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Kinetic theory and ergodic properties.

I Prigogine¹, A P Grecos, C George

¹Faculté des Sciences, Université Libre de Bruxelles, 1050 Brussels, Belgium.

Proceedings of the National Academy of Sciences of the United States of America

|June 1, 1976

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Kinetic theory

Area of Science:

Statistical Mechanics
Dynamical Systems Theory

Background:

The justification of kinetic theory is often linked to ergodic theory.
Ergodic theory provides a framework for understanding the long-term average behavior of dynamical systems.

Purpose of the Study:

To investigate the foundational assumptions of kinetic theory.
To determine the role of the collision operator in the kinetic description of dynamical systems.
To challenge the necessity of ergodic theory for justifying kinetic theory.

Main Methods:

Analysis of the properties of the collision operator.
Derivation of asymptotic behavior for system states and observables.
Comparison with stability concepts in classical Hamiltonian systems.

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Main Results:

The asymptotic behavior of states and observables is determined by collisional invariants.
This determination is independent of the ergodicity of the system.
The collision operator's properties are key, not system ergodicity.

Conclusions:

Ergodic theory is not the sole or primary justification for kinetic theory.
Collisional invariants dictate long-term behavior in these dynamical systems.
Findings relate to stability concepts in Hamiltonian mechanics.