Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14339
DOI : http://dx.doi.org/10.1007/978-3-319-25040-3_57You do not have permission to access embedded form.

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We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat, and momentum as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. Finally, we conclude on Balian Gauge theory of Thermodynamics compatible with Souriau’s Model.

Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics

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        <identifier identifierType="DOI">10.23723/11784/14339</identifier><creators><creator><creatorName>Frédéric Barbaresco</creatorName></creator></creators><titles>
            <title>Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2015</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><subjects><subject>Information geometry</subject><subject>Symplectic Geometry</subject><subject>Momentum Map</subject><subject>Cartan-Poincaré Integral</subject><subject>Invariant Lie Group</subject><subject>Thermodynamics</subject><subject>Geometric Mechanics</subject><subject>Euler-Poincaré Equation</subject><subject>Gibbs Equilibrium</subject><subject>Fisher Metric</subject><subject>Maximum Entropy</subject><subject>Gauge Theory</subject></subjects><dates>
	    <date dateType="Created">Sun 8 Nov 2015</date>
	    <date dateType="Updated">Wed 31 Aug 2016</date>
            <date dateType="Submitted">Tue 16 Jan 2018</date>
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We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat, and momentum as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. Finally, we conclude on Balian Gauge theory of Thermodynamics compatible with Souriau’s Model.

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www.thalesgroup.com OPEN Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics FRÉDÉRIC BARBARESCO ADVANCED RADAR CONCEPTS BUSINESS UNIT THALES AIR SYSTEMS www.thalesgroup.com OPEN FISHER METRIC: Fréchet Clairaut Equation & Koszul-Vinberg Characteristic Function 3 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Fisher Metric and Information Geometry (IG) ▌ IG could be introduced with Koszul-Vinberg Characteristic Function: with and are dual convex cones ▌ Density is given by Solution of Maximum Entropy: ▌ The inversion is given by Legendre transform based on :      xde(x) x , * ,    *    dpp   * )(log)()(*               dpdpdppMax p *** )(.and1)(such that)(log)(     )(loglog)(where )( )(with)( * 1 1 , xdex dx xd x dξe e p x Ω ξ, ξ, *                  )(-1  **,*** andlog)( Φ with)(,)( *     xxdex dx (x)d xxxxx x*  ...,)( 2 , 2 *   uuxKux(x)u)(x   4 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Fisher Metric and Information Geometry (IG) ▌ Maurice Fréchet, studying “distinguished functions” (densities with estimator reaching the Fréchet-Darmois bound), have also observed that solution should verify the Alexis Clairaut Equation: ▌ Fisher Metric appears as hessian of characteristic function logarithm:      xxxxxxx /)()()(,)( **1*1*** 2 2 2 2 )(log )(,)(log x (x) x p xxp x x             )( log )( )(log )( 22 2 2 2 2 2 2      VarEE x (x) xI x (x) x p ExI x                 1 2* *2 2 2             xx www.thalesgroup.com OPEN SOURIAU MODEL OF INFORMATION GEOMETRY: Lie Group Thermodynamics 6 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Covariant Gibbs Equilibrium ▌ Jean-Marie Souriau has observed in 1966 in « Définition covariante des équilibres thermodynamiques » that Classical Gibbs Equilibrium is not covariant with respect to Dynamic Groups (Gallileo Group in classical Mechanic or Poincaré Group in Relativity). Classical thermodynamics corresponds to the case of Time translation. ▌ To solve this incoherency, Souriau has extended definition of Canonical Gibbs Ensemble to Symplectic Manifolds on which a Lie Group has a Symplection Action: (Planck) Temperature is an element of the Dynamic Group Lie Algebra Heat is an element of the Dynamic Group Dual Lie Algebra ▌ In case of non-commutative groups, specific properties appear: the symmetry is spontaneously broken, some cohomological type of relationships are satisfied in the algebra of the Lie group 7 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS ▌ 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: ▌ Solution: with 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 d )(U M )(p                   QdpU dp dppsArgMin M M Mp     )()( 1)( such that)(log)( )(  M dpps  )(log)(    Gibbs Equilibrium: Solution of Maximum Entropy      M U M U de deU Q     )(, )(, )( )(,)( )(   U ep      M U de   )(, log)( )(,)(   QQs 8 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS ▌ This value is a strict minimum of s ▌ 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, with ▌ Identifying E with its bidual: )( Qdd .   