Koszul/Souriau Fisher Metric Spaces & Optimization by Maximum - - PowerPoint PPT Presentation
Koszul/Souriau Fisher Metric Spaces & Optimization by Maximum Entropy: Hessian Information Geometry, Lie Group Thermodynamics & Poincar'Marle'Souriau Equation Frdric
Frédéric BARBARESCO, THALES AIR SYSTEMS Senior Scientist & Advanced Studies Manager Advanced Radar Concepts Dept. 21/11/2014
2 / 2 /
Preamble
matrices of stationary time series (THPD matrix: Toeplitz Hermitian Positive Definite matrix)
Problem 2: How to define « Ordered Statistics » for covariance
matrices, knowing that there is no « total orders » for these matrices
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Toeplitz Definite Positive , det Hermitian ?? ) / ( ξ ξ ξ ξ ξ ξ ξ > = =
+
p
n
ξ ξ ξ ξ ξ ξ ξ ≤ ≤ ≤ > − ⇒ < .... : Order Global no But Definite Positive : Order Local
2 1 1 2 2 1
3 / 3 /
Preamble
Solution to Problem 1: Koszul/Souriau Solution of Maximum Entropy Entropy as Legendre Transform of Generalized Characteristic Function (Laplace
Transform on Convex Cone with Inner Product given by Cartan-Killing form)
Density of Probability as Souriau Covariant Solution of Maximum Entropy Solution to Problem 2: Frechet Median Barycenter Metric given by Koszul Hessian Geometry (hessian of Entropy) & Souriau Lie Group
Thermodynamics (metric defined by symplectic cocycle and Geometric temperature)
Geometric Median by Fréchet barycenter in Metric space, solved by Karcher Flow
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( ) ( )
− −
Θ − Θ −
=
*
ξ,
dξ e e p
ξ ξ ξ ξ
1 1
) (
) (
1 ξ −
Θ = x dx x d x ) ( ) ( Φ = Θ = ξ ξ ξ ξ ξ
ξ
d p
∫
Ω
=
*
) ( .
∫
Ω −
− = Φ
*
,
log ) ( ξ
ξ d
e x
x
( )
) (
,
y xad
ad Tr y x
θ
=
∑
=
=
n i i median
d Min Arg
1
) , ( ξ ξ ξ
ξ 2 , 2 2 2
*
log ) ( log ) ( x d e x p E x I
x x
∂ ∂ = ∂ ∂ − =
∫
Ω −
ξ ξ
ξ ξ
4 / 4 /
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Koszul'Vinberg Characteristic Function
François Massieu in 1869 demonstrated that some thermal
properties of physical systems could be derived from “characteristic functions”.
This idea was developed by Gibbs and Duhem with the notion of
potentials in thermodynamics, and introduced by Poincaré in probability.
We will study generalization of this concept by Jean-Louis Koszul in Mathematics Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex
cones will be presented as cornerstone of “Information Geometry” theory:
defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF
(their gradients defining mutually inverse diffeomorphisms)
Fisher Information Metrics as hessian of these dual functions. Koszul proved that these metrics are invariant by all
automorphisms of the convex cones.
5 / 5 /
Paul LEVY (general use of characteristic function in Probability) Henri POINCARE (Introduction of characteristic function Ψ in Probability)
ψ φ
φ
log
= = e ψ
1869 François MASSIEU (introduction of characteristic function in Thermodynamic: Gibbs-Duhem Potentials)
T T S / 1 . 1 ∂ ∂ − = φ φ
« je montre, dans ce mémoire, que toutes les propriétés d’un corps peuvent se déduire d’une fonction unique, que j’appelle la fonction caractéristique de ce corps»
Koszul Characteristic: Massieu/Poincaré/Levy/Balian
Roger BALIAN (metric for quantum states by hessian metric from Von- Neumann Entropy)
Roger Balian, 1986 DISSIPATION IN MANY-BODY SYSTEMS: A GEOMETRIC APPROACH BASED ON INFORMATION THEORY
X D X F D S ˆ , ˆ ) ˆ ( ) ˆ ( − = X Tr X F ˆ exp ln ) ˆ ( =
D d D d Tr S d ds ˆ ln . ˆ
2 2
= − =
6 / 6 /
Thermodynamic Duhem'Massieu Potentials
Thermodynamique », Annales Scientifiques de l’Ecole Normale Supérieure, 3e série, tome VIII, p. 231, 1891
“Nous avons fait de la Dynamique un cas particulier de la Thermodynamique, une
Science qui embrasse dans des principes communs tous les changements d’état des corps, aussi bien les changements de lieu que les changements de qualités physiques “
four scientists were credited by Duhem with having carried out
“the most important researches on that subject”:
function and its partial derivatives”
J.W. Gibbs had shown that Massieu’s functions “could play the role of potentials
in the determination of the states of equilibrium” in a given system.
generality” based on general duality concept in “Die thermodynamischen Beziehungen antithetisch entwickelt“, St. Petersburg 1885
W TS E G + − = Ω ) (
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Influence of Massieu on Poincaré
[M. Massieu showed that, if we make choice for independent variables of v and T or of p and T, there is a function, moreover unknown, of which three functions of variables, p, U and S in the first case, v, U and S in the second, can be deducted
[Because from functions of M. Massieu, we can deduct the other functions of variables, all the equations of the Thermodynamics can be written not so as to contain more than these functions and their derivatives; it will thus result from it, in certain cases, a large simplification. We shall see soon an important application of these functions.]
2nd edition of Poincaré Lecture on « Thermodynamics »
8 / 8 /
1908
Massieu Characteristic Function by Henri Poincaré
9 / 9 /
1912
Characteristic Function by Henri Poincaré in Probability
Probability with LAPLACE TRANSFORM not with FOURIER TRANSFORM
10 / 10 /
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Koszul'Vinberg Characteristic Function
Jean-Marie Souriau has extended the Characteristic Function in
Statistical Physics:
looking for other kinds of invariances through co-adjoint action of a group on its
momentum space
defining physical observables like energy, heat and momentum as pure
geometrical objects.
In covariant Souriau model, Gibbs equilibriums states are indexed
by a geometric parameter, the Geometric Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector (a vector valued temperature of Planck), giving to the metric tensor a null Lie derivative.
Fisher Information metric appears as the opposite of the derivative
Geometric Capacity or Specific Heat.
We will synthetize the analogies between both Koszul and Souriau
models, and will reduce their definitions to the exclusive “Inner Product” selection using symmetric bilinear “Cartan-Killing form” (introduced by Elie Cartan in 1894).
