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From instability to irreversibility.

Y Elskens1, I Prigogine

  • 1Service de Chimie Physique II, C.P. 231, Université Libre de Bruxelles, Bvd, du Triomphe, B-1050 Bruxelles, Belgium.

Proceedings of the National Academy of Sciences of the United States of America
|August 1, 1986
PubMed
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This study explores the physical meaning of transforming unitary evolution groups into contracting semigroups for unstable dynamical systems. It introduces Boltzmann ensembles, which evolve to equilibrium, validating Boltzmann's intuition and advancing microscopic irreversibility research.

Area of Science:

  • Statistical Mechanics
  • Dynamical Systems Theory
  • Mathematical Physics

Background:

  • Recent work developed a transformation from unitary evolution groups U(t) to contracting semigroups W(t) for phase-space ensembles in Kolmogorov dynamical systems.
  • This transformation is crucial for understanding the behavior of unstable dynamical systems where trajectories with similar future asymptotic behavior are indistinguishable.

Purpose of the Study:

  • To investigate the physical meaning of the U(t) to W(t) transformation for phase-space ensembles.
  • To demonstrate how unstable dynamical systems can be described using bundles of converging trajectories.
  • To establish a link between these descriptions and Boltzmann's foundational concepts in statistical mechanics.

Main Methods:

  • Focusing on unstable dynamical systems with finite precision phase-space identification.

Related Experiment Videos

  • Utilizing the concept of bundles of converging trajectories as fundamental objects.
  • Representing ensembles with finite measure support as distribution functions (Boltzmann ensembles).

    • Main Results:

    • Demonstrated that Boltzmann ensembles evolve to equilibrium under a markovian semigroup.
    • Showed that standard Gibbs-Koopman ensembles obeying the Liouville equation emerge as a singular limit.
    • Validated Boltzmann's intuition for a specific class of unstable dynamical systems.

    Conclusions:

    • The transformation provides a framework for understanding irreversibility in unstable dynamical systems.
    • This approach represents a step towards deriving microscopic equations that exhibit irreversibility.
    • The concept of Boltzmann ensembles offers a new perspective on equilibrium in statistical mechanics.