Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics

07/11/2017
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We introduce poly-symplectic extension of Souriau Lie group Thermodynamics based on higher-order model of statistical physics introduced by R.S. Ingarden. This extended model could be used for small data analytics

Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics

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application/pdf Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics Frédéric Barbaresco
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We introduce poly-symplectic extension of Souriau Lie group Thermodynamics based on higher-order model of statistical physics introduced by R.S. Ingarden. This extended model could be used for small data analytics
Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics
application/pdf Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics

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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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adfa, p. 1, 2011. © Springer-Verlag Berlin Heidelberg 2011 Poly-Symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics F. Barbaresco, Advanced Radar Concepts, Thales Air Systems, France Abstract. We introduce poly-symplectic extension of Souriau Lie group Ther- modynamics based on higher-order model of statistical physics introduced by R.S. Ingarden. This extended model could be used for small data analytics. Keywords: Higher Order Thermodynamics, Lie Group Thermodynamics 1 Preamble As early as 1966, Souriau applied his theory of geometric mechanics to statisti- cal mechanics, developed in the Chapter IV of his book “Structure of Dynamical Sys- tems” [1,2], what he called “Lie group thermodynamics”. Using Lagrange’s viewpoint, in Souriau statistical mechanics, a statistical state is a probability measure on the man- ifold of motions. Souriau observed that Gibbs equilibrium is not covariant with re- spect to dynamic groups of Physics. To solve this braking of symmetry, Souriau in- troduced a new “geometric theory of heat” where the equilibrium states are indexed by a parameter  with values in the Lie algebra of the group, generalizing the Gibbs equilibrium states, where  plays the role of a geometric (Planck) temperature. We will generalize Souriau theory [4][5] in the framework of higher order thermodynam- ics as introduced by R.S. Ingarden [9-11] for mesoscopic systems. The Gibbs canoni- cal state results from the Maximum Entropy principle when the statistical mean value of energy is supposed to be known. Polish School has studied the maximum entropy inference with higher-order moments of energy (when not only mean values but also statistical moments of higher order of some physical quantities are taken into account). Ingarden in 1992 and Jaworski in 1981 have introduced the concept of second and higher-order temperatures, by assuming a distribution function which includes infor- mation not only on the average of the energy but also on higher-order moments, in particular 2nd moment related to fluctuations. This case should be considered in situa- tions where fluctuations are not negligible, such as near phase transitions or critical points, in metastable states in systems with a small number of degrees of freedom. Ingarden idea is that if we can measure more details, such as the first n cumulants of the energy, we can then introduce n high-order temperature, as the Lagrange multipli- ers when we maximize the Entropy with respect to these values:        2 2 1 0 2 2 1 2 1 . . 