Koszul Information Geometry & Souriau Lie Group 4Thermodynamics

21/09/2014
Publication MaxEnt 2014
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Koszul Information Geometry & Souriau Lie Group 4Thermodynamics

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application/pdf Koszul Information Geometry and Souriau Lie Group Thermodynamics
application/pdf Koszul Information Geometry & Souriau Lie Group 4Thermodynamics Frédéric Barbaresco

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Optimal matching between curves in a manifold
Drone Tracking Using an Innovative UKF
Jean-Louis Koszul et les structures élémentaires de la Géométrie de l’Information
Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics
Session Geometrical Structures of Thermodynamics (chaired by Frédéric Barbaresco, François Gay-Balmaz)
Opening and closing sessions (chaired by Frédéric Barbaresco, Frank Nielsen, Silvère Bonnabel)
GSI'17-Closing session
GSI'17 Opening session
Démonstrateur franco-britannique "IRM" : gestion intelligente et homéostatique des radars multifonctions
Principes & applications de la conjugaison de phase en radar : vers les antennes autodirectives
Nouvelles formes d'ondes agiles en imagerie, localisation et communication
Compréhension et maîtrise des tourbillons de sillage
Wake vortex detection, prediction and decision support tools
Ordonnancement des tâches pour radar multifonction avec contrainte en temps dur et priorité
Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics
Reparameterization invariant metric on the space of curves
Probability density estimation on the hyperbolic space applied to radar processing
SEE-GSI'15 Opening session
Lie Groups and Geometric Mechanics/Thermodynamics (chaired by Frédéric Barbaresco, Géry de Saxcé)
Opening Session (chaired by Frédéric Barbaresco)
Invited speaker Charles-Michel Marle (chaired by Frédéric Barbaresco)
Koszul Information Geometry & Souriau Lie Group 4Thermodynamics
MaxEnt’14, The 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering
Koszul Information Geometry & Souriau Lie Group Thermodynamics
Robust Burg Estimation of stationary autoregressive mixtures covariance
Koszul Information Geometry and Souriau Lie Group Thermodynamics
Koszul Information Geometry and Souriau Lie Group Thermodynamics
Oral session 7 Quantum physics (Steeve Zozor, Jean-François Bercher, Frédéric Barbaresco)
Opening session (Ali Mohammad-Djafari, Frédéric Barbaresco)
Tutorial session 1 (Ali Mohammad-Djafari, Frédéric Barbaresco, John Skilling)
Prix Thévenin 2014
SEE/SMF GSI’13 : 1 ère conférence internationale sur les Sciences  Géométriques de l’Information à l’Ecole des Mines de Paris
Synthèse (Frédéric Barbaresco)
POSTER SESSION (Frédéric Barbaresco)
ORAL SESSION 16 Hessian Information Geometry II (Frédéric Barbaresco)
Information/Contact Geometries and Koszul Entropy
