Axiomatics of Classical Statistical Mechanics by Rudolf Kurth

Author:Rudolf Kurth
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2019-04-27T16:00:00+00:00

at time t. Hence, by the assumption of , the instant has been distinguished in a way which does not necessarily have any objective foundation.

This objection appears irrefutable. On the other hand, it is hard to see how that separation (or some other way of distinguishing the instant ) could be avoided. Hence it seems best to accept as a working hypothesis, but as nothing more.

15.5. Thus we have been led to the second fundamental problem of § 15.2. Can it justifiably be expected that the logical consequences drawn from a subjectively assumed probability distribution are objectively true? In the answer, two distinctions are to be made, one with respect to the initial probability distributions, the other one with respect to the conclusions drawn from them.

Initial probability distributions express either:

(a) exclusively certain knowledge, or

(b) include supplementary hypothetical assumptions.

Conclusions drawn from given initial probability distributions may be

(α) topological,

(ß) measure-theoretical, or

(γ) statistical,

in their mathematical character.

Examples of (a) and (b) have been given in § 15.3 and § 15.4. The alternative (α) means that if the initial point x of a given mechanical system is known to belong to a certain Lebesgue subset of the phase space Γ of finite positive measure, and if, according to the Rule 15.1, the initial probability density is defined by

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