Description of Experiments in Physics: A Dynamical Approach
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A way of describing repetitive experiments can be founded on the notions of the theory of dynamical systems. Dynamical invariants of a stationary system decompose its set of trajectories. Asymptotic statistical behaviour is uniform within a component which is not further decomposed by an invariant. The time average probabilities of classical systems become limits of relative frequencies when time and state space are discretized. A complete set of invariants is replaced by a parametric family of probability distributions. The parameters are precisely those factors the control of which specifies statistical laws of the repetitive experiment. Various ways of interpreting stationary probabilities are suggested, these interpretations depending on whether parameters exist, have been identified, or are randomized over.
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- Proofs of all the results of ergodic theory to, execpt for the ergodic decomposition theorem, can be found in e.g. Farquhar, Ergodic Theory in Statistical Mechanics (Wiley, New York 1964). For the latter result, see references in Cornfield, Fomin and Sinai, Ergodic Theory (Springer, Berlin 1982). I have developed further my ideas on probability on dynamical systems in my essays Ergodic Theory and the foundationf of probability, pp. 257-278 in Skyrms and Harper (eds.), Causation, Chance, and Credence, vol. 1 (Reidel, Dordrecht 1988). For Kelper motion see Stenberg, Celestial Mechanics, Part I, chapter II, p. 108 ff. particularly.Google Scholar