Page - 45 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 Remark18. 1. Thephysicalmeaningof theparameter εwhichappears in the expressionof thematrixb is clearlyapparent inexpression (10)of 〈Ji,b〉: ε=− 1 kT , Tbeing theabsolute temperatureandk theBoltzmann’s constant. 2. The same expression (10) shows that the relativemotionof thegaswith respect to themovingvessel in which it is contained, seen by an observer linked to thatmoving vessel, is described by aHamiltonian system inwhich the kinetic andpotential energies of the i-th particle are, respectively, 1 2mi ‖−→pi0‖2 and mifi( −→ri0). This result canbe obtained inanotherway: byderiving theHamiltonianwhichgoverns the relativemotionof amechanical systemwith respect toamoving frame, asusedby Jacobi [63] todetermine the famous Jacobi integral of the restrictedcircular three-bodyproblem(inwhich twobigplanetsmoveon concentric circular orbits aroundtheir commoncenter ofmass, anda thirdplanet ofnegligiblemassmoves in thegravitationalfield createdby the twobigplanets). 3. ThegeneralizedGibbs state of the system imposes to thevariousparts of the system, i.e., to thevarious particles, to be at the same temperature T =− 1 kε and to be statistically at rest in the samemoving reference frame. 7.3.5. ThreeExamples 1. Letusset−→ω =0and−→β =0. Themotionof themovingvessel containingthegas (withrespect to thesocalledfixedreference frame) isa translationataconstantvelocity −→ δ ε . The function fi( −→ri0) is thena constant. In themoving reference frame,which is an inertial frame,we recover the thermodynamicequilibriumstateofamonoatomicgasdiscussed inSection6.3.1. 2. Let us set now−→ω = 0 and−→δ = 0. The motion of the moving vessel containing the gas (with respect to the socalledfixed reference frame) isnowanuniformlyaccelerated translation, withacceleration −→ β ε . The function fi( −→ri0)nowis fi( −→ri0)=−→ri0 · −→ β ε . In the moving reference frame, which is no more inertial, we recover the thermodynamic equilibriumstateofamonoatomicgas inagravityfield−→g =− −→ β ε discussed inSection6.3.2. 3. Letusnowset−→ω =ω−→ez ,−→β = 0and−→δ = 0. Themotionof themovingvessel containing the gas (withrespect to thesocalledfixedreference frame) isnowarotationaroundthecoordinatez axisataconstantangularvelocity ω ε . The function fi( −→ri0) isnow fi( −→ri0)=−ω 2 2ε2 ‖−→ez ×−→ri0‖2 . The lengthΔ= ‖−→ez ×−→ri,0‖ is thedistancebetween the i-thparticle and theaxisof rotationof themoving frame (the coordinate zaxis). Moreover,wehave seen that ε= −1 kT . Therefore in thegeneralizedGibbs state, theprobabilitydensity ρi(b)ofpresenceof the i-thparticle in its symplecticmanifoldofmotionMi,ωi,withrespect to theLiouvillemeasureλωi, is ρi(b)= 1 Pi(b) exp (−〈Ji,b〉)=Constant ·exp(− 12mikT ‖−→pi0‖2+ mi2kT (ω ε )2 Δ2 ) . 45