Soshichi Uchii, Kyoto University

7. Ergodic Hypothesis

Boltzmann tried to resolve the problem by means of "the Ergodic Hypothesis." The intuitive idea of this hypothesis may be informally explained this way. We have already seen that the assumption of equiprobability of all micro-states leads to the conclusion that the state of equilibrium is most probably realized in any gas, thereby supporting the second law of thermodynamics in a probabilistic way. So if we can justify this equiprobability assumption by mechanical means, Boltzmann's problem will disappear. The Ergodic Hypothesis is supposed to do such a job in a very sophisticated manner.

This hypothesis states that a system of molecules will assume, in the long run, all conceivable micro-states that are compatible with the conservation of energy. And if we can assume this hypothesis, then there is a unique time-average for any macro-state; indeed, we can calculate its value within the Newtonian mechanics. To be more specific, remember that (1) the time-average was a mechanical concept, and that (2) Boltzmann defined probability in terms of this mechanical concept. This means that the mechanical concept of time-average has the same mathematical structure as that of probability. Thus on the assumption of Ergodic Hypothesis, we can calculate the value of probability, within the framework of mechanics. And the H-Theorem should now be understood as saying that the time-average of the equilibrium state or those states which are close to the equilibrium is as large as being almost identical to one. [Actually, the concept of ergodicity has raised many difficult problems; but for simplicity we will ignore them. The philosophical import of the Ergodic Hypothesis can be grasped without getting involved with such problems. If you have a difficulty with the preceding formulation, the following version may help: "A system of molecules will go, in the long run, arbitrarily close to every conceivable micro-state."]

8. Division between Mechanics and Probability

6. Can we define Probability in the Kinetic Theory?

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March 1, 1999. (c) Soshichi Uchii