2 1 , , 1 U H H U H H e e Z P                  (2) Ingarden proposed that if we can measure the second cumulant of the energy (the fluctuation of the energy), the equilibrium state is not the canonical state, but would need two temperatures. Ingarden argues that for a macroscopic system there is very little difference between the two states, and that we would need a mesoscopic or microscopic system to be able to detect the higher temperature. W. Jaworski [7,8] has shown that the contribution to the total entropy, arising from the extra information corresponding to the higher-order moments, is o(N) , when N tends to infinity and N/V ratio is constant, with N the number of particles and V the volume. The main result of W. Jaworski is that from a purely thermodynamic point of view, the information cor- responding to the higher-order moments of extensive physical quantities is not essen- tial and can be neglected in the maximum entropy procedure. Jaworski showed that the maximum entropy inference has a certain stability property with respect to infor- mation corresponding to higher order moments of extensive quantities. It can serve as an argument in favor of the maximum entropy method in statistical physics and to understand better why these methods are successful. R.F. Streater [3] has prefered to say that the states with generalized temperatures are not in equilibrium, assuming that the final state, at large times, will be the canonical or grand canonical state depending on mixing properties. R. F. streater [3] intends that this occur even for a mesoscopic system, such as a few atoms, adding that his approach is equivalent to Ingarden model if the relaxation time from the state with generalized temperatures to the final equilibrium is very long. 2 Model of Souriau Lie Groups Thermodynamics In 1970, Souriau [1-2] introduced the concept of co-adjoint action of a group on its momentum space, based on the orbit method works, that allows to define physical observables like energy, heat and momentum or moment as pure geometrical objects. The moment map is a constant of the motion and is associated to symplectic coho- mology. In a first step to establish new foundations of thermodynamics, Souriau has defined a Gibbs canonical ensemble on a symplectic manifold M for a Lie group ac- tion on M. In classical statistical mechanics, a state is given by the solution of Liou- ville 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 the Liouville measure by the scalar function given by the generalized partition function ) ( , ) (    U e   defined by the energy U (defined in the dual of the Lie algebra of this dynamical group) and the geometric temperature  , where  is a normalizing constant such the mass of prob- ability is equal to 1,      M U d e     ) ( , log ) ( . Jean-Marie Souriau then generalizes the Gibbs equilibrium state to all symplectic manifolds that have a dynamical group. Souriau has observed that if we apply this theory for non-commutative group (Galileo or Poincaré groups), the symmetry has been broken. For each temperature  , element of the Lie algebra g, Souriau has introduced a tensor   ~ , equal to the sum of the co- cycle  ~ and the heat coboundary (with [.,.] Lie bracket):     ) ( , , ~ , ~ 2 2 1 2 1 1 Z ad Q Z Z Z Z Z     (3) This tensor   ~ has the following properties: Y X Y X ), ( ) , ( ~    where the map  is the symplectic one-cocycle of the Lie algebra g with values in * g , with   ) ( ) ( e X T X e   where  the one-cocycle of the Lie group G.   Y X, ~  is constant on M and the map       g g : , ~ Y X is a skew-symmetric bilinear form, and is called the symplectic two-cocycle of Lie algebra g associated to the moment map J , with the following properties:     Map Moment the with , ) , ( ~ , J J J J Y X Y X Y X    (4)       0 ) , , ( ~ ) , , ( ~ ) , , ( ~       Y X Z X Z Y Z Y X (5) where X J linear application from g to differential function on M : X J X R M C    ), , ( g and the associated differentiable application J , called mo- ment(um) map g g*    X X x J x J x J x M J X , ), ( ) ( such that ) ( , :  . The geometric temperature, element of the algebra g , is in the the kernel of the tensor   ~ :     ~ Ker such that   , 0 , ~ g         . The following symmetric ten- sor          2 1 2 1 , , ~ , , , Z Z Z Z g        , defined on all values of   ,. (.)    