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.
Similar content being viewed by others
References
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).
I. Prigogine, C. George, F. Hennin, and L. Rosenfeld,Chem. Scr. 4:5–32 (1973).
Y. G. Sinai,Funct. Anal. Appl. 6:35 (1972).
M. Aizenman, S. Goldstein, and J. L. Lebowitz,Commun. Math. Phys. 39:289–301 (1975).
Y. G. Sinai,Usp. Mat. Nauk 27:137 (1972).
G. Gallavotti and D. Ornstein,Commun. Math. Phys. 38:83–101 (1974).
Y. G. Sinai,Sov. Math. Dokl. 4:1818–1822 (1963).
D. Anosov,Proc. Steklov Inst. No. 90 (1967).
D. Ornstein,Ergodic Theory, Randomness and Dynamical Systems (Yale University Press, New Haven, Connecticut, 1974).
B. Misra,Proc. Natl. Acad. Sci. USA 75:1627–1631 (1978).
K. Goodrich, K. Gustafson, and B. Misra, On a converse to Koopman's lemma,Physica (Utrecht) 102A:379–388 (1980).
E. B. Dynkin,Markov Processes (Springer, New York, 1965).
V. I. Arnold and A. Avez,Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968).
M. Courbage, C. Coutsomitros, and B. Misra, On the isomorphisms of Markov processes associated with Bernoulli systems, submitted toAnn. Inst. Henri Poincaré.
B. Simon,The P(φ) 2 Euclidean (Quantum) Fluid Theory (Princeton University Press, Princeton, New Jersey, 1974), proof of Theorem I.13.
M. Courbage and B. Misra, On the equivalence between Bernoulli dynamical systems and stochastic Markov processes, to appear inPhysica.
Author information
Authors and Affiliations
Additional information
Dr. Goldstein's research was supported in part by NSF Grant No. PHY78-03816.
Rights 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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF01008481