Soshichi Uchii, Kyoto University


2. What is the problem with the Kinetic Theory?

I was led to this question of reduction, and other related questions as well, by reading Dr. Tomonaga's last work, What is Physics? (2 vols., Iwanami, 1979). Dr. Shin'ichiro Tomonaga (1906-1979) was a famous physicist, Nobel Prize Winner; he is known for many excellent works in the field of quantum mechanics; but his last work was on historical and philosophical analysis of physics. [By the way, his father was a professor of philosophy at Kyoto Univertsity, a long time ago.] And I was particularly impressed by his analysis of the kinetic theory of gases, not because this part was written on his death-bed, but because this is undoubtedly the best part of his book. In this part, he begins with Maxwell's adventure of introducing statistical method into physics, then explains Boltzmann's attempt to define entropy in terms of mechanical concepts, and finally comes to the crucial question of irreversibility.

Why is the last question crucial? There are two interrelated problems: (1) First, the basis of the kinetic theory is of course Newtonian mechanics, which is a typical example of deterministic theories; then how can we reconcile this deterministic character with the probabilistic or statistical method, which is also essential in the kinetic theory? To put it crudely, within Newtonian mechanics, there is no room for probability, since every motion of any particle is uniquely determined, given its initial condition, by the laws of motion. Then why and where do we need probability, or is it permissible at all that we introduce probability in order to handle the behavior of a bunch of molecules? Dr. Tomonaga's analysis makes it clear how these questions came to be realized, to be deepened, and to be answered.

(2) Second, the laws of motion are all reversible, in the sense that if a particle moves in a certain way then the reverse motion is also possible according to the same laws. But the second law of thermodynamics is not reversible in this sense; it prohibits, for many thermodynamic processes, the reverse change. In technical words, the entropy of a closed system cannot decrease (if you don't know this, I'll explain it in a moment). Then, the question is, why and how can such an irreversible law derivable, in the kinetic theory, from the mechnical laws which are all reversible? Dr. Tomonaga argues that Boltzmann's attempt to answer this irreversibility question were on the right track, and we can reconstruct it as giving satisfactory answers to both of the preceding two questions.


3. A Quick Review of Thermodynamics

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March 1, 1999; last modified March 26. (c) Soshichi Uchii

suchii@bun.kyoto-u.ac.jp