Q 02 2     )( )(Q )(Q dQds . Q s    )(,)(   QQs    M U deUU   )(, )()( Gibbs Canonical Ensemble 9 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Gibbs Canonical Ensemble on Symplectic Manifold ▌ 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 extended this notion of 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. 10 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Gibbs Canonical Ensemble on Symplectic Manifold ▌ In classical statistical mechanics, a state is given by the solution of Liouville equation on the phase space, the partition function. ▌ As 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 energy (defined in dual of Lie Algebra of the dynamic group) the geometric temperature (defined in Lie Algebra of the dynamic group) a normalizing constant such the mass of probability is equal to 1 ▌ The Gibbs equilibrium state is extended to all Symplectic manifolds with a dynamic 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. )(,)(  U e  U  )(  11 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Gibbs Ensemble of a Dynamic Group ▌ Let M a Symplectic Manifold with a Dynamic Group G: is a Lie Group that acts on by « Symplectomorphisms » Let transformation of by the element : Let an element of Lie Algebra of Dynamic Group , we can associate a vectors Field defined on , that characterize infinitesimal action of Dynamic Group has a moment map, a differentiable application from the manifold to the dual of Lie Algebra : . The moment map is characterized by the equation: • Symplectic form of • arbitrary variation of point on • associate variation of to )( : xxg MMG g g   g Gg  G gZ g G MZ M G G * g UMx M * g g   )()(, ZUxZx M    M x x M U U x M M      )(exp)(exp xtZZxtZ t MMM    12 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Geometric (Planck) Temperature in the Lie Algebra ▌ Let a Group and a Manifold with a moment map , the Geometric (Planck) Temperature is all elements of Lie Agebra of such that the following integrals converges in a neighborhood of : notes the duality of and is the Liouville density on ▌ Theorem: The function is infinitly differentiable in (the largest open proper subset of ) and is nth derivative for all , the tensor integral is convergent: ▌ To each temperature , we can associate probability law on with distribution function (such that the probability law has a mass equal to 1): with and The set of these probalities law is Gibbs Ensemble of the Dynamic Group,  is the Thermodynamic Potential and Q is the Geometric Heat MG g J  G     M U deI   , 0 )( U, g * g Md 0I  C  g     M U n dUeI n   , )(  M )(,)(  U e       M U deI   )(, 0 loglog)( 0 1)(,)( )( I I UdeQ M U      * gQ 13 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Geometric Fisher Metric: Geometric Heat Capacity ▌ We can observe that the Geometric Heat is function of Geometric Temperature in Dual Lie Algebra : ▌ We have: ▌ Its derivative is a 2nd order symmetric tensor: ▌ This quatratic form is positive, and positive definite for each unless there exist a non null element such that (means that the moment varies in an affine sub-manifold of ) 0 1)(,)( )( I I UdeQ Q M U        * gg  Q  C  * g   Q QQ I I I II I IQ      1 2 0 11 0 2           M U dQUQUe Q    )(,)( Mx gZ 0,  ZQU U * g 0     Q 2 2        Q 14 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Geometric Entropy and Legendre Transform ▌ We have the inequality: ▌ The application is injective and as is derivative is invertible, then this application is a diffeomorphism of on the open ▌ We can then apply the Legendre Transform: from which we obtain the Shannon Entropy: 01001 ,  Q Q  Q Q  * g*      ,Q   )(,)( )(,)( with )(.log)(,)(,     U MM U ep QsdppdUe        15 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Legendre Transform ▌ We have the reciprocal formula: ▌ For Classical Thermodynamics (Time translation only), we recover the definition of Boltzmann Entropy:       ,)(Qs   Q Q s    s Q s Q     ,)( T dQ ds T Q s            1   16 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Example of Gallileo Group ▌ 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. Math. 59, 1954, pp 1–46) has proven that the symplectic cohomology space of the Galilean group is one-dimensional. ▌ Lie Algebra of Gallileo Group                                1100 10 1 ' ' t x e wuR t x  )3( ,and, ' ..' 