# Koszul information geometry and Souriau Lie group thermodynamics

## Abstract

The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from 'Characteristic Functions', was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by 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, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, 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 (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, 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,more »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22390874

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: AIP Conference Proceedings; Journal Volume: 1641; Journal Issue: 1; Conference: MAXENT 2014: Conference on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Clos Luce, Amboise (France), 21-26 Sep 2014; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CAPACITY; ENTROPY; EQUILIBRIUM; GEOMETRY; HEAT; INDEXES; LIE GROUPS; METRICS; POTENTIALS; PROBABILITY; SPACE-TIME; SPECIFIC HEAT; THERMODYNAMICS; VECTORS

### Citation Formats

```
Barbaresco, Frédéric, E-mail: frederic.barbaresco@thalesgroup.com.
```*Koszul information geometry and Souriau Lie group thermodynamics*. United States: N. p., 2015.
Web. doi:10.1063/1.4905965.

```
Barbaresco, Frédéric, E-mail: frederic.barbaresco@thalesgroup.com.
```*Koszul information geometry and Souriau Lie group thermodynamics*. United States. doi:10.1063/1.4905965.

```
Barbaresco, Frédéric, E-mail: frederic.barbaresco@thalesgroup.com. Tue .
"Koszul information geometry and Souriau Lie group thermodynamics". United States.
doi:10.1063/1.4905965.
```

```
@article{osti_22390874,
```

title = {Koszul information geometry and Souriau Lie group thermodynamics},

author = {Barbaresco, Frédéric, E-mail: frederic.barbaresco@thalesgroup.com},

abstractNote = {The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from 'Characteristic Functions', was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by 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, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, 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 (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, 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. These elements has been developed by author in [10][11].},

doi = {10.1063/1.4905965},

journal = {AIP Conference Proceedings},

number = 1,

volume = 1641,

place = {United States},

year = {Tue Jan 13 00:00:00 EST 2015},

month = {Tue Jan 13 00:00:00 EST 2015}

}