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Elie Cartan by Henri Poincaré
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« A l’Exception d’Henri Poincaré qui écrivit peu avant sa mort un rapport sur les travaux d’Elie Cartan à l’occasion de la candidature de celui-ci à la Sorbonne, les mathématiciens français ne voyaient pas l’importance de l’œuvre. » Paulette Libermann La géométrie différentielle d’Elie Cartan à Charles Ehresmann et André Lichnerowicz Géométrie au XXième siècle, HERMANN, 2005
12 / 12 /
Cartanian Filiation: J.L Koszul & J.M. Souriau
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2 2 2 2
log ) ( log x (x) x p E
x
∂ ∂ = ∂ ∂ −
Ω
ψ ξ
ξ
Ω ∈ ∀ = ∫
Ω − Ω
x d e x
x
) (
*
,
ξ ψ
ξ
( ) ( )
) ( . , ,
2 2 1 2 1
1 Z
ad Q Z Z f Z Z f
Z
+ =
β
[ ] ( ) ( ) ()
( ) [
]
,. . Im , , , , ,
2 1 2 1 2 1
β β
β β β
= ∈ ∀ ∈ ∀ = ad Z Z Z Z f Z Z g
∂ ∂ − = ∈ ∈
−
Q Q C f Ker e Temperatur
Capacity Heat
, , : ) ( β β β
β
∫
∈ − −
=
*
x x x
d e e p ξ ξ
ξ ξ , ,
/ ) ( G e D f to associated cocycle with ) )( ( θ θ =
( )
[ ] ( )
Map Moment Souriau with , , . ) ( ), (
2 1 2 1 , 2 , 1
µ µ ξ ξ σ Z Z f Z Z Z Z
M M
+ =
( )
( )
Involution Cartan , where ) , ( with ) ( , , g ad ad Tr y x B y x B y x
y x
∈ = − = η η β ψ ∂ ∂ − = ∂ ∂ =
Ω
Q x (x) I Fisher
2 2 log
Koszul Forms Koszul Characteristic Function Koszul Hessian Metric Jean-Louis Koszul
[ ] [ ] ( ) [ ] ( )
2 1 2 1
, , , , , Z Z f Z Z g β β β
β β
= (x) d
Ω
= ψ α log
Ω
= = ψ α log
2
d D g
Jean-Marie Souriau Souriau Moment Map Souriau Geometric Temperature/Heat Capacity Souriau/Fisher Metric from Symplectic Cocycle Elie CARTAN
Leçons sur les invariants intégraux, Hermann, 1922 La théorie des groupes finis et continus et la géométrie différentielle (Written by J. Leray)
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Jean'Marie Souriau and his Lie Group Thermodynamics
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J.M. Souriau, “Sur la Stabilité des Avions,” ONERA Publ., 62, vi+94, 1953 Engines could be positionned everywhere and a stable command could be defined J.M. Souriau, Calcul linéaire, P.U.F., Paris, 1964. Multilinear Algebra Le Verrier-Souriau Algorithm Computation of Matrix Characteristic Equation J.M. Souriau, Structure des systèmes dynamiques, Dunod, Paris, 1970 Symplectic Geometry Structure of Classical & Quantum Mechanics
Theorem
Theorem
cohomology of the action of the Galilean group (not for Poincaré Group in Relativity) J.M. Souriau, Définition covariante des équilibres thermodynamiques, Supp. Nuov. Cimento,1,4, p.203-216, 1966 J.M. Souriau, Thermodynamique et géométrie. In diff. Geo. methods in mathematical physics, II,
Lie Group Thermodynamics (Gibbs Equilibrium is not covariant by Gallileo/Poincaré Groups)
AIRBUS BOEING J.M. Souriau, Les groupes comme universaux, Géométrie au XXième siècle, Hermann, 2005 J.M. Souriau, Grammaire de la Nature, 2007 Souriau theorem revisited by AIRBUS/BOEING
« La masse totale d’un système dynamique isolé est la classe de cohomologie du défaut d’équivariance de l’application moment »
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Souriau Books
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http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm Chapter 4 on « Statistical Mechanics » http://www.jmsouriau.com/Publications/JMSouriau-SSD-Ch4.pdf
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Souriau Model for Gaussian Law
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J.M. Souriau, Structure des systèmes dynamiques, Chapitre 4 « Mécanique Statistique »
Souriau « Geometric Temperature » idea come from his book « Calcul Linéaire » (chap. on «Multilinear Algebra »
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Souriau Mechanical Statistics
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J.M. Souriau, Structure des systèmes dynamiques, Chapitre 4 « Mécanique Statistique »
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Covariant Souriau Statistical Mechanics
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J.M. Souriau, Structure des systèmes dynamiques, Chapitre 4 « Mécanique Statistique »
Classical Gibbs Equilibrium is not covariant according to Dynamic Group of Mechanics (Gallileo Group and Poincaré Group) !!!
18 / 18 /
!"
Hirohiko Shima Book, « Geometry of Hessian
Structures », world Scientific Publishing 2007, dedicated to Jean-Louis Koszul
Hirohiko Shima Keynote Talk at GSI’13 http://www.see.asso.fr/file/5104/download/9914
http://repmus.ircam.fr/_media/brillouin/ressources/une
information.pdf
Jean-Louis Koszul J.L. Koszul, « Sur la forme hermitienne canonique des espaces homogènes complexes », Canad. J. Math. 7, pp. 562-576., 1955 J.L. Koszul, « Domaines bornées homogènes et orbites de groupes de transformations affines », Bull. Soc. Math. France 89, pp. 515-533., 1961 J.L. Koszul, « Ouverts convexes homogènes des espaces affines », Math. Z. 79, pp. 254-259., 1962 J.L. Koszul, « Variétés localement plates et convexité », Osaka J. Maht. 2,
J.L. Koszul, « Déformations des variétés localement plates », .Ann Inst Fourier, 18 , 103-114., 1968
Jean'Louis Koszul and his Hessian Geometry
20 / 20 / Projective Legendre Duality and Koszul Characteristic Function
LEGENDRE TRANSFORM FOURIER/LAPLACE TRANSFORM
ENTROPY= LEGENDRE(- LOG[LAPLACE]) Ψ = Φ − =
2 2 log
d d g
INFORMATION GEOMETRY METRIC
S d d g
2 * 2 *
= Ψ =
Ω −
− = Φ − = Ψ
*
,
log ) ( log ) ( dy e x x
y x
) ( , ) (
* * *
x x x x Ψ − = Ψ ξ ξ ξ d p p
x x
Ω
− = Ψ
*
) ( log ) (
* Φ(x) x,ξ
ξ,x x
e dξ e e p
*
+ − − −
= =
/ ) (ξ
Ω
=
*
) ( .
*
ξ ξ ξ d p x
x Legendre Transform of minus logarithm of characteristic function (Laplace transform) = ENTROPY !!!
ds2=d2ENTROPY ds2=-d2LOG[LAPLACE]
21 / 21 / Koszul'Vinberg Characteristic Function/Metric of convex cone
J.L. Koszul and E. Vinberg have introduced an affinely invariant
Hessian metric on a sharp convex cone through its characteristic function.
dimension on (a convex cone is sharp if it does not contain any full straight line).