ad is positive definite, and defines extension of classical Fisher metric in Information Ge- ometry (as hessian of the logarithm of partition function):          . Im , , , ~ , , 2 1 2 1 2 1     ad Z Z Z Z Z Z g       g (6) with      . Im , , 0 , 2 1 2 1   ad Z Z Z Z g    (7) These equations are universal, because they are not dependent on the symplectic manifold but only on the dynamical group G, the symplectic two-cocycle  , the tem- perature  and the heat Q . Souriau called it “Lie groups thermodynamics”. Theorem (Souriau Theorem of Lie Group Thermodynamics). Let  be the larg- est open proper subset of g , Lie algebra of G, such that   M U d e    ) ( , and   M U d e     ) ( , . are convergent integrals, this set  is convex and is invariant under every transformation (.) g Ad . Then, the fundamental equations of Lie group thermo- dynamics are given by the action of the group:  Action of Lie group on Lie algebra: ) (  g Ad  (8)  Characteristic function after Lie group action:    , 1      g (9)  Invariance of entropy with respect to action of Lie group: s s  (10)  Action of Lie group on geometric heat:   g Q Ad Q g a Q g     ) ( ) , ( * (11) In the framework of Lie group action on a symplectic manifold, equivariance of moment could be studied to prove that there is a unique action a(.,.) of the Lie group G on the dual * g of its Lie algebra for which the moment map J is equivariant, that means for each M x :     ) ( ) ( )) ( , ( ) ( * g x J Ad x J g a x J g g      (12) Jean-Louis Koszul has analyzed Souriau model in his book “Introduction to symplec- tic geometry” [6]. Defining classical operation g     a G s sas a Ads , , 1 ,   g g,    b a b a b ada , , and G s Ad Ad t s   , 1 - s * with classical properties   g    a ad Ad a a , exp exp or   g   a ad Ad a t a , exp * exp , we can consider: M x sx x  ,  , * g  M :  , we have   v ax a v d , ), (    . If we study * g   M Ad s s M : *     , we have: a Ad d a d Ad a Ad d s s s 1 , , , * *              ) ( , ), ( , , ), ( 1 1 v a s d a sv d sv asx v asx s a Ad v d M s             a s d a Ad d M s , , *      and then prove that 0 , *   a Ad s d s M     (13) If we develop the cocycle given by G s x Ad sx s s    , ) ( ) ( ) ( *    , we can study G t s Ad s st s    , , (t) ) ( ) ( *       . If we note   g   b a b a , , ), ( d b a, c    ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( * * * * * t Ad s x Ad Ad tx Ad s x Ad stx st s t s s st                    By developing g     a M x a d x ad ax d a t , , ) ( ) ( ) (     , we obtain:     g      b a M x (x) b b a b a x b ax d , , , , , a , ), ( d , ), ( ), (       (14) We have then       g     b a b a b a b , , ), ( d , , , , a , b a, c       And the property             g     c b a b a c c a c b c , , , 0 , , , , c , b a, c    (15) If the moment map is transform as   b a b a c b a c , , ) , ( ) , ( ' '            (16) By considering this action of the group on dual Lie algebra ) ( ) , ( , * s Ad s s G s           * * g g We have the property that M x G s s x Ad x s sx s       , , ) ( ) ( ) ( ) ( *      where the cocycle is given by ) ( ) ( ) ( * x Ad sx s s      We can verify the following properties:                    ) ( ) ( ) ( , , * x x e Ad e e G e  * * g g (17)     * g                              , , , ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 1 2 * * 2 1 2 * 1 * * 2 1 * 2 1 2 1 1 2 1 2 1 G s s s s s s Ad Ad s s s Ad s Ad Ad s s Ad s s s s s s s s s (18) Finally using       g     b a b a b a b , , ), ( d , , , , a , b a, c       :                  c b a b a c b a b a b a , ) , ( , , , , ) ( ), ( * * * *     (19) 3 Poly-Sympectic Higher-Order Lie Groups Thermodynamics As observed by Jean-Marie Souriau, the Gausian density is a maximum entropy density of 1st order. This remark is clear if we replace z and ) , ( R m by  and  :                                                                                                                                      T T T T T T T T T T T R m z R z z R m m R m n m z R m z n R m E p p E I Fisher d e S m R m (R) π) ( n Z zz H za Tr Hz z z a H a R m R mm R m zz E z E zz z e Z p z p e e R e R z p T T T T                                          ˆ ˆ ) ( log ) ( log ) ( log ) ( : log ) ( log ) ( with ) ˆ ( and ) ( ) ( ˆ ) ( , ˆ ) ˆ ( and 2 1 det log 2 1 2 log 2 log with , where 2 1 and ˆ , with 1 ) ( ) ( ) det( 2 1 ) det( 2 1 ) ( 2 2 , 1 1 1 1 , ˆ ) , ( 2 1 2 1 2 / 1 2 / ) ( 2 1 2 / 1 2 / ) , ( * 1 1 1 1 (20) As soon as 1963, R. S. Ingarden has introduced the concept of higher order tem- peratures for statistical systems such as thermodynamics. In physics, the concept of temperature is connected with the mean value of kinetic energy of molecules in an ideal gas. For a general physical system with interactions among particles (non-ideal gas, liquid or solid), an equilibrium probability distribution is assumed to depend on temperature T as the only statistical parameter of the Gibbs state:   ) ( . 