Page - 128 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 254 7. Planck’sPotentialofaContinuum Now, letusreveal the linkbetweenthepreviousrelativistic thermodynamicsofcontinuaandLie groupstatisticalmechanics in theclassicalGalileancontextand, tosimplify, inabsenceofgravitation. Inotherwords,howtodeduceT fromM andζ fromz?Wework insevensteps: • Step1: defining theorbit.Tobeginwith,weconsider themomentumasanGalileantensor, i.e., its componentsarmodifiedonlybytheactionofGalileantransformations. Inorder tocalculate the integral (10), theorbit isparameterizedthanks toamomentummap.Calculating the infinitesimal generatorsZ=(dC,dP)bydifferentiationof (12): dC= ( dτ0 dk ) , dP= ( 0 0 du j(d ) ) , where j(d )v= d ×v, thedualpairing(4) reads: μZ= l ·d −q ·du+p ·dk−edτ0 . (26) Themostgeneral formof theaction(6) itemizes in: p=Rp′+mu, q=R(q′−τ0p′)+m(k−τ0u) , (27) l=Rl′−u×(Rq′)+k×(Rp′)+mk×u , (28) e= e′+u ·(Rp′)+ 1 2 m‖u‖2 . (29) where theorbit invariantmoccuring in thesymplecticcocycleθ isphysically interpretedas the particlemass. In [3] (Theorem11.34,p. 151), thecocycleofGalileo’sgroup isderived froman explicit formof thesymplectic form.Analternativemethodtoobtain itusingonly theLiegroup structure isproposedin[2] (Theorem16.3,p. 329andTheorem17.4,p. 374). Taking intoaccount (3), the transformation law(6)of theGalileanmomentumtensorμ reads: K=K′P−1+Km(C,P), L=(PL′+CK′)P−1+Lm(C,P) , (30) whereKm andLm are thecomponentsofθ. Inparticular,onehas: Km(C,P)=m ( −1 2 ‖u‖2,uT ) . (31) • Step2: representing theorbit byequations.Toobtain them,wehavetodeterminea functionalbasis. Thefirst step is to calculate their number. Westart determining the isotropygroupofμ. The analysiswillberestrictedtomassiveparticles:m =0. Thecomponents p,q, l,ebeinggiven,we havetosolve the followingsystem: p=Rp+mu , (32) q=Rq−τ0(Rp+mu)+mk , (33) l=Rl−u×(Rq)+k×(Rp)+mk×u , (34) u ·(Rp)+ 1 2 m‖u‖2=0 , (35) withrespect toτ0,k,R,u. Owingto (32), theboostucanbeexpressedwithrespect to therotation Rby: u= 1 m (p−Rp) , (36) 128