Page - 33 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 still has ameaningwhen the restmassm of the particle is 0. In an orthonormal reference frame, theequationsofmotionof aparticlewhosemotion ismathematicallydescribedbyaHamiltonian systemwithHamiltonian H= cp= c √ p12+p22+p32 are ⎧⎪⎪⎨⎪⎪⎩ dxi dt = ∂H ∂pi = c pi p dpi dt =−∂H ∂xi =0, (1≤ i≤3) , which shows that theparticlemovesona straight line at thevelocityof light c. It seems therefore reasonable todescribeagasofNphotons inavesselofvolumeV at rest inan inertial reference frame byaHamiltoniansystem,with theHamiltonian H= c N ∑ i=1 ‖−→pi‖= c N ∑ i=1 √ pi12+pi22+pi32 . With thesamenotationsas thoseused in theprevioussection, thepartition functionPof thegas takes thevalue, foreachb>0, P(b)= ∫ D exp ( −bc N ∑ i=1 ‖−→pi‖ ) N ∏ i=1 (d−→xid−→pi)= ( 8πV c3b3 )N . Theprobabilitydensityof thecorrespondingGibbsstate,withrespect to theLiouvillemeasure λω=∏Ni=1(d −→xid−→pi), is ρb= N ∏ i=1 ( c3b3 8πV ) exp(−bc‖−→pi‖) . This formulaappears in thebooksbySynge[54]andSouriau[14]. Physicistsconsider itasnot adequate for thedescriptionofagasofphotonscontained inavesselat thermalequilibriumbecause thenumberofphotons in thevessel, at anygiven temperature, cannotbe imposed: it results from theprocessesofabsorptionandemissionofphotonsbythewallsof thevessel,heatedat the imposed temperature,whichspontaneouslyoccur. Inotherwords, thisnumber isastochastic functionwhose probability lawis imposedbyNature. Souriauproposes, inhisbook [14], away toaccount for the possible variation of the number of photons. Instead of using the phase space of the systemof N massless relativisticparticlescontainedinavessel,heuses themanifoldofmotionsMN of that system (which is symplectomorphic to itsphase space). Heconsiders that themanifoldofmotionsMofa systemofphotons in thevessel is thedisjointunion M= ⋃ N∈N MN , of all themanifolds ofmotions MN of a systemof N massless relativistic particles in the vessel, for all possible values of N ∈ N. Fo N = 0 the manifold M0 is reduced to a singleton with, asLiouvillemeasure, themeasurewhichtakes thevalue1ontheonlynonemptypartof thatmanifold (thewholemanifoldM0).Moreover, sinceanyphotoncannotbedistinguishedfromanyotherphoton, twomotionsof the systemwith the samenumberNofmasslessparticleswhichonlydifferby the labelling of these particlesmust be considered as identical. Souriau considers too that since the numberNofphotons freelyadjusts itself, thevalueof theparameterb= 1 kT must,at thermodynamic equilibrium,be thesameinallpartsMN of thesystem,N∈N.Heuses too the fact thataphotoncan havetwodifferentstatesof (circular)polarization.Withtheseassumptions thevalueatanybof the partitionfunctionof thesystemis 33