1 ) ( x H e Z x P      with T k  1  and   q p H x H , ) (  where p is position, q the mechanical momentum and  k the Boltzmann constant (a factor to insure that H .  is dimensionless). In case of no stochastic interactions between particles (ideal gas), partition function Z is integrable and we obtain Gauss distribution in the momentum space which corresponds to the result of the limit theorem for large N. Boltzmann ideal gas model can fail if the number of particles is not large enough (rnesoscopic systems), and if the interactions between particles are not weak enough. Gibbs hy- pothesis can also fail if stochastic interactions with the environment are not sufficient- ly weak. As remarked by R.S. Ingarden, nobody has never observed thermal equilib- rium of Gibbs in large and complex systems (Earth's atmosphere, cosmic systems, biological organisms), but only flows, turbulence or pumping, replacing classical approach by the concept of local temperature and thermodynamic flows (thermo- hydrodynamics and non-equilibrium thermodynamics), that is non-coherent with the concept of temperature which is global/intensive by definition and does not depend on position. R.S. Ingarden propose to consider the stationary case by means of the con- cept of higher order temperatures defined by:        n n n U x H U x H x H n e Z x P        ) ( ... ) ( ) ( . 1 ,..., 2 2 1 1 ,..., 1 ) (        (21) where   H E U  is the mean energy introduced to preserve the invariance of the total energy with respect to an arbitrary additive constant, and   n Z    ,..., log 1 0   the normalizing constant. The new constants k  are said to be -temperatures of order k. ) (x H is usually a quadratic function of x (for ideal gas only of p, for ideal solid of p and q). The probability distribution is fixed uniquely by all (independent and not contradictory) statistical moments which should be experimentally measured. But if the number of values is too large to make this method practical, we can measure only the lowest moments up to some order (if the higher orders do not change the result to a given accuracy), and to fix the respective -temperatures as Lagrange multipliers by maximization of entropy of distribution     dx x P x P S n n ) ( log ) ( ,..., ,..., 1 1        , with the given moments as additional conditions. R.S. Ingarden observed that the entropy maximization randomizes higher moments in a symmetric way, and it liquidates any possible bias with respect to their special values, and it gives the best estimate to a given accuracy. The values of  can be found by:   k k k Z x E          log 0 with             dx x P x dx e x Z x E n n k k k k x k k ) ( ,..., 1 1 1    (22) dx e Z n k k k x      1  and the relation:             n k k k n k k k Z x E S 1 0 0 1 log      (23) R.S. Ingarden has applied this model for linguistic statistics, assuming the appearance of higher order temperatures since there occur rather strong statistical correlations between phonemes and words as elements of these statistics. He argued his choice observing that in the case of word statistics, the existence of strong correlations is given by grammatical or semantical studies [9]. R. S. Ingarden made the conjecture that his high order thermodynamics is the model of statistically interacting, small systems, and biological living systems, although the calculation/observation are more difficult. We have seen that Souriau has replaced classical Maximum Entropy ap- proach by replacing Lagrange parameters by only one geometric “temperature vector” as element of Lie algebra. In