N. N. Bogolyubov's quantum method of generating functionals in statistical physics: The current Lie algebra, its representations, and functional equations

N. N. Bogolyubov; A. K. Prikarpatskii

doi:10.1007/bf01056817

N. N. Bogolyubov's quantum method of generating functionals in statistical physics: The current Lie algebra, its representations, and functional equations

Ukrainian Mathematical Journal ◽

10.1007/bf01056817 ◽

1986 ◽

Vol 38 (3) ◽

pp. 245-249

Author(s):

N. N. Bogolyubov ◽

A. K. Prikarpatskii

Keyword(s):

Lie Algebra ◽

Functional Equations ◽

Statistical Physics ◽

Quantum Method

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Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

10.20944/preprints201608.0078.v1 ◽

2016 ◽

Cited By ~ 1

Author(s):

Frédéric Barbaresco

Keyword(s):

Lie Algebra ◽

Lie Group ◽

Statistical Physics ◽

Geometric Theory ◽

Adjoint Action ◽

Information Geometry ◽

Affine Group ◽

Exponential Families ◽

Natural Exponential Families ◽

Fisher Metric

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 moment space, defining physical observables like energy, heat, and moment 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. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. 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. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of Information Geometry for the action of an affine Group for exponentiel families, and provide some illustration of use cases for multivariate Gaussian densities. Information Geometry is presented in the context of seminal work of Fréchet and his Clairait-Legendre equation. Souriau model of Statistical Physics is validated as compatible with Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

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Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

10.20944/preprints201608.0078.v2 ◽

2016 ◽

Author(s):

Frédéric Barbaresco

Keyword(s):

Lie Algebra ◽

Lie Group ◽

Statistical Physics ◽

Geometric Theory ◽

Adjoint Action ◽

Information Geometry ◽

Affine Group ◽

Exponential Families ◽

Natural Exponential Families ◽

Fisher Metric

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 moment space, defining physical observables like energy, heat, and moment 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. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. 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. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of Information Geometry for the action of an affine Group for exponentiel families, and provide some illustrations of use cases for multivariate Gaussian densities. Information Geometry is presented in the context of seminal work of Fréchet and his Clairaut-Legendre equation. Souriau model of Statistical Physics is validated as compatible with Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

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Structure of the Extended Schrödinger–Virasoro Lie Algebra $\widetilde{\mathfrak{sv}}$

Algebra Colloquium ◽

10.1142/s1005386709000522 ◽

2009 ◽

Vol 16 (04) ◽

pp. 549-566 ◽

Cited By ~ 18

Author(s):

Shoulan Gao ◽

Cuipo Jiang ◽

Yufeng Pei

Keyword(s):

Field Theory ◽

Conformal Field Theory ◽

Lie Algebra ◽

Lie Algebras ◽

Statistical Physics ◽

Two Dimensional ◽

Universal Central Extension ◽

Central Extensions ◽

Infinite Dimensional ◽

Conformal Field

We study the derivations, the central extensions and the automorphism group of the extended Schrödinger–Virasoro Lie algebra [Formula: see text], introduced by Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that [Formula: see text] is an infinite-dimensional complete Lie algebra, and the universal central extension of [Formula: see text] in the category of Leibniz algebras is the same as that in the category of Lie algebras.

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