3 SOR ReRwux ett wtuxRx                          xxso RR          :)3( ,and , 000 00 3 17 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Broken Symmetry and cocycle ▌ For Classical Thermodynamics, the group G (Group of time translation) leaves unchanged Gibbs equilibrium states. ▌ This is not true in the general case: the symmetry is broken. ▌ If we consider a Gibbs state, the probability law , its image by with is a probability law. ▌ To compute , we have to compute where is the moment map ▌ There exist an application such that: ▌ verifies the equality, proving that it is -cocycle of Group  Gg   g 1 gJ  J * g UMxJ : * gG:     )()())(,()( * gxJAdxJgaxJ gg    1*   gg AdUUAd   * g G GggggAdgg g  2112 * 21 ,,)()()( 1  0)( e )()( *1 gAdg g   g 18 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Souriau Moment(um) map ▌ is a linear application from to differential function on : ▌ the associated differentiable application , called moment(um) map: ▌ If instead of , we take the following momentum map: ▌ where is constant, the symplectic cocycle is replaced by ▌ where is one-coboundary of with values in . ▌ We have also properties XJ g M XJX RMC    ),(g J gg*  XXxJxJxJxMJ X ,),()(such that)(with:  J MxQxJxJ  ,)()(' * gQ  QAdQgg g * )()('  QAdQ g * '  G * g 0)(with)()()( 12 * 21 1  eggAdgg g  19 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Lie Group Action on Symplectic Manifold G g e * g  x *  )(g g )(xg )(xZM * gAd )(xJ ))(( xJ g     )()())(,()( * gxJAdxJgaxJ gg  20 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Distribution of probability by Group action ▌ The distribution density under the action of the Lie Group is given by: ▌ The set of Geometric Temperature is invariant by the adjoint action of ▌ If we use , we have the constraint ▌ By derivation of (**), we have: U e ,* ** :            gAdg g , , * 1**    )(*  gAd  G   )(  gAdg    Q 0,   Q (**)     0,,, ~  ZQZ    YX YX ),(YX, :, ~    gg  )()( eXTX e )()( *1 gAdg g  21 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Geometric (Planck) Temperature ▌ We have previously observed that: ▌ is called the Symplectic Cocycle of Lie algebra associated to the momentum map where linear application from to differential function on : and the associated differentiable application , called moment(um) map: ▌ is a 2-form of and verify: ▌ If we define: ▌ We can observe that :     0,,, ~  ZQZ   YX , ~  g J       MapMomenttheandBracketPoisson.,.with,),( ~ , JJJJYX YXYX  g YX, ~        0),,( ~ ),,( ~ ),,( ~  YXZXZYZYX      21222121 ,)(with)(,, ~ , ~ 11 ZZZadZadQZZZZ ZZ    ~ Ker   ,0, ~ g  XJ g M XJX RMC    ),(g J gg*  XXxJxJxJxMJ X ,),()(such that)(with:  22 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Fisher Metric of Souriau Lie Group Thermodynamics ▌ Souriau has introduced the Riemannian metric ▌ This metric is an extension of Fisher metric, an hessian metric: If we differentiate the relation ▌ The Fisher Metric is then a generalization of “Heat Capacity”:       2121 ,, ~ ,,, ZZZZg         21222121 ,)(with)(,, ~ , ~ 11 ZZZadZadQZZZZ ZZ    ~ Ker    gQAdAdQ gg   )()( *           ,., ~ ),.(,,., ~ ,., 1.11 1    ZAdQZZ Q Z              212.2121 ,, ~ ),(,,, ~ .,, 1 ZZZAdQZZZZ Q Z              21212 2 ,, ~ ,,, ZZZZg Q           kT 1  T Q kT T kT T QQ K                  2 1 )/1(  DC T Q T DCt T .with .        