Soshichi Uchii, Kyoto University

9. Probability Applicable when we are Ignorant

But with respect to the last point, you may raise a question: What is the nature of such a justification? In order to answer this, Dr. Tomonaga suggests the idea of incomplete knowledge. We often speak of chance when our knowledge is not sufficient for precise predictions (this idea is essentially due to Laplace, and endorced by Maxwell); and this applies here. We cannot solve a vast number of mechanical equations; moreover, we do not even know the exact initial conditions which are indispensable for precise predictions. Thus in the kinetic theory, we have to content ourselves with rougher knowledge of distribution functions, which correspond to thermodynamic quantities observable to us humans.

And, only in this sort of context, the concept of "the time-average in the long run," which is originally a mechanical concept but, functionally, exactly like probability, can be related to probability. That is, their relation is this: The probability that the moment we observe a given set of molecules happens to be within the time in which the set has a certain distribution, is equal to the time-average of that distribution in the long run. [op. cit., 127]

Thus, Dr. Tomonaga asserts that both concepts of probability, one mechanical, the other epistemic or inductive, are necessary in order to give a full picture of the second law of thermodynamics. He is suggesting, in other words, that both the mechanical point of view and the observational point of view are necessary; the latter being, within the context of the kinetic theory, nothing but the statistical point of view. Notice that the condition of our ignorance is essential, in order to combine the two concepts of probability by the preceding relation. For if we had sufficient knowledge about the behavior of molecules, we would be able to predict, with a high probability, when we can find a gas-state with a low entropy, i.e. a state with a very small value of time-average. Thus we can deliberately break the preceding relation of probability with time-average.

Getting back to Tomonaga's interpretation, he emphasizes that the grounds of the Ergodic Hypothesis and its use in deriving a unique value of the time-average are contained in the mathematical structure of the Newtonian mechanics. Without this basis, Tomonaga asserts, it would be impossible to bring in probability in between the behavior of a set of molecules and the humans observing their behavior. In this way, the use of non-mechanical probability and the explanation of irreversibility in terms of it are justified on the mechanical basis, provided that we are ignorant of the behavior of the molecules. This is how I understand Dr. Tomonaga's analysis. [And, whether or not Boltzmann's theory is completely satisfactory, Tomonaga's reconstruction is, it seems to me, one of the most sympathetic readings of Boltzmann's purposes.]

Aside from Dr. Tomonaga's argument, the need for the observational or statistical point of view may be understood in another way. The validity of the second law of thermodynamics of course depends (not exclusively, but essentially) on a vast number of observational data. Empirically, we are quite sure that, without some artificial device (like air-conditioner or refrigerator), heat does not flow from a cold body to a hot body. Now, we do not want to throw away this evidential support of second law when we try to reconstruct thermodynamics in the kinetic theory. Then, we have not only to look for a theoretical device for reconstructing the second law, but also to assure ourselves about its connection with the observational data. But such data are almost all of the statistical nature; that is, our device for measurement is such that it can collect nothing but average values or statistical means of a vast number of mechanical movements. Thus, although we have in the kinetic theory introduced the mechanical or microscopic point of view, this point of view must be supplemented by another point of view, as far as we want to be in touch with observational data. In other words, the need for the observational or statistical point of view is nothing but another way to express the indispensability of genuinely probabilistic concept, not reducible to any mechanical concept.

ｪｬ 10. The Case for or against Reductionism?

ｪｩ 8. Division between Mechanics and Probability

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March 1, 1999; last modified March 4. (c) Soshichi Uchii