Let the Lebesgue measure on dual space of , the following
integral: is called the Koszul-Vinberg characteristic function
Ω E R
*
Ω Ω ξ d
*
E E Ω ∈ ∀ = ∫
Ω − Ω
x d e x
x
) (
*
,
ξ ψ
ξ
22 / 22 / Koszul'Vinberg Characteristic Function/Metric of convex cone
We can define a diffeomorphism by:
with
When the cone
is symmetric, the map is a bijection and an isometry with a unique fixed point (the manifold is a Riemannian Symmetric Space given by this isometry): , and
Ω
= ψ log
2
d g
( )
log log 2 1 log log ) ( log
2 u 2 2 2
− + = = dudv dudv d d du du d du d x d
v u v u v u u u u
ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ
) ( log
*
x d x
x Ω
− = − = ψ α ) ( ) ( ), ( tu x f dt d x f D u x df
t u
+ = =
=
Ω
x
x α − =
*
x x =
* *)
( n x x =
*
, cste x x =
Ω Ω
) ( ) (
*
*
ψ ψ
*
x
n y x y y x = Ω ∈ = , , / ) ( min arg
* *
ψ
*
x
n y x y = Ω ∈ , ,
* *
Ω
Ω − Ω −
=
* *
, , *
/ . ξ ξ ξ
ξ ξ
d e d e x
x x
23 / 23 /
Koszul Entropy via Legendre Transform
we can deduce “Koszul Entropy” defined as Legendre Transform of minus
logarithm of Koszul-Vinberg characteristic function : with and where
Demonstration:
we set Using and we can write: and
) ( , ) (
* * *
x x x x Φ − = Φ Φ =
x
D x*
*
*Φ
=
x
D x ) ( log ) ( x x
Ω
− = Φ ψ Ω ∈ ∀ = ∫
Ω − Ω
x d e x
x
) (
*
,
ξ ψ
ξ
Ω − Ω −
=
* *
, , *
/ . ξ ξ ξ
ξ ξ
d e d e x
x x
Ω − Ω − Ω
− = = −
* *
, , *
/ , ) ( log , ξ ξ ξ ψ
ξ ξ
d e d e h x d h x
x x h
Ω − Ω − −
= −
* *
, , , *
/ . log , ξ ξ
ξ ξ ξ
d e d e e x x
x x x
Ω − Ω − − Ω − Ω − Ω − Ω − Ω − −
− = Φ + − = Φ
* * * * * * *
, , , , , * * , , , , * *
/ . log log . ) ( log / . log ) ( ξ ξ ξ ξ ξ ξ ξ
ξ ξ ξ ξ ξ ξ ξ ξ ξ
d e d e e d e d e x d e d e d e e x
x x x x x x x x x
24 / 24 /
Koszul'Vinberg Characteristic Function Legendre Transform
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− = Φ = − = Φ − = Φ − = Φ + − = Φ − = Φ
Ω Ω − − Ω − − Ω Ω − − Ω Ω − − − Ω Ω − − Ω − Ω Ω − − − Ω − Ω − Ω − − Ω − Ω − Ω − Ω − Ω − −
* * * * * * * * * * * * * * * * * * * *
, , , , * * , , , , , , , , * * , , , , * * , , , , , * * , , , , * * *
log . ) ( 1 with . log . log ) ( . log log ) ( / . log log . ) ( log / . log ) ( , ) ( ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
d d e e d e e x d d e e d d e e e d d e e d e x d d e e e d e x d e d e e d e d e x d e d e d e e x x x x
x x x x x x x x x x x x x x x x x x x x x x x x x
25 / 25 /
[ ] ( )
[ ] [ ]
∫
Ω
Φ = Φ ≥ Φ ⇒ Φ − ≥ Φ Φ ≤ Φ ⇒ Φ
* * *
) ( ) ( ) ( ) ( ) ( , ) ( : Transform Legendre ) ( conv. : Ineq. Jensen
* * * * * * * *
ξ ξ ξ ξ ξ ξ E d p x x x x x E E
x
Koszul Entropy via Legendre Transform
We can then consider this Legendre transform as an entropy, that we could
named “Koszul Entropy”: With and
ξ ξ ξ ξ ξ ξ
ξ ξ ξ ξ
d p p d d e e d e e
x x x x x x
Ω Ω − − Ω Ω − −
− = − = Φ
* * * *
) ( log ) ( log
, , , , * Φ(x) x,ξ dξ e x,ξ
ξ,x x
e e dξ e e p
*
*
+ − ∫ − − − −
= = =
−
log
/ ) (ξ
( ) ξ
ξ ξ ξ ξ ξ d e d ξ.e d p Φ D x
ξ Φ
x,ξ x x
* *
− Ω + − Ω
= = = =
* *
. ) ( .
* [ ]
1 log ) ( ) ( log log ) (
* * * * * * *
) ( ) ( ) ( ) ( ,
= ⇒ − Φ = Φ − = − = Φ
∫ ∫ ∫ ∫
Ω Φ − Ω Φ − Ω Φ + Φ − Ω −
ξ ξ ξ ξ
ξ ξ ξ ξ
d e d e x x d e d e x
x x
[ ]
[ ] ( )
ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
ξ ξ
E E d p d p x d p d p p e e p
x x x x x x x x * * * * * * * * ) ( ) ( ,
) (
) ( . ) ( ) ( if
and if ) ( ) ( ) ( ) ( ) ( log ) ( log log ) ( log
* * * * *
Φ = Φ Φ = Φ Φ = Φ = − ⇒ Φ − = = =
∫ ∫ ∫ ∫
Ω Ω Ω Ω Φ − Φ + −
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Barycentre & Koszul Entropy
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*
*
) ( , ) (
* * *
x x x Sup x
x
Φ − = Φ
( ) ξ
ξ ξ ξ ξ ξ d e d ξ.e d p Φ D x
ξ Φ
x,ξ x x
* *
− Ω + − Ω
= = = =
* *
. ) ( .
*
Ω Ω
* *
* *
x x
Φ(x) x,ξ dξ e x,ξ
ξ,x x
e e dξ e e p
*
*
+ − ∫ − − − −
= = =
−
log
/ ) (ξ Barycenter of Koszul Entropy = Koszul Entropy of Barycenter
27 / 27 /
Koszul metric & Fisher Metric
To make the link with Fisher metric given by matrix
, we can
is given by:
We could then deduce the close interrelation between Fisher metric
and hessian of Koszul-Vinberg characteristic logarithm.
2 2 2 2 2 2 2 2 2 2 2 2 *
log ) ( log ) ( , ) ( ) ( log , ) ( ) ( ) ( log x (x) x (x) x p E x I x (x) x x x x p x x p
x x x
∂ ∂ = ∂ Φ ∂ − = ∂ ∂ − = ⇒ ∂ Φ ∂ = ∂ − Φ ∂ = ∂ ∂ − Φ = Φ − =
Ω
ψ ξ ξ ξ ξ ξ ξ
ξ 2 2 2 2
log ) ( log ) ( x (x) x p E x I
x
∂ ∂ = ∂ ∂ − =
Ω
ψ ξ
ξ
FISHER METRIC (Information Geometry) = KOSZUL HESSIAN METRIC (Hessian Geometry) ) (x I ) ( log ξ
x
p
28 / 28 /
Koszul Metric and Fisher Metric as Variance
We can also observed that the Fisher metric or hessian of KVCF
logarithm is related to the variance of :
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ξ
− − −
− = ∂ ∂ ⇒ =
* * *
e dξ e x (x) Ψ dξ e (x) Ψ . 1 log log log ξ + − − = ∂ ∂
− − − − 2 2 2 2 2
. . . 1 log
* * * *
e dξ e dξ e dξ e x (x) Ψ ξ ξ
2 2 2 2 2 2
) ( . ) ( . . . log − = − = ∂ ∂
− − − −
* * * * * *
ξ,x
ξ,x
p dξ p dξ dξ e e dξ dξ e e x (x) Ψ ξ ξ ξ ξ ξ ξ
) ( log ) ( log ) (
2 2 2 2 2 2
ξ ξ ξ ψ ξ
ξ ξ ξ
Var E E x (x) x p E x I
x
= − = ∂ ∂ = ∂ ∂ − =
Ω
29 / 29 /
New Definition of Maximum Entropy Density
How to replace by mean value of , in :
with
Legendre Transform will do this inversion by inversing We then observe that Koszul Entropy provides density of
Maximum Entropy with this general definition of density: with and where and
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( ) ( )
− −
Θ − Θ −
=
*
ξ,
dξ e e p
ξ ξ ξ ξ
1 1
) ( ) (
1 ξ −
Θ = x dx x d x ) ( ) ( Φ = Θ = ξ ξ ξ ξ ξ
ξ
d p
Ω
=
*
) ( .
Ω −
− = Φ
*
,
log ) ( ξ
ξ d
e x
x
x ξ ) (
*
x = ξ
Ω − −
=
*
, ,
) ( ξ ξ
ξ ξ
d e e p
x x x
Ω
=
*
) ( . ξ ξ ξ ξ d px dx x d ) ( Φ = ξ
30 / 30 /
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Cartan'Killing Form and Invariant Inner Product
It is not possible to define an ad(g)-invariant inner product for any
two elements of a Lie Algebra, but a symmetric bilinear form, called “Cartan-Killing form”, could be introduced (Elie Cartan PhD 1894)
This form is defined according to the adjoint endomorphism
Lie bracket:
The trace of the composition of two such endomorphisms defines
a bilinear form, the Cartan-Killing form:
The Cartan-Killing form is symmetric: and has the associativity property: given by:
x
ad g x g
y x y adx , ) ( =
y xad
ad Tr y x B = ) , ( ) , ( ) , ( x y B y x B =
z y x B z y x B , , , , =
[ ]
z y x B ad ad ad Tr z y x B ad ad ad Tr ad ad Tr z y x B
z y x z y x z y x
, , , , , , , ,
,
= = = =
31 / 31 /
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Cartan'Killing Form and Invariant Inner Product
Elie Cartan has proved that if is a simple Lie algebra (the Killing
form is non-degenerate) then any invariant symmetric bilinear form
The Cartan-Killing form is invariant under automorphisms
To prove this invariance, we have to consider:
Then
g g ) (g Aut ∈ σ g
y x B y x B , ) ( ), ( = σ σ
1 ) ( 1
rewritten ), ( ) ( , ) ( ) ( ), ( ,
− −
= = ⇒ = = σ σ σ σ σ σ σ σ σ
σ
x
ad ad z x z x y z y x y x
) , ( ) ( ), ( ) ( ), (
1 ) ( ) (
y x B ad ad Tr y x B ad ad Tr ad ad Tr y x B
y x y x y x
= = = =
−
σ σ σ σ σ σ
σ σ
32 / 32 /
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Cartan'Killing Form and Invariant Inner Product
A natural G-invariant inner product could be introduced by Cartan- Killing form:
Cartan Generating Inner Product: The following Inner product
defined by Cartan-Killing form is invariant by automorphisms of the algebra where is a Cartan involution (An involution on is a Lie algebra automorphism
identity).
