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From deterministic dynamics to probabilistic descriptions.

B Misra1, I Prigogine, M Courbage

  • 1Faculté des Sciences Université Libre de Bruxelles, Campus Plaine, Boulevard du Triomphe, 1050 Bruxelles, Belgium.

Proceedings of the National Academy of Sciences of the United States of America
|August 1, 1979
PubMed
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This study reveals that probabilistic Markov processes can arise from deterministic dynamics without information loss, using a "change of representation" for unstable systems. This finding offers new perspectives on statistical mechanics and irreversibility.

Area of Science:

  • Mathematical Physics
  • Statistical Mechanics
  • Dynamical Systems Theory

Background:

  • Deterministic dynamics typically require information loss (coarse-graining) to yield probabilistic descriptions.
  • Existing models often assume irreversible processes arise from information degradation.

Purpose of the Study:

  • To explore an alternative pathway for deriving probabilistic processes from deterministic dynamics.
  • To investigate if information-preserving transformations can link deterministic laws to stochastic Markov processes.
  • To analyze the role of system instability in this transformation.

Main Methods:

  • Developed a mathematical framework based on invertible, positivity-preserving, nonunitary similarity transformations.
  • Applied these transformations to convert unitary groups (deterministic dynamics) to contraction semigroups (stochastic Markov processes).

Related Experiment Videos

  • Explicitly constructed transformations for Bernoulli systems, introducing Lyapounov variables and an "internal time" operator.

    • Main Results:

    • Demonstrated that stochastic Markov processes can emerge from deterministic dynamics solely through a change of representation, without information loss.
    • Showcased this possibility for systems with high degrees of motion instability.
    • Constructed explicit transformations for Bernoulli systems, illustrating the method.

    Conclusions:

    • Deterministic dynamics can lead to probabilistic descriptions without information loss, challenging conventional understanding.
    • The approach provides a new perspective on irreversibility and entropy increase in physical systems.
    • The findings have significant implications for statistical mechanics and other physics domains.