lncs_8085_cover.pdf
Geometric Science of Information - GSI 2013 Proceedings
Médaille Ampère 2007

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www.thalesgroup.com Thales Air Systems Date Koszul Information Geometry & Souriau Lie Group Thermodynamics Frédéric BARBARESCO, THALES AIR SYSTEMS Senior Scientist & Advanced Studies Manager, Advanced Radar Concepts Dept. SEE Emeritus Member, Ampere Medal 2007 Aymé Poirson Prize 2014 (French Academy of Sciences) 2 /2 / Thales Air Systems Date 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. 3 /3 / Thales Air Systems Date 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 of Mean “Moment map” by geometric temperature, equivalent to a 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). 4 /4 / Geometric Science of Information Takeshi SASAKI W. BLASCHKE Eugenio CALABI Calyampudi R. RAO Nikolai N. CHENTSOV Hirohiko SHIMA Jean-Louis KOSZUL Von Thomas FRIEDRICH Y. SHISHIDO Homogeneous Convex Cones G. E.B. VINBERG Jean-Louis KOSZUL Homogeneous Symmetric Bounded Domains G. Elie CARTAN Carl Ludwig SIEGEL Probability in Metric Space Maurice R. FRECHET Information Theory Nicolas .L. BRILLOUIN Claude. SHANNON Probability on Riemannian Manifold Michel EMERY Marc ARNAUDON Geometric Science of Information KOSZUL-VINBERG METRIC (KOSZUL-VINBERG CHARACTERISTIC FUNCTION) FISHER METRIC Probability/G. on structures Y. OLLIVIER M. GROMOV Contact G. Vladimir ARNOLD 5 /5 / Hessian Geometry by J.L. Koszul 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 Prof. M. Boyom tutorial : http://repmus.ircam.fr/_media/brillouin/ressources/une -source-de-nouveaux-invariants-de-la-geometrie-de-l- 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, pp. 285-290., 1965 J.L. Koszul, « Déformations des variétés localement plates », .Ann Inst Fourier, 18 , 103-114., 1968 Foundation of Information Geometry: Jean-Louis Koszul Works 6 /6 / Projective Legendre Duality and Koszul Characteristic Function LEGENDRE TRANSFORM FOURIER/LAPLACE TRANSFORM ENTROPY= LEGENDRE(- LOG[LAPLACE]) Ψ=Φ−= 22 log ddg INFORMATION GEOMETRY METRIC Sddg 2*2* =Ψ= ∫ Ω − −=Φ−=Ψ * , log)(log)( dyexx yx )(,)( *** xxxx Ψ−=Ψ ξξξ dpp xx∫ Ω −=Ψ * )(log)(* Φ(x)x,ξξ,xξ,x x edξeep * +−−− == ∫/)(ξ ∫ Ω = * )(.* ξξξ dpx x Legendre Transform of minus logarithm of characteristic function (Laplace transform) = ENTROPY !!! ds2=d2ENTROPY ds2=-d2LOG[FOURIER] 7 /7 / 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. is a sharp open convex cone in a vector space of finite dimension on (a convex cone is sharp if it does not contain any full straight line). is the dual cone of and is a sharp open convex cone. Let the Lebesgue measure on dual space of , the following integral: is called the Koszul-Vinberg characteristic function Ω E R * Ω Ω ξd * E E Ω∈∀= ∫ Ω − Ω xdex x )( * , ξψ ξ 8 /8 / Koszul-Vinberg Characteristic Function/Metric of