) ( , , y x B y x θ − = g ∈ θ g θ g
33 / 33 /
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From Cartan'Killing Form to Koszul Information Metric
Involution Cartan , with ) ( , , Form Killing Cartan ) , ( g y x B y x ad ad Tr y x B
y x
∈ − = − = θ θ Ω ∈ ∀ − = Φ
Ω −
x d e x
x
log ) ( Function stic Characteri Koszul
*
,
ξ
ξ
− − Ω Ω
= = − = Φ Φ − = Φ
*
ξ,x x x x x
dξ e e p d p x d p p x x x x x ) ( Density Koszul ) ( . with ) ( log ) ( ) ( ) ( , ) ( Entropy Koszul
* *
* * * * * *
ξ ξ ξ ξ ξ ξ ξ
2 , 2 2 2 2 2
*
log ) ( ) ( log ) ( Metric Koszul x d e x (x) x I x p E x I
x x
∂ ∂ = ∂ Φ ∂ − == ∂ ∂ − =
Ω −
ξ ξ
ξ ξ
34 / 34 /
Koszul Density: Application for SPD matrices
We can then named this new density as “Koszul Density”:
With
Φ(x) x,ξ dξ e x,ξ
ξ,x x
e e dξ e e p
*
*
+ − ∫ − − − −
= = =
−
log
/ ) (ξ
( ) ξ
ξ ξ ξ ξ ξ d e d ξ.e d p Φ D x
ξ Φ
x,ξ x x
* *
− Ω + − Ω
= = = =
* *
. ) ( .
*
( )
( )
Ω − − + + −
= = =
− * 1
). ( . with det ) (
1 det log 2 1
ξ ξ ξ ξ ξ α ξ
ξ ξ α α ξ
d p e e p
x Tr x n x Tr x
+ = + = − = = = = ∈ ∀ =
− Ω + − Ω = Ω = Ω − Ω
1 * 2 1 dual
) ( , ,
2 1 det log 2 1 log ) ( det ) ( ) ( , , ,
* *
x n x d n d x I x d e x R Sym y x xy Tr y x
n n xy Tr y x x n
ψ ξ ψ ξ ψ
ξ
35 / 35 /
Relation of Koszul density with Maximum Entropy Principle
The density from Maximum Entropy Principle is given by: If we take such that: Then by using the fact that with equality if and
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= = −
Ω Ω Ω * (.)
* * *
) ( . 1 ) ( such ) ( log ) ( x d p d p d p p Max
x x x x px
ξ ξ ξ ξ ξ ξ ξ ξ
∫ − − − −
−
= =
*
*
dξ e x,ξ
ξ,x x
e dξ e e q
log
/ ) (ξ − − = = = =
Ω − ∫ − − − −
− *
, log
log , log ) ( log 1 / ). ( ξ ξ ξ ξ
ξ d
e x e q dξ e dξ e dξ q
x dξ e x,ξ x
*
* * *
1
1 log
−
− ≥ x x 1 = x ξ ξ ξ ξ ξ ξ ξ ξ d p q p d q p p
x x x x x x
Ω Ω
− − ≤ −
* *
) ( ) ( 1 ) ( ) ( ) ( log ) (
36 / 36 /
Relation of Koszul density with Maximum Entropy Principle
We can then observe that:
because
We can then deduce that: If we develop the last inequality, using expression of :
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) ( ) ( ) ( ) ( 1 ) (
* * *
= − = −
Ω Ω Ω
ξ ξ ξ ξ ξ ξ ξ ξ d q d p d p q p
x x x x x
1 ) ( ) (
* *
= = ∫
Ω Ω
ξ ξ ξ ξ d q d p
x x
ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ d q p d p p d q p p
x x x x x x x
Ω Ω Ω
− ≤ − ⇒ ≤ −
* * *
) ( log ) ( ) ( log ) ( ) ( ) ( log ) ( ) (ξ
x
q ξ ξ ξ ξ ξ ξ ξ
ξ
d d e x p d p p
x x x x
Ω Ω − Ω
− − − ≤ −
* * *
,
log , ) ( ) ( log ) (
Ω − Ω Ω
+ ≤ −
* * *
,
log ) ( . , ) ( log ) ( ξ ξ ξ ξ ξ ξ ξ
ξ d
e d p x d p p
x x x x
) ( , ) ( log ) (
*
*
x x x d p p
x x
Φ − ≤ − ∫
Ω
ξ ξ ξ ) ( ) ( log ) (
* *
*
x d p p
x x
Φ ≤ − ∫
Ω
ξ ξ ξ
39 / 39 /
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Covariant Definition of Thermodynamic Equilibriums
Jean-Marie Souriau , student of Elie Cartan at ENS Ulm in 1946,
has
given a covariant definition of thermodynamic equilibriums formulated statistical mechanics and thermodynamics in the framework
by use of symplectic moments and distribution-tensor concepts, giving a geometric status for:
Temperature Heat Entropy This work has been extended by C. Vallée & G. de Saxcé, P.
Iglésias and F. Dubois.
40 / 40 /
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Covariant Definition of Thermodynamic Equilibriums
The first general definition of the “moment map” (constant of the
motion for dynamical systems) was introduced by Jean-Marie Souriau during 1970’s
with geometric generalization such earlier notions as the Hamiltonian and the
invariant theorem of Emmy Noether describing the connection between symmetries and invariants (it is the moment map for a one-dimensional Lie group
In symplectic geometry the analog of Noether’s theorem is the
statement that the moment map of a Hamiltonian action which preserves a given time evolution is itself conserved by this time evolution.
The conservation of the moment of a Hamilotnian action was called
by Souriau the “Symplectic or Geometric Noether theorem”
considering phases space as symplectic manifold, cotangent fiber of
configuration space with canonical symplectic form, if Hamiltonian has Lie algebra, moment map is constant along system integral curves.
Noether theorem is obtained by considering independently each
component of moment map
41 / 41 /
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Souriau Covariant Model
Let be a differentiable manifold with a continuous positive
density and let E a finite vector space and a continuous function defined on with values in E. A continuous positive function solution of this problem with respect to calculus of variations:
is given by:
and and
Entropy can be stationary only if
there exist a scalar and an element belonging to the dual of E.
Entropy appears naturally as Legendre transform of :
M ω d ) (ξ U M ) (ξ p = = − =
Q d p U d p d p p s ArgMin
M M M p
ω ξ ξ ω ξ ω ξ ξ
ξ
) ( ) ( 1 ) ( such that ) ( log ) (
) ( ) ( . ) (
) (
ξ β β
ξ
U
e p
− Φ
=
−
− = Φ
M U
d e ω β
ξ β ) ( .
log ) (
− −
=
M U M U
d e d e U Q ω ω ξ
ξ β ξ β ) ( . ) ( .
) (
− =
M
d p p s ω ξ ξ ) ( log ) ( Φ β Φ ) ( . ) ( β β Φ − = Q Q s
42 / 42 /
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Souriau Covariant Model
This value is a strict minimum of s, and the
equation: has a maximum of one solution for each value of Q.
