Working PaperArticleVersion 1This version is not peer-reviewed
Lie Group Statistics and Lie Group Machine Learning Based on Souriau
Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New
Entropy Definition as Generalized Casimir Invariant Function in
Coadjoint Representation
Version 1
: Received: 4 March 2020 / Approved: 6 March 2020 / Online: 6 March 2020 (02:49:26 CET)
How to cite:
Barbaresco, F. Lie Group Statistics and Lie Group Machine Learning Based
on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher
Metric: New Entropy Definition as Generalized Casimir Invariant Function
in Coadjoint Representation. Preprints2020, 2020030099
Barbaresco, F. Lie Group Statistics and Lie
Group Machine Learning Based on Souriau Lie Groups Thermodynamics
& Koszul-Souriau-Fisher Metric: New Entropy Definition as
Generalized Casimir Invariant Function in Coadjoint Representation.
Preprints 2020, 2020030099
Cite as:
Barbaresco, F. Lie Group Statistics and Lie Group Machine Learning Based
on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher
Metric: New Entropy Definition as Generalized Casimir Invariant Function
in Coadjoint Representation. Preprints2020, 2020030099
Barbaresco, F. Lie Group Statistics and Lie
Group Machine Learning Based on Souriau Lie Groups Thermodynamics
& Koszul-Souriau-Fisher Metric: New Entropy Definition as
Generalized Casimir Invariant Function in Coadjoint Representation.
Preprints 2020, 2020030099
Abstract
In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in
Statistical Mechanics in the framework of Geometric Mechanics. This
Souriau’s model considers the statistical mechanics of dynamic systems
in their "space of evolution" associated to a homogeneous symplectic
manifold by a Lagrange 2-form, and defines in case of non null
cohomology (non equivariance of the coadjoint action on the moment map
with appearance of an additional cocyle) a Gibbs density (of maximum
entropy) that is covariant under the action of dynamic groups of physics
(eg, Galileo's group in classical physics). Souriau Lie Group
Thermodynamics was also addressed 30 years after Souriau by R. F.
Streater in the framework of Quantum Physics by Information Geometry for
some Lie algebras, but only in the case of null cohomology. Souriau
method could then be applied on Lie Groups to define a covariant maximum
entropy density by Kirillov representation theory. We will illustrate
this method for homogeneous Siegel domains and more especially for
Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by
using its Souriau’s moment map. For this case, the coadjoint action on
moment map is equivariant. For non-null cohomology, we give the case of
Lie group SE(2). Finally, we will propose a new geometric definition of
Entropy that could be built as a generalized Casimir invariant function
in coadjoint representation, and Massieu characteristic function, dual
of Entropy by Legendre transform, as a generalized Casimir invariant
function in adjoint representation, where Souriau cocycle is a measure
of the lack of equivariance of the moment mapping.
Subject Areas
Lie Groups Thermodynamics; Lie Group Machine Learning; Kirillov
Representation Theory; Coadjoint Orbits, Moment Map; Covariant Gibbs
Density; Maximum Entropy Density; Souriau-Fisher Metric; Generalized
Casimir Invariant Function
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Frédéric Barbaresco