23 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Associated Riemannian Metric: Geometric Fisher Metric ▌ We can compute the image of Geometric Heat by the Lie Group action: ▌ By tangential derivative to the orbite with respect to and by using positivity of , we find: ▌ is a 2-form of that verifies: ▌ Then, there exists a symmetric tensor defined on ▌ With the following invariances:  gQAdQ g  )(** gZ 0     Q        2121 ,, ~ ,,, ZZZZg             0,,,,, ~ ,, ~  ZZQZZZZ   ~ g       0),,( ~ ),,( ~ ),,( ~  YXZXZYZYX g )(Zad      )()( 2 2 2 12     I g AdI g            ))(()(  QsAdQs g  24 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Souriau Theorem of Lie Group Thermodynamics ▌ Let be the largest open proper subset of , Lie algebra of , such that and are convergent integrals, this set is convex and is invariant under every transformation , where is the adjoint representation of , such that with . . Let a unique affine action such that linear part is coadjoint representation of , that is the contragradient of the adjoint representation. It associates to each the linear isomorphism , satisfying, for each , :  g   M U de  )(,   M U de   )(, .  (.)gAd (.)gAdg  geg iTAd  1 :  ghghig  ** gg Ga : a G G G Gg  )(* * gGLAdg  * g gX )(,),( 1 * XAdXAd gg   )( gAd   1  g ss   gQAdQgaQ g  )(),( * 25 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Souriau Model of Lie Group Thermodynamics Gibbs canonical ensemble  *  g * g R R        ,1  g      QQs , Q )(gAd  e g G  gQAdg )(* TEMPERATURE In Lie Alebra HEAT In Dual Lie Alebra LOG OF CHARACTERISIC FUNCTION ENTROPY ENTROPY IS INVARIANT (Could be is defintion) 26 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Action of a Lie Group on a Symplectic Manifold ▌ For Hamiltonian, actions of a Lie group on a connected symplectic manifold, the equivariance of the momentum map with respect to an affine action of the group on the dual of its Lie algebra has been studied by Chales-Michel Marle & Paulette Libermann: ▌ https://www.agnesscott.edu/lriddle/WOMEN /abstracts/libermann_abstract.htm ▌ Paulette Libermann, Legendre foliations on contact manifolds, Differential Geometry and Its Applications, n°1, pp.57-76, 1991 27 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Continuous Medium Thermodynamics ▌ For Continuous Medium Thermodynamics, « Temperature Vector » is no longer constrained to be in Lie Algebra, but only contrained by phenomenologic equations (e.g. Navier equations, …). ▌ For Thermodynamic equilibrium, the « Temperature Vector » is a Killing vector of Space-Time. ▌ For each point X, there is a « Temperature Vector » , such it is an infinitesimal conformal transform of the metric of the univers : ▌ Conservation equations can be deduced for components of Impulsion- Energy tensor and Entropy flux : )(X ijijji g  ˆˆ ijg     vectoreTemperaturofcomponent: derivativecovariant:.ˆ j i  EquationKilling0  0ˆ  ij iT 0 j iSij T j S ijk k ijijji g  2 www.thalesgroup.com OPEN Euler-Lagrange Equation of Lie Group Thermodynamics 29 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Euler-Poincaré Equation of Lie Group Thermodynamics ▌ When a Lie algebra acts locally transitively on the configuration space of a Lagrangian mechanical system, Henri Poincaré proved that the Euler- Lagrange equations are equivalent to a new system of differential equations defined on the product of the configuration space with the Lie algebra. ▌ Euler-Poincaré equations can be written under an intrinsic form, without any reference to a particular system of local coordinates, proving that they can be conveniently expressed in terms of the Legendre and momentum maps of the lift to the cotangent bundle of the Lie algebra action on the configuration space. 30 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Euler-Poincaré Equation of Lie Group Thermodynamics ▌ The Lagrangian is a smooth real valued function defined on the tangent bundle . To each parameterized continuous, piecewise smooth curve , ,defined on a closed interval , with values in , one associates the value at of the action integral: ▌ The partial differential of the function with respect to its second variable , which plays an important part in the Euler-Poincaré equation, can be expressed in terms of the momentum and Legendre maps: with the moment map, the canonical projection on the 2nd factor, the Legendre transform, with and L TM   Mtt 10 ,:  10 ,tt M          1 0 )( )( t t dt dt td LI   gML: Ld2 2   Lt pLd * g  )( 2   LJLdpJ t  * g * :* gg* g Mp MTTM * :L )(),(/: xXXxTMM M  g  )(),()(/: *  JMMT M tt  * g 31 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Euler-Poincaré Equation of Lie Group Thermodynamics ▌ The Euler-Poincaré equation can therefore be written under the form: ▌ Following the remark made by Poincaré at the end of his note, the most interesting case is when the map only