The function is differentiable and we can write
and identifying E with its dual:
Uniform convergence of
proves that and that is convex.
Then, and are mutually inverse and differentiable,
where .
Identifying E with its bidual:
) ( . ) ( β β Φ − = Q Q s
− −
=
M U M U
d e d e U Q ω ω ξ
ξ β ξ β ) ( . ) ( .
) ( ) (β Φ Q d d . β = Φ β ∂ Φ ∂ = Q
−
⊗
M U
d e U U ω ξ ξ
ξ β ) ( .
) ( ) (
2 2
> ∂ Φ ∂ − β ) (β Φ − ) (β Q ) (Q β dQ ds . β = Q s ∂ ∂ = β
43 / 43 /
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Souriau'Gibbs Canonical Ensemble
In statistical mechanics, a canonical ensemble is the statistical
ensemble that is used to represent the possible states of a mechanical system that is being maintained in thermodynamic equilibrium.
Souriau has defined this Gibbs canonical ensemble on
Symplectic manifold M for a Lie group action on M
The seminal idea of Lagrange was to consider that a statistical
state is simply a probability measure on the manifold of motions
In Jean-Marie Souriau approach, one movement of a dynamical
system (classical state) is a point on manifold of movements.
For statistical mechanics, the movement variable is replaced by
a random variable where a statistical state is probability law on this manifold.
44 / 44 /
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Souriau'Gibbs Canonical Ensemble
Symplectic manifolds have a completely continuous measure,
invariant by diffeomorphisms: the Liouville measure
All statistical states will be the product of Liouville measure by the
scalar function given by the generalized partition function defined by the generalized energy (the moment that is defined in dual of Lie Algebra of this dynamical group) and the geometric temperature , where is a normalizing constant such the mass of probability is equal to 1,
Jean-Marie Souriau generalizes the Gibbs equilibrium state to all
Symplectic manifolds that have a dynamical group.
To ensure that all integrals could converge, the canonical Gibbs
ensemble is the largest open proper subset (in Lie algebra) where these integrals are convergent. This canonical Gibbs ensemble is convex.
the mean value of the energy a generalization of heat capacity Entropy by Legendre transform
λ
U
e
. β − Φ
U β Φ
−
− = Φ
M Ud
e ω
β.
log
β ∂ Φ ∂ = Q β ∂ ∂ − = Q K Φ − = Q s . β
45 / 45 /
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Souriau Lie Group Thermodynamic
For the group of time translation, this is the classical thermodynamic Souriau has observed that if we apply this theory for non-
commutative group (Galileo or Poincaré groups):
the symmetry has been broken Classical Gibbs equilibrium states are no longer invariant by this group This symmetry breaking provides new equations, discovered by
Jean-Marie Souriau.
For each temperature , Jean-Marie Souriau has introduced a
tensor , equal to the sum of cocycle and Heat coboundary (with [.,.] Lie bracket):
This tensor has the following properties:
definite:
β
β
f f
2 1 2 2 2 1 2 1
, ) ( with ) ( . , ,
1 1
Z Z Z ad Z ad Q Z Z f Z Z f
Z Z
= + =
β β
f
β
f
β
β f Ker ∈
β
g (.)
β
ad
2 1 2 1
, , , , , Z Z f Z Z g β β β
β β
=
46 / 46 /
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Souriau Lie Group Thermodynamic
Souriau equations are universal, because they are not dependent
the dynamical group G its symplectic cocycle the temperature the heat Souriau called this model “Lie Groups Thermodynamics”: “Peut-être cette thermodynamique des groups de Lie a-t-elle un intérêt
mathématique”.
For dynamic Galileo group (rotation and translation) with only one
axe of rotation:
this thermodynamic theory is the theory of centrifuge where the temperature
vector dimension is equal to 2 (sub-group of invariance of size 2)
these 2 dimensions for vector-valued temperature are “thermic conduction” and
“viscosity”, unifying “heat conduction” and “viscosity”.
2 1 2 2 2 1 2 1
, ) ( with ) ( . , ,
1 1
Z Z Z ad Z ad Q Z Z f Z Z f
Z Z
= + =
β β
β f Ker ∈
2 1 2 1
, , , , , Z Z f Z Z g β β β
β β
= β f Q
47 / 47 /
The Galileo group of an observer is the group of affine maps Matrix Form of Gallileo Group Symplectic cocycles of the Galilean group: V. Bargmann (Ann.
cohomology space of the Galilean group is one-dimensional.
Lie Algebra of Gallileo Group
Dynamic Gallileo Group
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= 1 1 1 1 ' ' t x e w u R t x
3 ( , and , ' . . '
3
SO R R e R w u x e t t w t u x R x ∈ ∈ ∈ + = + + =
+
× ∈ ∈ ∈
+
x x so R R
ω ε γ η ε γ η ω : ) 3 ( , and ,
3
48 / 48 /
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Fundamental Souriau Theorem
Let be the largest open proper subset of , Lie algebra of G,
such that and are convergent integrals
this set is convex and is invariant under every transformation ,
where is the adjoint representation of G, with:
and is the image under of the probability measure .
Rmq: is changed but with linear dependence to , then Fisher
metric is unchanged by dynamical group:
Ω
− M U
d e ω
ξ β ) ( .
− M U
d e ω ξ
ξ β ) ( .
. Ω
a ) (β β
→
) ( .
1
β θ β θ
a a + Φ = − Φ → Φ
−
s s →
) ( ) ( Q a a Q a Q
θ = + → ) (ς ς
+
→
M
a θ ) (ς
+ M
a
M
a ς Φ β
β β β β θ β I a a I = ∂ Φ ∂ − = ∂ − Φ ∂ − =
− 2 2 2 1 2
) (
49 / 49 /
Fundamental Souriau Theorem
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*
Ω
R
β Φ
) ( . β θ β
a + Φ
β β Φ − = Q Q s . Q ) (Q a
θ
β ) (β
ς ) (ς
+ M
a e a G
50 / 50 /
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Souriau Lie Group definition of Fisher Metric
Let be the derivative of (symplectic cocycle of G) at the identity
element and let us define: Then
such that: and that gives the structure of a positive Euclidean space
f θ
2 1 2 2 2 1 2 1
, ) ( with ) ( . , , ,
1 1
Z Z Z ad Z ad Q Z Z f Z Z f
Z Z
= + = Ω ∈ ∀
β
β
β
f
Ω ∈ ∀ = β β β
β
, , f
β
g
β
β
., (.) = ad
. Im , , , , ,
2 1 2 1 2 1 β β β β
ad Z Z Z Z f Z Z g ∈ ∀ ∈ ∀ =
. Im , , ,
2 1 2 1 β β
ad Z Z Z Z g ∈ ∀ ≥
51 / 51 /
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Koszul Information Geometry, Souriau Lie Group Thermodynamics
Koszul Information Geometry Model Souriau Lie Groups Thermodynamics Model Characteristic function
Ω ∈ ∀ − = Φ
∫
Ω −
x d e x
x
log ) (
*,
ξ
ξ
∀ − = Φ
∫
−
β ω β
ξ β
log ) (
) ( . M U
d e
Entropy
ξ ξ ξ d p p x
x x
∫
Ω
− = Φ
*) ( log ) ( ) (
* *
∫
− =
M
d p p s ω ξ ξ ) ( log ) (
Legendre Transform
) ( , ) (
* * *
x x x x Φ − = Φ ) ( . ) ( β β Φ − = Q Q s
Density
∫
− − + −
= =
*ξ,x x Φ(x) x,ξ x
dξ e e p e p ) ( ) ( ξ ξ
∫
− − Φ + −
= =
M U U U
d e e p e p ω ξ ξ
ξ β ξ β β β ξ β β ) ( . ) ( . ) ( ) ( .
) ( ) (
Dual Coordinate Systems
* *
and Ω ∈ Ω ∈ x x
∫ ∫ ∫
− Ω − Ω
= =
*ξ,x x
dξ e d e d p x
* *. ) ( .
