Soshichi Uchii, Kyoto University

6. Can we define Probability in the Kinetic Theory?

However, there is still a smell of question-begging in this reply by Boltzmann. For, in order for this reply to work, he has to show, first of all, that his use of probability in deriving the H-Theorem is indeed consistent with, or justifiable on the basis of, the Newtonian mechanics. But one of the key ideas for this was contained in his 1872 paper [actually, the same idea is already in Boltzmann 1868, 1871]. There, he had already suggested that the probability in the kinetic theory can be defined in terms of the time-average of a gas-state: Since a gas may be in a certain (macroscopic) state, we can measure the length of time during which the gas is in that state. And just as we can define the relative frequency of an event in the long run, we can define the time-average in the long run of each (macroscopic) state of the gas. Namely, the time-average of a state is the relative proportion of time (in the long run) in which the gas has that state. Then we can identify this time-average with the probability of that state.

probability of a gas-state = time-average of the state

= relative proportion of time during which the gas was in that state

In order to see the merit of this definition of probability, let us compare this with another proposal by Boltzmann (1877a) in which he explained the notion of probability in terms of the number of micro-states, or what he called "complexions." This may be explained in the following way: Let us distinguish a macro-state of gas from a micro-state of the molecules constituting the gas. The former corresponds to a thermodynamic state (which we can measure), the latter to a mechanical state (which we cannot usually measure); to be more precise, a micro-state (complexion) is a particular way to distribute a given amount of energy among a specified number of molecules. And a macro-state can be represented in the kinetic theory by a distribution function (i.e., a function determinig the statistical distribution of molecules to all possible values of position and velocity). For any given macro-state, there are many micro-states which correspond to it.

If you think this is too abstract, just imagine a deck of cards. A state of the deck in which all red cards are separated from the black cards is an example of a macro-state. In order to describe such a state, we don't have to refer to individual cards; at most the number of cards and their relative position are relevant (e.g., 26 red cards on the top, 26 black cards on the bottom). But if you refer to each individual card, including its order within the whole, and thereby list a particular combination of 52 cards, this is an example of a micro-state. Of course, there are thousands of such combinations corresponding to the preceding macro-state.

Now, for any given macro-state, we can count, theoretically, the number of micro-states corresponding to it. Or more exactly, we can determine its relative proportion to the whole possible micro-states. Then, Boltzmann's proposal is that, assuming that each micro-state is equally probable, we can measure the probability of a macro-state by this relative proportion. In this way, we can define the probability of a macro-state, and moreover we can show that the state of equilibrium is the most probable one; but clearly, since this presupposes the equiprobability of each micro-state, it fails to give the meaning of probability entirely in terms of physical concepts.

The strength of the former definition of probability in terms of time-average is that it has given the meaning of probability all in terms of mechanical concepts. We start from an actual gas or system of molecules; it goes through many changes by collisions etc., and we measure the time each macro-state assumes in the whole process. Thus the probability in this sense can be defined within the framework of Newtonian mechanics, without bringing in any dubious assumptions--- so it seems.

But still, there is a problem. You can define probability in this way, all right. But how can you assure that we can obtain a unique value of probability in this sense? In other words, how can you assure that a unique time-average exists for each macro-state? That was the crucial problem for Boltzmann.

7. Ergodic Hypothesis

5. Mechanics and Irreversibility

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March 1, 1999. (c) Soshichi Uchii