depends on its second variable . The Euler-Poincaré equation becomes:       )(),( )( with)(),()(),( 1 * )( tVt dt td tVtLdJtVtJad dt d tV            L   RMTHMTTMMTLH   ***11 :,:,,)()(,)( LLL  RML g: gX    0)(* )(        tVLdad dt d tV     )(),()(),( 12 * )( tVtLdJtVtLdad dt d tV            )(),()( 2 tVtLdtVLd  32 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Euler-Poincaré Equation of Lie Group Thermodynamics ▌ We can use analogy of structure when the convex Gibbs ensemble is homogeneous. We can then apply Euler-Poincaré equation for Lie Group Thermodynamics. Considering Clairaut equation: with , ▌ A Souriau-Euler-Poincaré equation can be elaborated for Souriau Lie Group Thermodynamics: and ▌ An associated equation on Entropy is: ▌ That reduces to: Due to      )(),(, 11 QQQQQs    * g     )(Q g  )(1 Q Qad dt dQ *    0* QAd dt d g dt d QadQ dt d dt ds   * ,,   dt d Q dt d dt ds   ,  0,,,, **    adQQadXadXad VV New interesting Equations for Thermodynamics 33 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Poincaré-Cartan Integral Invariant of Lie Group Thermodynamics ▌ Analogies between Geometric Mechanics & Geometric Lie Group Thermodynamics, provides the following similarities of structures: ▌ We can then consider a similar Poincaré-Cartan-Souriau Pfaffian form: ▌ This analogy provides an associated Poincaré-Cartan Integral Invariant: transforms in ▌ For Thermodynamics, we can then deduce an Euler-Poincaré Variational Principle: The Variational Principle holds on , for variations , where is an arbitrary path that vanishes at the endpoints, :      Qp q            ,. )()( )()( QsLqpH QspH qL                        Q q L p Q s p H dt dq q       dtdtsQdtsdtQdtHdqp ).(.,..,..     ba CC dtHdqpdtHdqp ....   ba CC dtdt ).().(  g   ,  )(t 0)()(  ba    0.)( 1 0  t t dtt 34 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Question 35 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Question These formulas are universal , in that they do not involve the symplectic manifold U , but only Group G , the symplectic cocycle f and the couple  and Q. Perhaps this " Lie group thermodynamics" could be of mathematical interest. www.thalesgroup.com OPEN Seminal Idea of Souriau in Statistical Physics: Study of Multivariate Gaussian Case 37 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Souriau Chapter on Statistical Physics: Multivariate Gaussian Law http://www.jmsouriau.com/structure_ des_systemes_dynamiques.htm 38 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Maximum Entropy / Gibbs Density for Multivariate Gaussian Law ▌ if we take vector with tensor components , components of will provide moments of 1st and 2nd order of the density of probability , that is defined by Gaussian law. In this particular case, we can write: ▌ By change of variable given by , we can then compute the logarithm of the Koszul characteristic function: ▌ We can prove that 1st moment is equal to and that components of variance tensor are equal to elements of matrix , that induces the 2nd moment. The Koszul Entropy, its Legendre transform, is then given by:         zz z   )( p )(andwith 2 1 , nSymHRaHzzzax nTT  aHzHz 2/12/1 '       2logdetlog 2 1 )( 11 nHaHax T   aH 1  1 H     enH .2logdetlog 2 1 )( 1*    www.thalesgroup.com OPEN Balian Gauge Theory of Thermodynamics 40 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Compatible Balian Gauge Theory of Thermodynamics ▌ Entropy is an extensive variable depending on n independent extensive/conservative quantities characterizing the system ▌ The n intensive variables are defined as the partial derivatives: ▌ Balian has introduced a non-vanishing gauge variable which multiplies all the intensive variables, defining a new set of variables: ▌ The 2n+1-dimensional space is thereby extended into a 2n+2-dimensional thermodynamic space spanned by the variables ,where the physical system is associated with a n+1-dimensional manifold in , parameterized for instance by the coordinates and . S  n qqSq ,...,10  ),...,1( niqi  i i n i q qqS    ),...,( 1  nipp ii ,...,1,.0   T niqp i i ,...,1,0with,  M T n qq ,...,1 0p 41 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Compatible Balian Gauge Theory of Thermodynamics ▌ the contact structure in 2n+1 dimension: ▌ is embedded into a symplectic structure in 2n+2 dimension, with 1-form, as symplectization: ▌ The n +1-dimensional thermodynamic manifolds are characterized by : . The 1-form induces then a symplectic structure on : ▌ The concavity of the entropy , as function of the extensive variables, expresses the stability of equilibrium states. It entails the existence of a metric structure in the n-dimensional space : ▌ which defines a distance between two neighboring thermodynamic states:   n i i i dqdq 1 0 .