*
ξ ξ ξ ξ ξ
∈ Q and β
∫ ∫ ∫
− −
= =
M U M U M
d e d e U d p U Q ω ω ξ ω ξ ξ
ξ β ξ β β ) ( . ) ( .
) ( ) ( ). ( heat Geometric
Map Moment Souriau
Mean : map Moment Souriau : e Temperatur Geometric Souriau : Q U β
Dual Coordinate Systems
x x x ∂ Φ ∂ = ) (
*
and
* * *
) ( x x x ∂ Φ ∂ = β ∂ Φ ∂ = Q
and
Q s ∂ ∂ = β
Hessian Metric
) (
2 2
x d ds Φ − =
( )
β Φ − =
2 2
d ds
Fisher metric
2 , 2 2 2 2 2
*log ) ( ) ( log ) ( x d e x (x) x I x p E x I
x x
∂ ∂ = ∂ Φ ∂ − = ∂ ∂ − =
∫
Ω −
ξ ξ
ξ ξ 2 ) ( . 2 2 2 2 2
log ) ( ) ( log ) ( β ω β β β β ξ β
ξ β β ξ
∂ ∂ = ∂ Φ ∂ − = ∂ ∂ − =
∫
− M U
d e ) ( I p E I Capacity Geometric Souriau : ) (
2 2
β β β β β ∂ ∂ − = ∂ ∂ − = ∂ Φ ∂ − = Q K Q ) ( I
52 / 52 /
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Geometric heat Capacity / Specific heat
We observe that the Information Geometry metric could be
considered as a generalization of “Heat Capacity”. Souriau called it the “Geometric Capacity”. This geometric capacity is related to calorific capacity.
β ∂ Φ ∂ = Q β β β β ∂ ∂ − = ∂ Φ ∂ − = Q ) ( I
2 2
) ( T Q kT T kT T Q Q K ∂ ∂ = ∂ ∂ ∂ ∂ − = ∂ ∂ − =
2
1 1 β kT 1 = β Q K U
2 2 2 2
) ( ). ( ) ( . ) ( ) ( − = − = ∂ ∂ − =
M M
d p U d p U U E U E Q I ω ξ ξ ω ξ ξ β β
β β ξ ξ
U E d p U β Φ Q
M ξ β
ω ξ ξ = = ∂ ∂ =
) ( ). (
53 / 53 /
FOURIER HEAT EQUATION & HEAT CAPACITY
D C T Q . = ∂ ∂ T Q T Q IFisher ∂ ∂ = ∂ ∂ − =
2
1 ) ( β β
1 2 1 1
) ( .
− − −
∆ = ∂ ∂ ⇒ ∆ = ∂ ∂ β β β κ β κ
Fisher
I t T D C t T
54 / 54 /
ΘερEός : Souriau « Lie Group » Thermometer
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= ∂ ∂ − = Entropy Geometric : s e Temperatur (Planck) Geometric : Heat Geometric : with U Var ) ( β β β Q Q I Fisher
∂ Φ ∂ = β Q
β β Φ − = Q Q s ,
∂ ∂ = Q s β
r g m
e r p
) (
β −
∝
55 / 55 /
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Koszul Information Geometry, Souriau Lie Group Thermodynamics
Koszul Information Geometry Model Souriau Lie Groups Thermodynamics Model Convex Cone
Ω ∈ x Ω convex cone Ω ∈ β Ω convex cone: largest open subset of ,
Lie algebra of G, such that ∫
− M U
d e ω
ξ β ) ( .
and ∫
− M U
d e ω ξ
ξ β ) ( .
.
are convergent integrals
Transformation
( )
Ω ∈ → Aut g gx x with ) (β β
→
Transformation
(non invariant) ( )
g x gx x det log ) ( ) ( ) ( + Φ = Φ → Φ
Ω Ω Ω
( )
( )
( )
( )β
θ β β β
1
) (
−
− Φ = Φ → Φ a a
Transformation
(invariant) ( ) ( )
* * * * *
* * *) ( x x gx x
Ω Ω Ω Ω
Φ = ∂ Φ ∂ Φ → Φ x x x ∂ Φ ∂ =
Ω
) ( with
*
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( )
( )
( )β
θ β β β θ β β θ β β β β β
1
' ' ) ( ) ( ' ' ' ' .with ) ( . ' ' '. ' '
−
− Φ = Φ = Φ = Φ + = ∂ + Φ ∂ = ∂ Φ ∂ = = = Φ − = Φ − = → a a a Q a a a a Q a Q s Q Q Q s Q s
Geometry Metric (invariant)
( )
( ) [ ]
( )
x I x x x g x gx I = ∂ Φ ∂ − = ∂ + Φ ∂ − =
Ω Ω 2 2 2 2
) ( det log ) (
( )
( )
( ) [ ]
( ) ( )
β β β β β θ β β I a a I = ∂ Φ ∂ − = ∂ − Φ ∂ − =
− 2 2 2 1 2
) (
56 / 56 /
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Invariance of Fisher Metric
In both Koszul and Souriau models, the Information Geometry
Metric and the Entropy are invariant respectively to:
the automosphisms
to adjoint representation of Dynamical group G acting on , the convex
cone considered as largest open subset of , Lie algebra of G, such that and are convergent integrals.
g Ω
Ω
− M U
d e ω
ξ β ) ( .
− M U
d e ω ξ
ξ β ) ( .
.
x I x x x g x gx I = ∂ Φ ∂ − = ∂ + Φ ∂ − =
Ω Ω 2 2 2 2
) ( det log ) (
β β β β β θ β β I a a I = ∂ Φ ∂ − = ∂ − Φ ∂ − =
− 2 2 2 1 2
) (
Ω ∈ → Aut g gx x with ) (β β
→
58 / 58 /
Notation
#
Let a Lie Group and tangent space of at its neutral
element
with
For with Curve from
tangent to : and transform by :
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G G Te G e
g e g e
i T Ad G g G T GL G Ad = ∈ →
1
:
−
ghg h ig
Ad Ad ad
Y X Y ad G T Y X G T End G T Ad T ad
X e e e e
, ) ( , ) ( : = ∈ → =
G ) (K GL G
n
= C R K
= ) (K M G T
n e
=
1
) ( ), (
−
= ∈ ∈ gXg X Ad G g K M X
g n
Y X YX XY Y Ad T Y ad K M Y X
X e X n
, ) ( ) ( ) ( ) ( , = − = = ∈ ) ( c I e
d =
= ) 1 ( c X = ) exp( ) ( tX Ad t = γ ) exp( ) ( tX t c = Ad YX XY tX Y tX dt d Y t dt d Y Ad T Y ad
t t X e X
− = = = =
= − = 1
) exp( ) exp( ) ( ) ( ) ( ) ( γ
59 / 59 /
Action of a Lie Group on a Symplectic Manifold
Let be an action of Lie Group G on differentiable
manifold M, the fundamental field associated to an element of Lie algebra of group G is the vectors field on M:
and if for all , the fundamental field is globally hamiltonian
There exist linear application from to differential function on We can then associate a differentiable application ,called
moment of action :
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M G → × Φ : X
M
X
, exp ) (
=
− Φ =
t M
x tX dt d x X
) , ( ) , ( ,
2 1 2 1
x g g x g g Φ = Φ Φ x x e = Φ ) , (
with and
Φ M Φ
X
M
X
X
J X R M C → →
∞
) , (
J
Φ J
= → X X x J x J x J x M J
X
, ), ( ) ( such that ) ( :
60 / 60 /
Action of a Lie Group on a Symplectic Manifold
We associate a bilinear and anti-symmetric form , Symplectic
Cocycle of Lie algebra :
If with constant , then:
[ ] {
Y X Y X
J J J Y X , ) , (
,
− = Θ Θ
Bracket Poisson : .,.
