internal energyE(b),whichdoesnotdependonthemassof theparticle. Evenmore: eachdegreeof freedom of eachparticle, i.e., eachof the the three components of the the linearmomentumof theparticle onthe threeaxesofanorthonormalbasis,has thesamecontribution 1 2b to the internalenergyE(b). This result isknowninphysicsunder thenameTheoremof equipartitionof the energyata thermodynamic equilibrium. It canbeeasilygeneralizedforpolyatomicgases, inwhichaparticlemaycarry, inaddition 30