Soshichi Uchii, Kyoto University

5. Mechanics and Irreversibility

Boltzmann generalized Maxwell's approach and succeeded in obtaining a formula which determines the changes in the distribution (of velocities as well as positions) in a gas, resulting from collision between molecules and from external forces. The formula is called "Boltzmann's transport equation" (1872). Moreover, he succeeded in defining the mechanical equivalent of the concept of entropy; and he proved the mechanical equivalent of the second law of thermodynamics. In other words, he showed that a gas in any arbitrary initial state will, as a result of collisions, tend to approach to the state of equilibrium, with highest entropy. This is called, by later people, "Boltzmann's H-Theorem."

But, still, there is something strange about this H-Theorem. For, as we have already seen, the mechanical laws are all reversible. Then by what magic could we derive a law which tells us that certain mechanical procecces are irreversible? But since the H-Theorem translates the second law of thermodynamics into mechanical language, it in effect says that certain mechanical processes are irreversible! [Maxwell was already aware of this strangeness about the second law of thermodynamics which can be "proved" within the kinetic theory; because "Maxwell's demon" introduced in his Theory of Heat (1871) is relevant to this problem. The point of this "demon" is that , if such a demon can exist, then the second law does not hold.] Thus the irreversibility of the second law poses a great difficulty to the kinetic theory.

Josef Loschmidt, a colleague of Boltzmann's, raised this question in his 1876 paper. In order to make it clear that the H-Theorem cannot be proved by mechanics alone, he put forward the following sort of consideration: given a gas in a certain state towards equilibrium, suppose the velocities of all molecules are reversed at some instant; then, clearly, the gas goes from a state of high entropy back to a state of low entropy. That is to say, if the moleculer movements towards equilibrium are possible, the reverse movements must be possible as well, according to the same mechanical laws. And this is against the H-Theorem.

Boltzmann quickly responded to this objection (1877). The essence of his replies is as follows: Indeed we cannot derive the H-Theorem from mechanical laws alone, and we must admit that the Theorem cannot be proved for all initial conditions of a gas; we employed probabilistic assumptions also in order to determine the changes in distribution caused by collisions of molecules. Thus whether or not the H-Theorem holds is a matter of probability. What H-Theorem really means is that it is overwhelmingly probable that the gas tends to approach the state of equilibrium, as time goes on, as the result of collisions of a vast number of molecules. The crucial point is that since there are vastly many more uniform than non-uniform distributions, the number of states which lead to uniform distributions is much greater than the number that lead to non-uniform ones.

Thus, according to Boltzmann, the H-Theorem does not contradict the reversibility of mechanical laws, and at the same time, it can explain the irreversibility of the second law of thermo-dynamics in terms of probability.

6. Can we define Probability in the Kinetic Theory?

4. Mechanics and Probability: Maxwell's Adventure

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March 1, 1999. (c) Soshichi Uchii