with
) , , ( ) , , ( ) , , ( = Θ + Θ + Θ Y X Z X Z Y Z Y X
with
+ = J J'
µ
Y X Y X Y X , , ) , ( ) , ( ' µ + Θ = Θ
With cobord of
Y X Y X , , ) , ( µ µ = ∂
61 / 61 /
Action of a Lie Group on a Symplectic Manifold
$
There exist a unique affine action such that linear part is
coadjoint representation and that induce equivariance of moment is called Cocycle associated to
The differential
element :
If then :
a ) ( ) , ( :
*
g Ad g a G a
g
θ ξ ξ + = → ×
X Ad X Ad
g g 1 *
, ,
−
= ξ ξ J
) ( ) ( )) ( , ( ) , (
*
g x J Ad x J g a x g J
g
θ + = = Φ J
G : θ e θ
e
T θ J
[ ] {
Y X Y X e
J J J Y X Y X T , ) , ( ), (
,
− = Θ = θ µ + = J J'
Y X Y X Y X , , ) , ( ) , ( ' µ + Θ = Θ µ µ θ θ
*
) ( ) ( '
g
Ad g g − + =
Where is cobord of G
µ µ
* g
Ad −
63 / 63 /
Seminal Paper of Poincaré 1901, revisited by Marle
%&'()($ *($)+ ) +,,,--) .) /012/.& )&13&
Heni Poincaré proved that when a Lie algebra acts locally transitively
Euler-Lagrange equations are equivalent to a new system of differential equations defined on the product of the configuration space with the Lie algebra
%4'+ 5* *)6(78 ($*($ 8) )41) &5/9) 43&/
Marle has written the Euler-Poincaré equations, under an intrinsic
form, without any reference to a particular system of local coordinates
Marle has proven that they can be conveniently expressed in terms of
the Legendre and momentum maps of the lift to the cotangent bundle
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64 / 64 /
Seminal Paper of Poincaré
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« Ayant eu l’occasion de m’occuper du mouvement de rotation d’un corps solide creux, dont la cavité est remplie de liquide, j’ai été conduit à mettre les équations générales de la mécanique sous une forme que je crois nouvelle et qu’il peut être intéressant de faire connaître » Henri Poincaré, CRAS, 18 Février 1901 « Elles sont surtout intéressantes dans le cas où U étant nul, T ne dépend que des η η η η » Henri Poincaré
65 / 65 /
Lagrangian Mechanical system with M is smooth configuration space Poincaré assumes that a finite dimensional Lie algebra
acts on the configuration manifold M
New expression of the functional If
is a parametrized continuous, piecewise differentiable curve in M, and any lift of to M g, we have:
EULER'POINCARE EQUATION by Lift of admissible Curve
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R TM L M t t dt dt t d L I
t t
→ → = ∫ : and , : with ) ( ) (
1
1
γ γ γ
( ) , ( ) , ( : x X X x X x TM M
M
= → × ϕ ϕ
moment Souriau to d (associate :
*
→ M M T
t
ϕ
: with and : with , ) ( ) , ( : and ) , ( , : with ) ( ) (
1
1
× = → × = = → × = = × → = ∫ M p V p M M p p V(t) dt dγ x X L X x R M L L V t M t t dt t L I
M M M t t
γ γ γ ϕ γ γ γ γ γ
γ γ ) ( ) ( γ γ I I =
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MARLE Intrinsic Expression of EULER'POINCARE EQUATION
J.M. Marle intrinsic expression (independant of any choice of local
coordinates) of the Euler-Poincaré equation:
Poincaré “This equation is useful mainly when
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× = Ω Ω = − M L d p t V t t V t L d ad dt d
t t V
: ) ( ), ( ) ( ), (
1 * 2 * ) (
γ γ
∈ − = → = = X V X ad X ad ad X,V
X ad
V V V V
and , , , , such that : and
* *
ξ ξ ξ variable 2nd its and 1st its respect to with : function the
als differenti partial the be : :
* 2 * 1
R M L M L d M T M L d → × → × → ×
:
*
× M p R M L → × :
X
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EULER'POINCARE'MARLE EQUATION
* *#5($ #**
Souriau Moment Map: The Legendre Map
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M T J
*
:
∈ = X M T X X J
M M
, , ) ( , ), (
*
ξ ξ π ξ ξ
X X
M M M
), ( ) ( ξ π ϕ ξ π =
) ( ), ( ) ( :
*
ξ ξ π ξ ϕ ξ ϕ J M M T
M t t
= × →
J p J
M t t
, π ϕ ϕ = =
al differenti (vertical
*
L d M T TM
vert
→
ϕ ϕ ϕ
L d p J p L d
t t
= ⇒ = =
2 2
with
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EULER'POINCARE'MARLE REDUCTION
Euler-Poincaré Equation with Legendre and Moment Maps: The Euler-Poincaré Equation and Reduction: Following the remark
made by Poincaré at the end of his note, let us now assume that the map only depends on its 2nd variable
The Euler-Poincaré Equation in Hamiltonian Formalism
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) ( ), ( ) ( ) ( ), ( ) ( ), (
1 * ) (
t V t dt t d t V t L d J t V t J ad dt d
t V
γ ϕ γ γ γ ϕ = = −
M L → × :
X
) (
* ) (
= − t V L d ad dt d
t V
R M T H M T TM M T L H → → ∈ − =
− − * * * 1 1
: : , ) ( ) ( , ) (
ξ ξ ξ ξ
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POINCARE'MARLE'SOURIAU EQUATION ON DUAL LIE ALGEBRA
Euler-Poincaré Equation in the Framework of Souriau Lie Group
Thermodynamics
Souriau Lie Group Thermodynamics: Poincaré-Marle-Souriau Equation:
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) (
* ) (
= − t V L d ad dt d
t V *
= ∂ Φ ∂ − β
β
ad dt d Q ad dt dQ
* β
=
) ( ), ( ,
1 1
Q Q Q Q Q s
− −
Θ Φ − Θ = Φ − = β β
Θ = ∈ ∂ Φ ∂ = Θ =
−
) ( ) (
1 Q
Q β β β β
∂ ∂ = Q Q s ) ( β
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POINCARE'MARLE'SOURIAU EQUATION ON DUAL LIE ALGEBRA
Recall that
has a natural Poisson structure called Kirillov- Kostant-Souriau structure, which allow to associate to any smooth function its Hamiltonian vector field :
If there exist a smooth function
such that , the parametrized curve satisfies the Hamilton differential equation on :
In Souriau Lie Group Thermodynamics:
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h →
− = ξ ξ ξ χ
ξ
, ) (
* ) ( dh h
ad R h →
J h H
=
1 0,
: t t J ς ξ
) ( ) (
* ) (
t ad dt t d
dh
ξ ξ
ξ
− =
β β Φ − = Q Q s ,
) ( ) (
* )) ( (
t Q ad dt t dQ
t Q ds
− = dQ ds β =
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INFORMATION GEOMETRY BASED ON GEOMETRIC MECHANICS
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Élie CARTAN est le fils de Joseph CARTAN maréchal-ferrant. Elie CARTAN le FORGERON (du Latin FABER) l’Homme qui bat la matière sur l’ENCLUME, pour lui imprimer la COURBURE pour la mettre en FORME)
GEOMETRIC MECHANICS
MECHANICS J.M. SOURIAU LIE GROUP THERMODYNAMICS J.M. SOURIAU
INFORMATION J.L. KOSZUL
Texte de Bergson - Homo faber "En ce qui concerne l'intelligence humaine, on n'a pas assez remarqué que l'invention mécanique a d'abord été sa démarche essentielle… . Si nous pouvions nous dépouiller de tout orgueil, si, pour définir notre espèce, nous nous en tenions strictement à ce que l'histoire et la préhistoire nous présentent comme la caractéristique constante de l'homme et de l'intelligence, nous ne dirions peut-être pas Homo sapiens, mais Homo faber. En définitive, l'intelligence, envisagée dans ce qui en paraît être la démarche originelle, est la faculté de fabriquer des objets artificiels, en particulier des
Henri Bergson, L’ Évolution créatrice (1907), Éd. PUF, coll. "Quadrige", 1996, chap. II, pp.138-140 NEW FOUNDATION OF INFORMATION THEORY (Sapiens) by GEOMETRIC MECHANICS (Faber)
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Legendre Transform and Minimal Surface
Legendre has introduced « Legendre Transform » to solve Minimal
Surface Problem
Classical “Legendre transform” with our previous notations:
( ) ( ) ( ) ( ) ( )
= = Φ = Φ Φ = = = = = = = Φ Φ − = Φ − = dQ ds dQ ds dQ ds d d d d d d Q y x q p Q Q Q q p Q s y x z Q Q Q s
2 1 2 1 2 1 2 1
and , ) , ( ) , ( with , . β β β β β β β ω β β β β β
dq d y dp d x q p y q x p y x z ω ω ω = = − + = and with ) , ( . . ) , (
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Legendre Transform and Minimal Surface
In the following relation, we recover the definition of Entropy : But also relation with Mean Curvature
= = ⇒ = = = + dQ ds ds dQ dq d y dp d x d dq y dp x β β ω ω ω . and . .
