Skip to main content
Log in

On intrinsic randomness of dynamical systems

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We discuss the problem of nonunitary equivalence, via positivity-preserving similarity transformations, between the unitary groups associated with deterministic dynamical evolution and semigroups associated with stochastic processes. Dynamical systems admitting such nonunitary equivalence with stochastic Markov processes are said to beintrinsically random. In a previous work, it was found that the so-called Bernoulli systems (discrete time) are intrinsically random in this sense. This result is extended here by showing that a more general class of dynamical systems—the so-calledK systems andK flows—are intrinsically random. The connection of intrinsic randomness with local instability of motion is briefly discussed. We also show that Markov processes associated through nonunitary equivalence tononisomorphic K flows are necessarily non-isomorphic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. Misra, I. Prigogine, and M. Courbage,Physica (Utrecht) 98A:1–26 (1979); see also an earlier shorter version inProc. Natl. Acad. Sci. USA 76:3607–3611 (1979).

    ADS  Google Scholar 

  2. I. Prigogine, C. George, F. Hennin, and L. Rosenfeld,Chem. Scr. 4:5–32 (1973).

    Google Scholar 

  3. Y. G. Sinai,Funct. Anal. Appl. 6:35 (1972).

    Article  Google Scholar 

  4. M. Aizenman, S. Goldstein, and J. L. Lebowitz,Commun. Math. Phys. 39:289–301 (1975).

    Article  ADS  Google Scholar 

  5. Y. G. Sinai,Usp. Mat. Nauk 27:137 (1972).

    Google Scholar 

  6. G. Gallavotti and D. Ornstein,Commun. Math. Phys. 38:83–101 (1974).

    Article  ADS  Google Scholar 

  7. Y. G. Sinai,Sov. Math. Dokl. 4:1818–1822 (1963).

    Google Scholar 

  8. D. Anosov,Proc. Steklov Inst. No. 90 (1967).

  9. D. Ornstein,Ergodic Theory, Randomness and Dynamical Systems (Yale University Press, New Haven, Connecticut, 1974).

    MATH  Google Scholar 

  10. B. Misra,Proc. Natl. Acad. Sci. USA 75:1627–1631 (1978).

    Article  ADS  Google Scholar 

  11. K. Goodrich, K. Gustafson, and B. Misra, On a converse to Koopman's lemma,Physica (Utrecht) 102A:379–388 (1980).

    ADS  MathSciNet  Google Scholar 

  12. E. B. Dynkin,Markov Processes (Springer, New York, 1965).

    Book  Google Scholar 

  13. V. I. Arnold and A. Avez,Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968).

    MATH  Google Scholar 

  14. M. Courbage, C. Coutsomitros, and B. Misra, On the isomorphisms of Markov processes associated with Bernoulli systems, submitted toAnn. Inst. Henri Poincaré.

  15. B. Simon,The P(φ) 2 Euclidean (Quantum) Fluid Theory (Princeton University Press, Princeton, New Jersey, 1974), proof of Theorem I.13.

    Google Scholar 

  16. M. Courbage and B. Misra, On the equivalence between Bernoulli dynamical systems and stochastic Markov processes, to appear inPhysica.

Download references

Author information

Authors and Affiliations

Authors

Additional information

Dr. Goldstein's research was supported in part by NSF Grant No. PHY78-03816.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goldstein, S., Misra, B. & Courbage, M. On intrinsic randomness of dynamical systems. J Stat Phys 25, 111–126 (1981). https://doi.org/10.1007/BF01008481

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01008481

Key words

Navigation