By R. Jancel
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Extra resources for Foundations of Classical and Quantum Statistical Mechanics
T This definition o f observed macroscopic quantities depends o n t h e fact that the recording devices always behave as integrators, whatever m a y b e t h e apparatus used. W e shall use another definition o f macroscopic quantities i n the second section, based o n t h e inherent inaccuracy o f any macroscopic o b servation (see Chapter V, § II. 1). 12 Ergodic Theory in Classical Statistical Mechanics The macroscopically small time Τ is generally assumed t o b e a very long one in relation t o t h e process of the return t o equil i b r i u m !
In what follows, we shall confine ourselves t o stressing two statements which appear t o us t o be essential: the first concerns the possibility of putting part of Khinchin's proof into t h e form of a theorem from t h e general theory of dynamics; the second concerns the difficulties encountered by Khinchin's theory because of the possible existence of primary integrals other than those of energy. In order t o be able t o discuss usefully these various points we shall first of all in this section analyse t h e role of the primary integrals of a Hamiltonian system in ergodic theory.
50). 67) express t h e complete dynamic content of Khinchin's m e t h o d ; they are valid for any definition of probability, provided it is invariant with respect t o the group Tt. However, in order that these results are applicable t o the ergodic theory, it is essential that this definition satisfies the following two conditions: 1. e. of the sum functions in Khinchin's theory) is small. 3* 35 Classical and Quantum Statistical Mechanics 2. That the probability for the occurrence of phases corresponding t o a set of very small measure is itself very small.