convex cone Koszul-Vinberg Metric : 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 is characterized by is the center of gravity of the cross section of : Ω= ψlog2 dg [ ] ( )loglog 2 1log log)(log 2 u 2 22 ∫∫ ∫∫ ∫ ∫ ∫ − +== dudv dudvdd du dud dudxd vu vuv u uu u ψψ ψψψψ ψ ψψ ψψ )(log* xdx x Ω−=−= ψα )()(),( 0 tuxf dt d xfDuxdf t u +== = Ω xx α−=* xx =** )( nxx =* , cstexx =ΩΩ )()( * *ψψ * x { }nyxyyx =Ω∈= ,,/)(minarg ** ψ * x { }nyxy =Ω∈ ,,* * Ω ∫∫ Ω − Ω − = ** ,,* /. ξξξ ξξ dedex xx 9 /9 / 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 )(,)( *** xxxx Φ−=Φ Φ= xDx* * * Φ= x Dx )(log)( xx Ω−=Φ ψ Ω∈∀= ∫ Ω − Ω xdex x )( * , ξψ ξ ∫∫ Ω − Ω − = ** ,,* /. ξξξ ξξ dedex xx ∫∫ Ω − Ω − Ω −==− ** ,,* /,)(log, ξξξψ ξξ dedehxdhx xx h ∫∫ Ω − Ω −− =− ** ,,,* /.log, ξξ ξξξ dedeexx xxx ∫∫∫∫ ∫∫∫ Ω − Ω −− Ω − Ω − Ω − Ω − Ω −−         −        =Φ +−=Φ **** *** ,,,,,** ,,,,** /.loglog.)( log/.log)( ξξξξ ξξξ ξξξξξ ξξξξ dedeededex dededeex xxxxx xxxx 10 /10 / Koszul-Vinberg Characteristic Function Legendre Transform Thales Air Systems Date                         −=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.)( 1with.log.log)( .loglog)( /.loglog.)( log/.log)(,)( ξ ξξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξξξξ ξξξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξξ ξξξξξ ξξξξ d de e de e x d de e d de e ed de e dex d de e edex dedeededex dededeexxxx x x x x x x x x x x x x x x xx xxxxx xxxx 11 /11 / [ ]( ) [ ] [ ]∫ Ω Φ=Φ≥Φ⇒ Φ−≥Φ Φ≤Φ⇒Φ * * * )()()()( )(,)(:TransformLegendre )(conv.:Ineq.Jensen *** *** ** ξξξξ ξξ Edpx xxxx EE x Koszul Entropy via Legendre Transform We can then consider this Legendre transform as an entropy, that we could named “Koszul Entropy”: With and ξξξξ ξξ ξ ξ ξ ξ dppd de e de e xxx x x x ∫ ∫ ∫ ∫ Ω Ω − − Ω Ω − − −=−=Φ * * * * )(log)(log , , , , * Φ(x)x,ξ dξex,ξ ξ,xξ,x x eedξeep * ξ,x * +− ∫−− −− === − ∫ log /)(ξ ( ) ξξξξξξ dedξ.edpΦDx ξΦΦ(x)x,ξ xx * * − Ω +− Ω ∫∫∫ ==== ** .)(.* [ ] 1log)()( loglog)( * * * * * * * )()( )()(, =⇒−Φ=Φ −=−=Φ ∫∫ ∫∫ Ω Φ− Ω Φ− Ω Φ+Φ− Ω − ξξ ξξ ξξ ξξ dedexx dedex xx [ ] [ ]( )ξξ ξξξξξξ ξξξξξξ ξξ ξξ EE dpdp xdpdpp eep xx xxx xx x ** ** *** *)()(, )(or )(.)()(ifonlyandif )()()()()(log )(loglog)(log ** ** * Φ=Φ         Φ=Φ Φ=Φ=−⇒ Φ−=== ∫∫ ∫∫ ΩΩ ΩΩ Φ−Φ+− 12 /12 / Barycentre & Koszul Entropy Thales Air Systems Date [ ] [ ]( )ξξ EE * )( * Φ=Φ [ ])(,)( *** xxxSupx x Φ−=Φ ( ) ξξξξξξ dedξ.edpΦDx ξΦΦ(x)x,ξ xx * * − Ω +− Ω ∫∫∫ ==== ** .)(.*         Φ=Φ ∫∫ ΩΩ ** )(.)()( ** ξξξξξξ dpdp xx Φ(x)x,ξ dξex,ξ ξ,xξ,x x eedξeep * ξ,x * +− ∫−− −− === − ∫ log /)(ξ Barycenter of Koszul Entropy = Koszul Entropy of Barycenter 13 /13 / Koszul metric & Fisher Metric To make the link with Fisher metric given by matrix , we can observe that the second derivative of 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 ExI x (x) x xx x p xxp x x x ∂ ∂ = ∂ Φ∂ −=      ∂ ∂ −=⇒ ∂ Φ∂ = ∂ −Φ∂ = ∂ ∂ −Φ=Φ−= Ωψξ ξξ ξξξ ξ 2 2 2 2 log)(log )( x (x) x p ExI x ∂ ∂ =      ∂ ∂ −= Ωψξ ξ FISHER METRIC (Information Geometry) = KOSZUL HESSIAN METRIC (Hessian Geometry) )(xI )(log ξxp 14 /14 / 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 : Thales Air Systems Date ξ ∫ ∫ ∫ − − − −= ∂ ∂ ⇒= * * * ξ,x ξ,x ξ,x dξe dξex (x)Ψ dξe(x)Ψ . 