Page - 29 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 configuration space is a smoothmanifoldN, described in theLagrangian formalismbya smooth time-independenthyper-regularLagarangianL :TN→Ror, in theHamiltonianformalism,bythe associatedHamiltonianHL : T∗N→R. Let t → −−→ x(t)beamotionof that system,−→x0 = −−→ x(t0) and−→x1 = −−→ x(t0)betheconfigurationsof thesystemfor thatmotionat times t0 and t1. Thereexistsanother motion t →−−→x′(t)of thesystemforwhich−−−→x′(t0)=−→x1 and −−−→ x′(t1)=−→x0: since theequationsofmotion are invariantby timereversal, themotion t →−−→x′(t) isobtainedsimplybytakingas initial conditionat time t0 −−−→ x′(t0)= −−→ x(t1)and d −−→ x′(t) dt ∣∣∣ t=t0 =−d −−→ x(t) dt ∣∣∣ t=t1 .Anothermoreseriousargumentagainstakind of thermodynamicbehaviourofLagarangianorHamiltoniansystemsrestsonthe famousrecurrence theoremduetoPoincaré [51]. This theoremasserts indeedthatwhentheusefulpartof thephasespace of thesystemisofafinite totalmeasure,almostallpoints inanarbitrarilysmallopensubsetof the phasespaceare recurrent, i.e., themotionstartingof suchapointat time t0 repeatedlycrosses that opensubsetagainandagain, infinitelymanytimeswhen t→+∞. Letusnowconsider, insteadofperfectlydefinedstates, i.e.,points inphasespace,statisticalstates, andaskthequestion:Whenat time t= t0 aHamiltoniansystemonasymplecticmanifold (M,ω) is in astatisticalstategivenbysomeprobabilitymeasureofdensityρ0withrespect to theLiouvillemeasure λω,does its statistical stateconverge,when t→+∞, towards theprobabilitymeasureofaGibbsstate? This question shouldbemademoreprecise by specifyingwhatphysicalmeaninghas a statistical stateandinwhatmathematical senseastatistical statecanconverge towards theprobabilitymeasure ofaGibbsstate. Apositivepartial answerwasgivenbyLudwigBoltzmannwhen,developinghis kinetic theoryofgases,heprovedhis famous(butcontroversed)Êta theoremstatingthat theentropy of thestatistical stateofagasofsmallparticles isamonotonously increasingfunctionof time. This question, linkedwithtimeirreversibility inphysics, is still thesubjectof important researches,both byphysicists andbymathematicians. The reader is referred to thepaper [50]byBalian foramore thoroughdiscussionof thatquestion. 6.3. ExamplesofThermodynamicEquilibria 6.3.1.ClassicalMonoatomic IdealGas Inclassicalmechanics,adilutegascontainedinavesselat rest inaGalileanreference frameis mathematicallydescribedbyaHamiltoniansystemmadebya largenumberofverysmallmassive particles,which interactbyverybrief collisionsbetweenthemselvesorwith thewallsof thevessel, whosemotionsbetweentwocollisionsare free. Letusfirstassumethat theseparticlesarematerial points and that no external field is acting on them, other than that describing the interactions by collisionswith thewallsof thevessel. TheHamiltonianofoneparticle inapartof thephasespace inwhich itsmotion is free is simply 1 2m ‖−→p‖2= 1 2m (p21+p 2 2+p 2 3) , with −→p =m−→v , wherem is themassof theparticle,−→v itsvelocityvectorand−→p its linearmomentumvector (in the consideredGalileanreference frame), p1, p2 and p3 thecomponentsof −→p inafixedorhtonormalbasis of thephysical space. LetNbethe totalnumberofparticles,whichmaynothaveall thesamemass.Weusea integer i∈{1, 2, . . . , N} to label theparticlesanddenotebymi,−→xi ,−→vi ,−→pi themassandthevectorsposition, velocityandlinearmomentumof the i-thparticle. TheHamiltonianof thegas is therefore H= N ∑ i=1 1 2mi ‖−→pi‖2+ terms involvingthecollisionsbetweenparticlesandwith thewalls. 29