Randomness and Probability in Dynamical Theories: On the Proposals of the Prigogine School

I discuss recent work in ergodic theory and statistical mechanics, regarding the compatibility and origin of random and chaotic behavior in deterministic dynamical systems. A detailed critique of some quite radical proposals of the Prigogine school is given. I argue that their conclusion regarding the conceptual bankruptcy of the classical conceptions of an exact microstate and unique phase space trajectory is not completely justified. The analogy they want to draw with quantum mechanics is not sufficiently close to support their most radical conclusion.

[1]  R. Lindsay,et al.  The Conceptual Foundations of the Statistical Approach in Mechanics , 1959 .

[2]  S. Goldstein,et al.  On intrinsic randomness of dynamical systems , 1981 .

[3]  V. I. Arnolʹd,et al.  Ergodic problems of classical mechanics , 1968 .

[4]  M. Kac.  Foundations of Kinetic Theory , 1956 .

[5]  Maurice Courbage,et al.  On the equivalence between Bernoulli dynamical systems and stochastic Markov processes , 1980 .

[6]  P. Shields.  The theory of Bernoulli shifts , 1973 .

[7]  John Earman,et al.  A Primer on Determinism , 1986 .

[8]  Joel L. Lebowitz.  MICROSCOPIC DYNAMICS AND MACROSCOPIC LAWS * , 1981 .

[9]  S. Zabell,et al.  Why Gibbs Phase Averages Work--The Role of Ergodic Theory , 1980, Philosophy of Science.

[10]  R. Batterman.  Irreversibility and Statistical Mechanics: A New Approach? , 1990, Philosophy of Science.

[11]  Nikolai SergeevichHG Krylov,et al.  Works on the foundations of statistical physics , 1979 .

[12]  I Prigogine,et al.  From deterministic dynamics to probabilistic descriptions. , 1979, Proceedings of the National Academy of Sciences of the United States of America.

[13]  D. Ornstein.  Ergodic theory, randomness, and dynamical systems , 1974 .

[14]  B. O. Koopman,et al.  Hamiltonian Systems and Transformation in Hilbert Space. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[15]  M. Rayner,et al.  Mathematics for the Physical Sciences , 1969 .

[16]  Lawrence B. Sklar.  Statistical Explanation and Ergodic Theory , 1973, Philosophy of Science.

[17]  K. Gustafson,et al.  On converse to Koopman's Lemma , 1980 .

[18]  M. Courbage,et al.  Intrinsic irreversibility of Kolmogorov dynamical systems , 1983 .

[19]  S. V. Fomin,et al.  Ergodic Theory , 1982 .

[20]  On the Foundations of Kinetic Theory , 1980 .