Page - 43 - in Differential Geometrical Theory of Statistics
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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
Differential Geometrical Theory of Statistics
- Title
- Differential Geometrical Theory of Statistics
- Authors
- Frédéric Barbaresco
- Frank Nielsen
- Editor
- MDPI
- Location
- Basel
- Date
- 2017
- Language
- English
- License
- CC BY-NC-ND 4.0
- ISBN
- 978-3-03842-425-3
- Size
- 17.0 x 24.4 cm
- Pages
- 476
- Keywords
- Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
- Categories
- Naturwissenschaften Physik