1log loglog ξ                 +−         −= ∂ ∂ ∫∫∫ ∫ −−− − 2 2 22 2 ... 1log *** * ξ,xξ,xξ,x ξ,x dξedξedξe dξe x (x)Ψ ξξ 2 2 2 2 2 2 )(.)(... log         −=             −= ∂ ∂ ∫∫∫ ∫ ∫ ∫ − − − − *** * * * xxξ,x ξ,x ξ,x ξ,x dξpdξpdξ dξe e dξ dξe e x (x)Ψ ξξξξξξ [ ] [ ] )( log)(log )( 22 2 2 2 2 ξξξ ψξ ξξξ VarEE x (x) x p ExI x =−= ∂ ∂ =      ∂ ∂ −= Ω 15 /15 / New Definition of Maximum Entropy We have then observed that Koszul Entropy provides density of Maximum Entropy: with and where and Thales Air Systems Date ( ) ( ) ∫ − − Θ− Θ− = * ξ, ξ, dξe e p ξ ξ ξ ξ 1 1 )( )(1 ξ− Θ=x dx xd x )( )( Φ =Θ=ξ ξξξξ ξ dp∫ Ω = * )(. ∫ Ω − −=Φ * , log)( ξξ dex x 16 /16 / Koszul Density: Application for SPD matrices We can then named this new density as “Koszul Density”: With Φ(x)x,ξ dξex,ξ ξ,xξ,x x eedξeep * ξ,x * +− ∫−− −− === − ∫ log /)(ξ ( ) ξξξξξξ dedξ.edpΦDx ξΦΦ(x)x,ξ xx * * − Ω +− Ω ∫∫∫ ==== ** .)(.* ( ) ( )[ ] ( ) ∫ Ω −− + +− === − * 1 ).(.withdet)( 1 detlog 2 1 ξξξξξαξ ξξααξ dpeep x Tr x n xTr x ( )           + = + =−== == ∈∀= − Ω + − Ω=Ω = Ω − Ω ∫ 1* 2 1 dual-self )(, , 2 1 detlog 2 1 log )(det)( )(,,, * * x n xd n dx Ixdex RSymyxxyTryx n n xyTryx x n ψξ ψξψ ξ 17 /17 / Thales Air Systems Date 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 of Symplectic Geometry 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. 18 /18 / Thales Air Systems Date 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 of symmetries). 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 19 /19 / Thales Air Systems Date 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 ωd )(ξU M )(ξp       = =       −= ∫ ∫ ∫ QdpU dp dppsArgMin M M Mp ωξξ ωξ ωξξ ξ )()( 1)( such that)(log)( )( )(.)( )( ξββ ξ U ep −Φ = ∫ − −=Φ M U de ωβ ξβ )(. log)( ∫ ∫ − − = M U M U de deU Q ω ωξ ξβ ξβ )(. )(. )( ∫−= M dpps ωξξ )(log)( Φ β Φ )(.)( ββ Φ−= QQs 20 /20 / Thales Air Systems Date 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: )(.)( ββ Φ−= QQs ∫ ∫ − − = M U M U de deU Q ω ωξ ξβ ξβ )(. )(. )( )(βΦ Qdd .β=Φ β∂ Φ∂ =Q ∫ − ⊗ M U deUU ωξξ ξβ )(. )()( 02 2 > ∂ Φ∂ − β )(βΦ− )(βQ )(Qβ dQds .β= Q s ∂ ∂ =β 21 /21 / Thales Air Systems Date 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. 22 /22 / Thales Air Systems Date 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 U de ωβ. log β∂ Φ∂ =Q β∂ ∂ −= Q K Φ−= Qs .β 23 /23 / Thales Air Systems Date 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: is a symplectic cocycle The following symmetric tensor , defined on all values of is positive definite: β βf f ( ) ( ) [ ]21222121 ,)(with)(.,, 11 ZZZadZadQZZfZZf ZZ =+=β βf βf ββ fKer∈ βg (.)