Seite - 43 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 For each time t ∈ R, the action of A(τ) = A ( t ε ) maps the fixed reference frame (O,−→ex ,−→ey ,−→ez) onto another affineEuclidean reference frame ( O(t),−→ex(t),−→ey(t),−→ez(t) ) ,whichwecall themoving reference frame. Thevelocity and the acceleration of the relativemotionof themoving reference framewithrespect to thefixedreference frameisgiven,at time t=0,by the fundamentalvectorfield associatedto theelementbof theLiealgebraGof theGalileangroup:wesee thateachpoint inspace hasamotioncomposedofarotationaroundtheaxis throughOparallel to−→ω , atanangularvelocity ‖−→ω‖ ε , andsimultaneouslyauniformlyacceleratedmotionof translationatan initialvelocity −→ δ ε and acceleration −→ β ε . At time t, thevelocityandaccelerationof themovingreference framewithrespect to its instantaneouspositionat that timecanbedescribedinasimilarmanner,but insteadofO,−→ω ,−→β and −→ δ wemustuse thecorrespondingtransformedelementsbytheactionofA(τ)=A ( t ε ) . 7.3.4.AGasContainedinaMovingVessel Weconsideramechanical systemmadebyagasofNpointparticles, indexedby i∈{1,2, . . . ,N}, contained inavesselwithrigid,undeformablewalls,whosemotion inspace isgivenbytheaction of the one-parameter subgroupG1 of theGalilean groupmadeby the A ( t ε ) , with t ∈R, above described.Wedenotebymi, −→ri (t)and−→vi(t) themass,positionvectorandvelocityvector, respectively, of the i-thparticle at time t. Since themotionof thevessel containing thegas ispreciselygivenby theactionofG1, theboundaryconditions imposedto thesystemare invariantbythataction,which leaves invariant the evolution spaceof themechanical system, isHamiltonianandprojects ontoa Hamiltonian actionofG1 on the symplecticmanifold ofmotions of the system. We can therefore consider thegeneralizedGibbsstatesof thesystem,asdiscussed inSection7.1.Wemustevaluate the momentummap Jof thatactionanditscouplingwith theelementb∈G. As inSection6.3.1wewill neglect, for thatevaluation, thecontributionsof thecollisionsof theparticlesbetweenthemselvesand with thewallsof thevessel. Themomentummapcanthereforebeevaluatedas ifallparticleswere free,anditscoupling 〈J,b〉withb is thesum∑Ni=1〈Ji,b〉of themomentummap Jiof the i-thparticle, consideredas free,withb.Wehave 〈 Ji( −→ri ,t,−→vi ,mi),b 〉 =mi (−→ω ·(−→ri ×−→vi)−(−→ri − t−→vi) ·−→β +−→vi ·−→δ − 12‖−→vi‖2ε) . FollowingSouriau[14],Chapter IV,pp. 299–303,weobserve that 〈Ji,b〉 is invariantbytheaction ofG1.Wecanthereforedefine −→ri0, t0 and−→vi0 bysetting⎛⎜⎝ −→ri0 −→vi0 t0 1 1 0 ⎞⎟⎠= exp(−t ε b )⎛⎜⎝ −→ri −→vi t 1 1 0 ⎞⎟⎠ andwrite 〈 Ji( −→ri ,t,−→vi ,mi),b 〉 = 〈 Ji( −→ri0,t0,−→vi0,mi),b 〉 . Thevectors−→ri0 and−→vi0 havea clearphysicalmeaning: theyare thevectors−→ri and−→vi as seen byanobservermovingwiththemovingaffineEuclideanreference frame ( O(t),−→ex(t),−→ey(t),−→ez(t) ) . Moreover,ascanbeeasilyverified, t0=0ofcourse.Wethereforehave 〈 Ji( −→ri ,t,−→vi ,mi),b 〉 =mi (−→ω ·(−→ri0×−→vi0)−−→ri0 ·−→β +−→vi0 ·−→δ − 12‖−→vi0‖2ε) =mi (−→vi0 ·(−→ω×−→ri0+−→δ )−−→ri0 ·−→β − 12‖−→vi0‖2ε) 43