parallel, Ingarden has introduced second and higher order temperature of the Gibbs state that could be extended to Souriau theory of ther- modynamics. Ingarden higher order temperatures could be defined in the case when no variation is considered, but when a probability distribution depending on more than one parameter. It has been observed by Ingarden, that Gibbs assumption can fail if the number of components of the sum goes to infinity and the components of the sum are stochastically independent, and if stochastic interactions with the environ- ment are not sufficiently weak. In all these cases, we never observe absolute thermal equilibrium of Gibbs type but only flows or turbulence. Non-equilibrium thermody- namics could be indirectly addressed by means of high order temperatures. Initiated by C. Gunther [12] and [13] based on n-symplectic model [14,15], it has been shown that the symplectic structure on the phase space remains true, if we re- place the symplectic form by a vector valued form, that is called polysymplectic. This extension defines an action of G over * ) ( * ... g g   n called n-coadjoint action:     n g g n n g n n n n g Ad Ad Ad g G Ad       * 1 * 1 ) ( * 1 * ) ( * * ) ( * ) ( * ,..., ,..., ... ... ... :                  g g g g (24) Let   n    ,..., 1  a poly-momentum, element of * ) ( * ... g g   n , we can define a n- coadjoint orbit   n    ,..., 1    at the point  , for which the canonical projection   k n n k     ,..., , ... : Pr 1 * * ) ( * g g g    induces a smooth map between the n-coadjoint orbit   and the coadjoint orbit k   :   k n k           ,..., 1 : that is a surjective submersion with   0 1    n k k KerT . Extending Souriau approach, equivariance of poly- moment could be studied to prove that there is a unique action a(.,.) of the Lie group G on * ) ( * ... g g   n for which the polymoment map   * ) ( * 1 ) ( ... : ,..., g g     n n n M J J J verifies M x and G g  :     ) ( ) ( )) ( , ( ) ( ) ( ) ( ) *( ) ( ) ( g x J Ad x J g a x J n n n g n g n      (25) with     n g g n n g J Ad J Ad x J Ad * 1 * ) ( ) *( ,..., ) (  and   ) ( ),..., ( ) ( 1 ) ( g g g n n     a poly- symplectic one-cocycle. We can also defined poly-symplectic two-cocycle   n n     ~ ,..., ~ ~ 1 ) ( with     k Y k X k Y X k k J J J Y X Y X , ), ( ) , ( ~ ,      where   ) ( ) ( e X T X k e k    . Finally, the poly-symplectic Souriau-Fisher metric is given by:              n k β ad Z Z ,Z Z Θ diag Z Z g k       ,..., , . Im , , ~ , , 1 2 1 2 1 2 1       g (26) with       ) ( , , ~ ,..., , ~ 2 2 1 1 2 1 1 Z ad Q Z Z Z Z Θ Z k k k n βk            (27) Compared to Souriau model, heat is replaced by previous polysymplectic model:   ... ,..., * ) ( * 1 g g     n n Q Q Q with           d e d e U Q M U M U k k n k n k k k n k k k                 1 1 ) ( , ) ( , 1 ). ( ) ,..., ( (28) with characteristic function:      d e M U n n k k k         1 ) ( , 1 log ) ,..., ( (29) We extrapolate Souriau results, who proved in [1][2] that     d e U M U k k k     ) ( , ). ( is locally normally convergent using multi-linear norm k U k U E Sup U ,   and where ... ) ( k U U U U k     is defined as a tensorial product (see [1] and Bourbaki). Entropy is defined by Legendre transform of Souriau-Massieu characteristic function:   ) ,..., ( , ,..., 1 1 1 n n k k k n Q Q Q S         where k n k Q Q Q S    ) ,..., ( 1  (30) The Gibbs density could be then extended with respect to high order temperatures by:             d e e e p M U U U Gibbs n k k k n k k k n k k k n                 1 1 1 1 ) ( , ) ( , ) ( , ,..., ) ( (31) 4 References 1. Souriau J.-M, Structures des systèmes dynamiques, Dunod, Paris, 1970 2. Souriau, J.-M., Mécanique statistique, groupes de Lie et cosmologie, Colloques int. du CNRS numéro 237, Géométrie symplectique et physique mathématique, 1974, pp. 59–113. 3. Nencka, H.; Streater, R.F. Information Geometry for some Lie algebras. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1999, 2, 441–460. 4. Barbaresco, F. Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry, Entropy 2016, 18, 386. 5. Marle, C.-M. 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