βad [ ][ ]( ) [ ]( )2121 ,,,,, ZZfZZg βββ ββ = 24 /24 / Thales Air Systems Date Souriau Lie Group Thermodynamic Souriau equations are universal, because they are not dependent of the symplectic manifold but only of: 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”. ( ) ( ) [ ]21222121 ,)(with)(.,, 11 ZZZadZadQZZfZZf ZZ =+=β ββ fKer∈ [ ][ ]( ) [ ]( )2121 ,,,,, ZZfZZg βββ ββ = β f Q 25 /25 / Thales Air Systems Date 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: where is the cocycle associated with the group G and the moment, 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 g ∫ − M U de ωξβ )(. ∫ − M U de ωξ ξβ )(. . Ω ga gaa a )(ββ ga→ ( ) ( ) )(.1 βθβθ gaaa +Φ=−Φ→Φ − ss → ( ) )()( QaaQaQ ** gg θ θ =+→ )(ςς + → Ma θ )(ς+ Ma Ma ς Φ β ( ) ( )[ ] ( )β ββ βθ β I a aI = ∂ Φ∂ −= ∂ −Φ∂ −= − 2 2 2 12 )(g 26 /26 / Fundamental Souriau Theorem Thales Air Systems Date Gibbs canonicalGibbs canonicalGibbs canonicalGibbs canonical ensembleensembleensembleensemble Ω * Ω g * g R R ( )βΦ ( ) ( ) )(. βθβ gaa+Φ ( ) ( )ββ Φ−= QQs . Q )(Qa * gθ β )(βga ς )(ς+ Ma e a G 27 /27 / Thales Air Systems Date 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 is a symplectic cocycle of ,that is independent of the moment of G There exists a symmetric tensor defined on the image of such that: and that gives the structure of a positive Euclidean space f θ ( ) ( ) [ ]21222121 ,)(with)(.,,, 11 ZZZadZadQZZfZZf ZZ =+=Ω∈∀ ββ βf g ( ) Ω∈∀= ββββ ,0,f βg [ ]ββ .,(.) =ad [ ]( ) ( ) ()( ).Im,,,,, 212121 βββ β adZZZZfZZg ∈∀∈∀= g ( ) ()( ).Im,,0, 2121 ββ adZZZZg ∈∀≥ 28 /28 / Thales Air Systems Date Koszul Information Geometry, Souriau Lie Group Thermodynamics Koszul Information Geometry Model Souriau Lie Groups Thermodynamics Model Characteristic function Ω∈∀−=Φ ∫ Ω − xdex x log)( * , ξξ g∈∀−=Φ ∫ − βωβ ξβ log)( )(. M U de Entropy ξξξ dppx xx∫ Ω −=Φ * )(log)()( ** ∫−= M dpps ωξξ )(log)( Legendre Transform )(,)( *** xxxx Φ−=Φ )(.)( ββ Φ−= QQs Density of probability ∫ − − +− = = * ξ,x ξ,x x Φ(x)x,ξ x dξe e p ep )( )( ξ ξ ∫ − − Φ+− = = M U U U de e p ep ω ξ ξ ξβ ξβ β βξβ β )(. )(. )()(. )( )( Dual Coordinate Systems ** and Ω∈Ω∈ xx ∫ ∫ ∫ − Ω − Ω == * ξ,x ξ,x x dξe de dpx * * . )(.* ξξ ξξξ * gg ∈∈ Qandβ ∫ ∫ ∫ − − == M U M U M de deU dpUQ ω ωξ ωξξ ξβ ξβ β )(. )(. )( )().( heatGeometricor MapMomentSouriauofMean: mapMomentSouriau: eTemperaturGeometricSouriau: Q U β Dual Coordinate Systems x x x ∂ Φ∂ = )(* and * ** )( x x x ∂ Φ∂ = β∂ Φ∂ =Q and Q s ∂ ∂ =β Hessian Metric )(22 xdds Φ−= ( )βΦ−= 22 dds Fisher metric 2 ,2 2 2 2 2 * log )( )(log )( x de x (x) xI x p ExI x x ∂ ∂ = ∂ Φ∂ −==       ∂ ∂ −= ∫ Ω − ξ ξ ξ ξ 2 )(.2 2 2 2 2 log )( )(log )( β ω β β β β ξ β ξβ β ξ ∂ ∂ = ∂ Φ∂ −=         ∂ ∂ −= ∫ − M U de )( I p EI CapacityGeometricSouriau: )( 2 2 β ββ β β ∂ ∂ −= ∂ ∂ −= ∂ Φ∂ −= Q K Q)( I 29 /29 / Thales Air Systems Date 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. is related to the mean, and is related to the variance of β∂ Φ∂ =Q ββ β β ∂ ∂ −= ∂ Φ∂ −= Q)( I 2 2 )( T Q kTT kT T QQ K ∂ ∂ =             ∂ ∂ ∂ ∂ −= ∂ ∂ −= 2 1 1 βkT 1 =β Q K U [ ] [ ] 2 222 )().