Seite - 63 - in Differential Geometrical Theory of Statistics
Bild der Seite - 63 -
Text der Seite - 63 -
Entropy2016,18, 386
Foreachtemperatureβ, elementof theLiealgebrag, Souriauhas introducedatensor ÎËβ, equal to
thesumof thecocycle ÎËandtheheatcoboundary(with [.,.] Liebracket):
ÎËβ(Z1,Z2)= ÎË(Z1,Z2)+ âŠ
Q,adZ1(Z2) âŞ
with adZ1(Z2)= [Z1,Z2] (29)
This tensor ÎËβ has the followingproperties:
⢠ÎË(X,Y)= ăÎ(X),Yăwhere themapÎ is theone-cocycleof theLiealgebragwithvalues ingâ,
withÎ(X)=Teθ(X(e))where θ theone-cocycleof theLiegroupG. ÎË(X,Y) is constantonM
andthemap ÎË(X,Y) : gĂgâ isaskew-symmetricbilinear form,andiscalled the symplectic
cocycle ofLie algebragassociatedto themomentmap J,with the followingproperties:
ÎË(X,Y)= J[X,Y]â{JX, JY}with {., .} PoissonBracketand J theMomentMap (30)
ÎË([X,Y] ,Z)+ÎË([Y,Z] ,X)+ÎË([Z,X] ,Y)=0 (31)
where JX linear application from g to differential function on M: gâCâ(M,R)
Xâ JX and the
associateddifferentiableapplication J, calledmoment(um)map:
J :Mâ gâ suchthat JX(x)= ăJ(x),Xă , Xâ g
x â J(x) (32)
If insteadof Jwetake the followingmomentmap: Jâ˛(x)= J(x)+Q , xâM
whereQâ gâ is constant, thesymplecticcocycleθ is replacedbyθâ˛(g)= θ(g)+QâAdâgQ
where θâ˛âθ =QâAdâgQ is one-coboundaryofGwithvalues in gâ. Wealsohaveproperties
θ(g1g2)=Adâg1θ(g2)+θ(g1)andθ(e)=0.
⢠Thegeometric temperature,elementof thealgebrag, is in the thekernelof the tensor ÎËβ:
βâKer ÎËβ, suchthat ÎËβ(β,β)=0 , âβâ g (33)
⢠Thefollowingsymmetric tensorgβ,deďŹnedonallvaluesof adβ(.)= [β, .] ispositivedeďŹnite:
gβ([β,Z1] , [β,Z2])= ÎËβ(Z1, [β,Z2]) (34)
gβ([β,Z1] ,Z2)= ÎËβ(Z1,Z2) , âZ1â g,âZ2â Im (
adβ(.) )
(35)
gβ(Z1,Z2)âĽ0 , âZ1,Z2â Im (
adβ(.) )
(36)
where the linearmap adX â gl(g) is the adjoint representation of the Lie algebra gdeďŹned
by X,Yâ g(=TeG) â adX(Y)= [X,Y], and the co-adjoint representation of the Lie algebra
g the linearmap adâX â gl(gâ)which satisďŹes, for each Ξ â gâ and X,Y â g:ăadâX(Ξ),Yă =
ăΞ,âadX(Y)ăThese equations are universal, because they are not dependent on the symplectic
manifoldbutonlyonthedynamicalgroupG, thesymplecticcocycleÎ, the temperatureβand
theheatQ. SouriaucalledthismodelâLiegroups thermodynamicsâ.
Wewillgive themaintheoremofSouriaufor thisâLiegroupthermodynamicsâ:
Theorem1(SouriauTheoremofLieGroupThermodynamics).LetΊbe the largest openproper subset
of g,Lie algebraofG, such that
M eâăβ,U(Ξ)ădÎťand
M Ξ ¡eâăβ,U(Ξ)ădÎťare convergent integrals, this setΊ is
convexand is invariantunder every transformationAdg(.),where g âAdg(.) is theadjoint representationof
G, such thatAdg=Teigwith ig : h â ghgâ1 . Let a :GĂgââ gâ auniqueafďŹneaction a such that linear
63
Differential Geometrical Theory of Statistics
- Titel
- Differential Geometrical Theory of Statistics
- Autoren
- FrĂŠdĂŠric Barbaresco
- Frank Nielsen
- Herausgeber
- MDPI
- Ort
- Basel
- Datum
- 2017
- Sprache
- englisch
- Lizenz
- CC BY-NC-ND 4.0
- ISBN
- 978-3-03842-425-3
- Abmessungen
- 17.0 x 24.4 cm
- Seiten
- 476
- SchlagwĂśrter
- Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
- Kategorien
- Naturwissenschaften Physik