Page - 44 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 wherewehaveusedthewellknownpropertyof themixedproduct −→ω ·(−→ri0×−→vi0)=−→vi0 ·(−→ω×−→ri0) . Letusset −→ U∗= 1 ε (−→ω×−→ri0+ −→ δ ) . Using−→vi0−−→U∗ and−→U∗ insteadof−→vi0,wecanwrite 〈 Ji( −→ri ,t,−→vi ,mi),b 〉 =miε ( −1 2 ‖−→vi0−−→U∗‖2−−→ri0 · −→ β ε + 1 2 ‖−→U∗‖2 ) . Weobserve that thevector −→ U∗onlydependson ε,−→ω ,−→δ ,whichareconstantsonce theelement b∈G is chosen,andof−→ri0,noton−→vi0. Ithasaclearphysicalmeaning: it is thevalueof thevelocityof themovingaffinereference framewithrespect to thefixedaffinereference frame,atpoint−→ri0 seenby anobserver linkedto themovingreference frame. Therefore thevector−→wi0=−→vi0−−→U∗ is the relative velocityof the i-thparticlewithrespect to themovingaffinereferenceframe,seenbyanobserver linked to themovingreference frame. The threecomponentsof−→ri0 andthe threecomponentsof−→pi0=mi−→wi0makeasystemofDarboux coordinatesonthesix-dimensional symplecticmanilold (Mi,ωi)ofmotionsof the i-thparticle.Witha slightabuseofnotations,wecanconsider themomentummap Ji asdefinedonthespaceofmotionsof the i-thparticle, insteadofbeingdefinedontheevolutionspaceof thisparticle, andwrite 〈 Ji( −→ri0,−→pi,0),b 〉 =−ε ( 1 2mi ‖−→pi0‖2+mifi(−→ri0) ) ,−→pi0=mi−→wi0=mi(−→vi0−−→U∗) , (10) and fi( −→ri0)=−→ri0 · −→ β ε − 1 2ε2 ‖−→ω×−→ri0‖2− −→ δ ε · (−→ω ε ×−→ri0 ) − 1 2ε2 ‖−→δ ‖2 . Equation(10) iswell suitedfor thedeterminationofgeneralizedGibbsstatesof thesystem.Let usset Pi(b)= ∫ Mi exp (−〈Ji,b〉)dλωi , EJi(b)= 1Pi(b) ∫ Mi Jiexp (−〈Ji,b〉)dλωi . The integrals in therighthandsidesof theseequalitiesconverge ifandonly if ε<0. Itmeans that thematrixbbelongstothesubsetΩoftheone-dimensionalLiealgebraoftheconsideredone-parameter subgroupG1oftheGalileangrouponwhichgeneralizedGibbsstatescanbedefinedifandonlyifε<0. Assumingthatconditionsatisfied,wecanuseDefinitions19. ThegeneralizedGibbsstatedetermined bybhasthesmoothdensity,withrespect totheLiouvillemeasure∏Ni=1λωi onthesymplecticmanifold ofmotionsΠNi=1(Mi,ωi), ρ(b)= N ∏ i=1 ρi(b) , withρi(b)= 1 Pi(b) exp (−〈Ji,b〉) . Thepartitionfunction,whoseexpression is P(b)= N ∏ i=1 Pi(b) , can be used,with the help of the formulae given in Section 7.2, to determine all the generalized thermodynamic functionsof thegas inageneralizedthermodynamicequilibriumstate. 44