()(.)()(         −=−= ∂ ∂ −= ∫∫ MM dpUdpUUEUE Q I ωξξωξξ β β ββξξ [ ]UEdpU β Φ Q M ξβ ωξξ == ∂ ∂ = ∫ )().( 30 /30 / Thales Air Systems Date 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 g , Lie algebra of G, such that ∫ − M U de ωξβ )(. and ∫ − M U de ωξ ξβ )(. . are convergent integrals Transformation ( )Ω∈→ Autggxx with )(ββ ga→ Transformation of Potential (non invariant) ( )gxgxx detlog)()()( +Φ=Φ→Φ ΩΩΩ ( ) ( ) ( ) ( )βθβββ 1 )( − −Φ=Φ→Φ aag Transformation of Entropy (invariant) ( ) ( )***** *** )( x x gx x Ω Ω ΩΩ Φ=      ∂ Φ∂ Φ→Φ x x x ∂ Φ∂ = Ω )( with * ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )βθβββ θ β βθ β ββ ββ 1 '' )( )( ' ' ' ' .with )(.'''.'' − −Φ=Φ=Φ=Φ += ∂ +Φ∂ = ∂ Φ∂ = = =Φ−=Φ−=→ aa aQa a aa Q a QsQQQsQs g g g g g * Information Geometry Metric (invariant) ( ) ( )[ ] ( )xI x x x gx gxI = ∂ Φ∂ −= ∂ +Φ∂ −= ΩΩ 2 2 2 2 )(detlog)( ( ) ( ) ( )[ ] ( ) ( )β β β β βθβ β I a aI = ∂ Φ∂ −= ∂ −Φ∂ −= − 2 2 2 12 )(g 31 /31 / Thales Air Systems Date Invariance of Fisher Metric In both Koszul and Souriau models, the Information Geometry Metric and the Entropy are invariant respectively to: the automosphisms of the convex cone 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 ga Ω g ∫ − M U de ωξβ )(. ∫ − M U de ωξ ξβ )(. . ( ) ( )[ ] ( )xI x x x gx gxI = ∂ Φ∂ −= ∂ +Φ∂ −= ΩΩ 2 2 2 2 )(detlog)( ( ) ( ) ( )[ ] ( ) ( )β β β β βθβ β I a aI = ∂ Φ∂ −= ∂ −Φ∂ −= − 2 2 2 12 )(g ( )Ω∈→ Autggxx with )(ββ ga→ 32 /32 / Thales Air Systems Date 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 with where is a Cartan involution (An involution on is a Lie algebra automorphism of whose square is equal to the identity). The Cartan-Killing form is invariant under automorphisms of the algebra : ( ))(,, yxByx θ−= g∈θ g θ g ( )yxadadTryxB =),( )(gAut∈σ g ( ) ( )yxByxB ,)(),( =σσ 33 /33 / Thales Air Systems Date From Cartan-Killing Form to Koszul Information Metric ( ) ( ) InvolutionCartan,with )(,, FormKillingCartan ),( g yxByx adadTryxB yx ∈ −= − = θ θ Ω∈∀−=Φ ∫ Ω − xdex x log)( FunctionsticCharacteriKoszul * , ξξ ∫ ∫ ∫ − − Ω Ω = = −=Φ Φ−=Φ * ξ,x ξ,x x x xx dξe e p dpx dppx xxxx )( DensityKoszul )(.with )(log)()( )(,)( EntropyKoszul * * * ** *** ξ ξξξ ξξξ 2 ,2 2 2 2 2 * log )( )(log )( MetricKoszul x de x (x) xI x p ExI x x ∂ ∂ = ∂ Φ∂ −==       ∂ ∂ −= ∫ Ω − ξ ξ ξ ξ 34 /34 / Thank you for your attention Thales Air Systems Date 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 on pourrait conclure de cette différence même, 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 ou 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