# Lie-admissible structure of statistical mechanics

## Abstract

The objective of this paper is to identify the algebraic structure of statistical mechanics for Newtonian systems as they actually occur in nature, that is, with forces nonderivable from a potenial. For this objective, we first review the definition of dissipative forces, nonconservative forces, and, more generally, variationally nonselfadjoint forces, the latter being the forces which violate the integrability conditions for the existence of a potential. We then review Liouville's theorem in its original formulation, and show that this theorem is indeed capable of incorporating variationally nonselfadjoint forces. By using this background, we then pass to the identification of a number of properties of the statistical description of Newtonian systems of the class considered. Our major result consists of the proof of a theorem according to which the time evolution law of densities for the statistical description of variationally nonselfadjoint Newtonian systems can always be represented in terms of brackets verifying the laws of the Lie-admissible algebras. Since these algebras are an algebraic covering of the Lie algebras, the conventional Lie-algebra formulation of statistical mechanics is recovered as a subcase when all forces are derivable from a potential. This result establishes the direct universality of the Lie-admissible algebras in statisticalmore »

- Authors:

- (Univ. of Orleans, France)

- Publication Date:

- OSTI Identifier:
- 6759033

- Alternate Identifier(s):
- OSTI ID: 6759033

- Report Number(s):
- CONF-7908175-

Journal ID: CODEN: HAJOD

- Resource Type:
- Conference

- Resource Relation:
- Journal Name: Hadronic J.; (United States); Journal Volume: 3:1; Conference: 2. workshop on Lie-admissible formulations, Cambridge, MA, USA, 1 Aug 1979

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; LIE GROUPS; ALGEBRA; STATISTICAL MECHANICS; ALGEBRAIC FIELD THEORY; CLASSICAL MECHANICS; LIOUVILLE THEOREM; POTENTIALS; AXIOMATIC FIELD THEORY; FIELD THEORIES; MATHEMATICS; MECHANICS; QUANTUM FIELD THEORY; SYMMETRY GROUPS 658000* -- Mathematical Physics-- (-1987)

### Citation Formats

```
Fronteau, J., Tellez-Arenas, A., and Santilli, R.M.
```*Lie-admissible structure of statistical mechanics*. United States: N. p., 1979.
Web.

```
Fronteau, J., Tellez-Arenas, A., & Santilli, R.M.
```*Lie-admissible structure of statistical mechanics*. United States.

```
Fronteau, J., Tellez-Arenas, A., and Santilli, R.M. Sat .
"Lie-admissible structure of statistical mechanics". United States.
doi:.
```

```
@article{osti_6759033,
```

title = {Lie-admissible structure of statistical mechanics},

author = {Fronteau, J. and Tellez-Arenas, A. and Santilli, R.M.},

abstractNote = {The objective of this paper is to identify the algebraic structure of statistical mechanics for Newtonian systems as they actually occur in nature, that is, with forces nonderivable from a potenial. For this objective, we first review the definition of dissipative forces, nonconservative forces, and, more generally, variationally nonselfadjoint forces, the latter being the forces which violate the integrability conditions for the existence of a potential. We then review Liouville's theorem in its original formulation, and show that this theorem is indeed capable of incorporating variationally nonselfadjoint forces. By using this background, we then pass to the identification of a number of properties of the statistical description of Newtonian systems of the class considered. Our major result consists of the proof of a theorem according to which the time evolution law of densities for the statistical description of variationally nonselfadjoint Newtonian systems can always be represented in terms of brackets verifying the laws of the Lie-admissible algebras. Since these algebras are an algebraic covering of the Lie algebras, the conventional Lie-algebra formulation of statistical mechanics is recovered as a subcase when all forces are derivable from a potential. This result establishes the direct universality of the Lie-admissible algebras in statistical mechanics, in the sense that these algebras occur without redefinition of the local variables and physical quantities, as well as they hold for all systems of the class considered. We then tackle the problem of quantization of the statistics of dissipative phenomena, with particular reference to an algebraic reinterpretation of the approach by Prigogine and his collaborators. The paper ends with a number of comments indicating the physical relevance of our analysis.},

doi = {},

journal = {Hadronic J.; (United States)},

number = ,

volume = 3:1,

place = {United States},

year = {Sat Dec 01 00:00:00 EST 1979},

month = {Sat Dec 01 00:00:00 EST 1979}

}