~    n i i i dqp 0 . M 0 T   n i i i dqdpd 0   n qqS ,...,1 iq     n ji ji ji dqdq qq S Sdds 1, 2 22     n j j jii dq qq S d 1 2     n i i i n i ii dqdp p dqdds 001 2 1  42 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS Compatible Balian Gauge Theory of Thermodynamics ▌ We can observe that this Gauge Theory of Thermodynamics is compatible with Souriau Lie Group Thermodynamics, where we have to consider the Souriau vector : transformed in a new vector 1            n    ii pp .0    .0 0 10 p p p p n                www.thalesgroup.com OPEN Links with Natural Exponential Families Invariant by a Group: Casilis and Letac 44 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS NEF (Natural Exponential Families): Letac & Casalis ▌ Let a vector space of finite size, its dual. Duality braket with . Positive Radon measure on , Laplace transform is : ▌ Transformation defined on ▌ Natural exponential families are given by: ▌ Injective function (domian of means): ▌ And the inverse function: ▌ Covariance operator: E * E   EEx  * , x, E    E x dxeLEL )()(with,0: ,*         LEDu ,ofinterior)( * )(log)(   Lk  )(k   )(),()(,)( )(,      dxedxPF kx   E dxxPk )(,)('    )('Imwith)(:    kMM FF     FF MmmmkmV   ,)()()( 1'''   45 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS NEF (Natural Exponential Families): Letac & Casalis ▌ Measure generetad by a familly : ▌ Let an exponential family of generated by and with automorphisms of and , then the familly is an exponential familly of generated by ▌ Definition: An exponential familly is invariant by a group (affine group of ), if : (the contrary could be false) )()('such that,),()'()( ,* dxedxREbaFF bxa    F F E   vxgx : )(EGLg  E Ev    )(,),()(   PF E )( F G E FFG  )(,   )()(,  FF  46 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS NEF (Natural Exponential Families): Letac & Casalis ▌ Theorem (Casalis): Let an exponential familly of and affine group of , then is invariant by if and only: ▌ When is a linear subgroup, is a character of , could be obtained by the help of Cohomology of Lie groups . )(FF  E G E F G                 )())((, ,''' '' ,', :such that,:,: )(),( 1 1 2 * dxedxG vgabbb aaga G RGbEGa bxa t                    G b G a 47 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS NEF (Natural Exponential Families): Letac & Casalis ▌ If we define action of on by: we can verify that: ▌ the action is an inhomogeneous 1-cocycle: , let the set of all functions from to , called inhomogenesous n-cochains, then we can define the operators: G * E *1 ,,. ExGgxgxg t     )()(. 12121 gagaggga  a 0n n G * E  * , EGn     *1* ,,: EGEGd nnn              n n n i nii i nn n gggF gggggFggFgggFd ,,,1 ,,,,,1,,.,, 21 1 1 12112111         48 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS NEF (Natural Exponential Families): Letac & Casalis ▌ Let , with inhomogneous n-cocycles , the quotient is the Cohomology Group of with value in . We have:        1** Im,,,   nnn dEGBdKerEGZ n Z      *** ,/,, EGBEGZEGH nnn  G * E    xxggx EGEd   . ,: **0   GgxxgExZ  ,.;*0         )()(.,, ,,: 1212121 11 *2*1 gFggFgFgggFdFdF EGEGd          2 2112121 *1 ,),()(.;, GgggFgFgggFEGFZ    xxggFExEGFB  .)(,;, **1 49 OPEN Cedocumentnepeutêtrereproduit,modifié,adapté,publié,traduit,d'unequelconquefaçon,entoutou partie,nidivulguéàuntierssansl'accordpréalableetécritdeThales-©Thales2015TousDroitsréservés. Geometric Science of Information 2015 THALES AIR SYSTEMS NEF (Natural Exponential Families): Letac & Casalis ▌ When the Cohomology Group then Then if is an exponential familly invariant by , verifies ▌ For all compact Group, and we can express  cgIgaGgEc t d 1* )(,such that,     0, *1 EGH    *1*1 ,, EGBEGZ  )(FF  G    )()(, )(,, 1 dxedxgGg gbxgcxc       )()(with)()(, , 0 ,)(, dxedxdxeedxegGg xcxcgbxc     0, *1 EGH a )()(, )(: 1 gagAAg EGAGA t gg       GA(E)GA AAAGgg gggg ofgroup-subcompact)( ,', '' 2   cgIgacgacgcAGg t d t g 11 )()()(,pointfixed  