Soshichi Uchii, Kyoto University

8. Division between Mechanics and Probability

However, there are still problems. First of all, obviously, there are certain initial conditions of gas molecules for which the Ergodic Hypothesis does not hold. Then how should we treat such cases? To this, Boltzmann's answer would be that they are really exceptional cases as far as thermodynamics is concerned; we may confine ourselves to the vast majority of cases in which the hypothesis holds. [Actually, there are a number of difficult problems surrounding the Ergodic Hypothesis, but we will ignore them because they do not seem to be relevant in this context.] The second problem is more essential. Since the concept of time-average is mechanical, even if we can define probability in terms of it, the problem of irreversibility still remains. Granting that Boltzmann is justified in assuming the Ergodic hypothesis, how can he explain the irreversibility of the second law? Does he establish irreversibility on the basis of all reversible laws?

Let us recall Boltzmann's answer to Loschmidt's objection. There, the concept of probability has played an essential role for explaining the irreversibility, consistently with the reversibility of mechanical laws. Thus the irreversibility of the second law is explained in terms of the probabilistic asymmetry between high and low entropy. This asymmetry must now be understood in terms of larger or smaller time-average. Now, the question is, is it sufficient for explaining the irreversibility of thermodynamic processes?

It is here that Dr. Tomonaga's perceptive observation comes in. According to his interpretation of what Boltzmann was doing, the probability as time-average is connected with another concept of probability when we make observations or measurements of the thermodynamic processes of a gas.

You will notice the following: By our paraphrasing, we must have replaced probabilistic terms by mechanical terms; but still, Boltzmann's answer to Loschmidt has revived itself in our explanation. And furthermore, we must notice that its revival is closely connected with such an assertion as this: "if we are to observe the distribution at an arbitrary moment, we must expect there is almost no chance that we come across such a distribution as having the time-average very close to zero."

That we have here such words as "expect" or "chance" means that we have brought in the theory of probability again. And, therefore, Boltzmann's assertion against Loschmidt, that "the H-Theorem is established not by mechanical laws alone but by employing the theory of probability," revives itself again. [Tomonaga 1979, vol. 2, 124-125; my translation]

Now, what does this mean? Dr. Tomonaga is saying that, when we speak of "our expectation" or "chance," we are referring to another concept of probability not reducible to mechanical concepts. We may call this probability "epistemic" or "inductive" (these are not Dr. Tomonaga's words, but mine). Then, Dr. Tomonaga is also saying that we can explain the irreversibility of the second law in terms of this epistemic or inductive probability. However, in this new explanation, the probability theory occupies a completely different place from that in Boltzmann's or Maxwell's early theories. Namely, "it occupies a place where it is impossible to collide with the laws of Newtonian mechanics." Because,

when we say that we, at the moment of our measurement, come across such a distribution as having the time-average nearly equal to zero, we are talking about our own situations in which we make observations, not about the behavior of the given set of molecules. [op. cit., 125]

In short, the set of molecules move according to the deterministic laws of mechanics; but when we speak of our chances of finding such and such distributions within the set, this is a statement outside of, and entirely independent from, the Newtonian mechanics. [If you want to assert that this non-mechanical probability may be defined in terms of frequency or other physical concepts, this does not affect Tomonaga's point. Because, in such a definition, you still need some non-mechanical concepts like "randomness" or "stochastic process," which must therefore be brought in from other sources.]

Thus, according to Dr. Tomonaga, the Ergodic Hypothesis has accomplished three things: First, it has enabled us to use only mechanical concepts when we make "probabilistic" calculations within the kinetic theory. Secondly, by means of that, it has enabled us to separate a genuine probabilistic part from the basic mechanical part. And thirdly and finally, it has enabled us to justify the former by means of the latter.

9. Probability Applicable when we are Ignorant

7. Ergodic Hypothesis

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March 1, 1999. (c) Soshichi Uchii