Φ = − − = ⇒ − = ⇒ >> − = ⇒ << = Φ + Φ = +
Φ Φ Φ
β β β β β β β β β d d Q Q Q Q H I Q Q Q Q Q d dQ Q Q d d Q H I Q H d d d d d d Q Q d d with 1 . 2 ) ( . . 1 2 ) ( 1 . 2 1 1
2 2 2 2 2 2
2nd Principle of Thermodynamics
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References
!" [1] Koszul J.L., Variétés localement plates et convexité, Osaka J. Math. , n2, p.285-290, 1965 [2] Koszul J.L., Domaines bornées homogènes et orbites de groupes de transformations affines, Bull. Soc. Math. France 89, pp. 515- 533., 1961 [3] Koszul J.L., Ouverts convexes homogènes des espaces affines, Math. Z. 79, pp. 254-259., 1962 [4] Koszul J.L., Déformations des variétés localement plates, .Ann Inst Fourier, 18 , 103-114., 1968
[6] Souriau J.M., Structure of dynamical systems, volume 149 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA,
[7] Souriau J.M., Thermodynamique et géométrie. In Differential geometrical methods in mathematical physics, II, vol. 676 of Lecture Notes in Math., pages 369-397. Springer, Berlin, 1978 [8] Souriau J.M., Géométrie Symplectique et physique mathématique, CNRS 75/P.785, Dec. 1975 [9] Souriau J.M., Mécanique classique et géométrie symplectique, CNRS CPT-84/PE-1695, Nov. 1984 [10] Souriau J.M., On Geometric Mechanics, Discrete and continuous Dynamical systems, V.ol 19, N. 3, pp. 595–607, Nov. 2007
Fibration and Fréchet Median, Matrix Information Geometry, Springer, 2013 [12] Barbaresco F., Eidetic Reduction of Information Geometry through Legendre Duality of Koszul Characteristic Function and Entropy, Geometric Theory of Information, Springer, 2014 [13] Barbaresco F., Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics,, MDPI Entropy, n16, pp. 4521-4565, August 2014. http://www.mdpi.com/1099-4300/16/8/4521/pdf
6
[14] Shima H. , Geometry of Hessian Structures, world Scientific Publishing 2007 [15] Gromov M., IHES Six Lectures on Probabiliy, Symmetry, Linearity. October 2014, Jussieu. http://www.ihes.fr/~gromov/PDF/probability-huge-Lecture-Nov-2014.pdf https://www.youtube.com/watch?v=hb4D8yMdov4&index=5&list=PLx5f8IelFRgGo3HGaMOGNAnAHIAr1yu5W
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Past Conferences
French/Indian Workshop on « Matrix Information Geometry », Ecole Polytechnique &
Thales Research & Technology, 23-25th February 2011 (with Prof. Rajendra Bhatia)
http://www.lix.polytechnique.fr/~schwander/resources/mig/slides/
SMAI’11 Congress, Mini-Symposium on « Information Geometry », 23-27th Mai 2011
http://smai.emath.fr/smai2011/programme_detaille.php
Symposium on « Information Geometry & Optimal Transport », hosted at Institut Henri
Poincaré, 12th February 2012 (with GDR CNRS MIA)
https://www.ceremade.dauphine.fr/~peyre/mspc/mspc-thales-12/
SMF/SEE Conference on « Geometric Science of Information », Ecole des Mines de
Paris, August 2013
http://www.see.asso.fr/gsi2013
SMF/SEE Conference on MaxEnt with special Issue "Information, Entropy and their
Geometric Structures", Clos Lucé in Amboise, sponsored by Jaynes Foundation, 21- 26th Sept. 2014
https://www.see.asso.fr/node/10784
Leon Brillouin Seminar on « Geometric Science of Information », Institut Henri
Poincaré & IRCAM, launched by THALES since December 2009, with Ecole Polytechnque & INRIA/IRCAM
http://repmus.ircam.fr/brillouin/past-events
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GSI’13 Proceedings
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http://www.springer.com/computer/image+processing/ book/978-3-642-40019-3
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MaxEnt’14, Amboise, Clos Lucé, September 2014
34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering Château Clos Lucé, Parc Leonardo Da Vinci, Amboise, France https://www.see.asso.fr/maxent14 http://djafari.free.fr/MaxEnt2014/Program_MaxEnt2014.html Sponsored by Jaynes Foundation Keynote speakers: Misha Gromov (IHES, Abel Prize 2009) Daniel Bennequin (Institut Mathématique de Jussieu) Roger Balian (CEA, French Academy of science) Stefano Bordoni (Bologna University) https://www.see.asso.fr/node/10784
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GSI’15 « Geometric Science of Information » 28th '30th October 2015 Ecole Polytechnique
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http://www.gsi2015.org
Authors will be solicited to submit a paper in a special Issue "Differential Geometrical Theory
international and interdisciplinary open access journal of entropy and information studies published monthly online by MDPI. http://www.mdpi.com/journal/entropy/special_is sues/entropy-statistics
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Léon Nicolas Brillouin Seminar
+:;
Hosting lab:
IRCAM, Salle Igor Stravinsky Projet INRIA/CNRS/Ircam MuSync (Arshia Cont)
Animation : Arshia Cont (IRCAM & INRIA), Frank Nielsen (Ecole
Polytechnique), F. Barbaresco (Thales)
Web Site: http://repmus.ircam.fr/brillouin/home Abstracts, Videos & Slides: http://repmus.ircam.fr/brillouin/past-events
http://repmus.ircam.fr/brillouin/home
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Thank you for your attention
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Nous avouerons qu’une des prérogatives de la géométrie est de contribuer à rendre l’esprit capable d’attention: mais on nous accordera qu’il appartient aux lettres de l’étendre en lui multipliant ses idées, de l’orner, de le polir, de lui communiquer la douceur qu’elles respirent, et de faire servir les trésors dont elles l’enrichissent, à l’agrément de la société. Joseph de Maistre Si on ajoute que la critique qui accoutume l’esprit, surtout en matière de faits, à recevoir de simples probabilités pour des preuves, est, par cet endroit, moins propre à le former, que ne le doit être la géométrie qui lui fait contracter l’habitude de n’acquiescer qu’à l’évidence; nous répliquerons qu’à la rigueur
que la critique donne, au contraire, plus d’exercice à l’esprit que la géométrie: parce que l’évidence, qui est une et absolue, le fixe au premier aspect sans lui laisser ni la liberté de douter, ni le mérite de choisir; au lieu que les probabilités étant susceptibles du plus et du moins, il faut, pour se mettre en état de prendre un parti, les comparer ensemble, les discuter et les peser. Un genre d’étude qui rompt, pour ainsi dire, l’esprit à cette opération, est certainement d’un usage plus étendu que celui où tout est soumis à l’évidence; parce que les occasions de se déterminer sur des vraisemblances
probabilités, sont plus fréquentes que celles qui exigent qu’on procède par démonstrations: pourquoi ne dirions –nous pas que souvent elles tiennent aussi à des objets beaucoup plus importants ? Joseph de Maistre