Soshichi Uchii, Kyoto University

4. Mechanics and Probability: Maxwell's Adventure

Now getting back to Dr. Tomonaga's analysis, let us tackle the first problem; that is, "Is it all right that we apply probabilistic method on the basis of the deterministic theory of Newtonian mechanics?" Or, maybe we can divide this question into two: (a) Is it consistent to combine probabilistic method with the deterministic theory of Newtonian mechanics? (b) Is the use of probability justified by the Newtonian mechanics?

Now, in order to get better understanding of these problems, we need to know more about the kinetic theory of gases. As I said before, Maxwell began a conceptual adventure of introducing probabilistic method into physics ["Illustrations of the Dynamical Theory of Gases," 1860]. The basic idea of the kinetic theory is this: we can attribute pressure in gases to the random impacts of molecules against the walls of the container; likewise we may be able to explain other important properties of gases, or heat processes in general, by referring to the mechanical movements of a vast number of molecules. In order to make this idea feasible, Maxwell introduced the idea of the statistical distribution of velocities in a gas at uniform pressure (which means, in equilibrium). Of course many molecules are moving with various velocities within the container; but instead of considering each molecule individually, we count the number of molecules within a given range of velocity.

Technically, we can do it this way: let the components of molecular velocity in three axes be x, y, z. Then we can take a small interval for each variable, say x+dx, y+dy, and z+dz. And since the number of all molecules in the container is finite, we can count, in principle, how many molecules have velocities witin this small range. In this way, we can register the number of molecules for each velocity range, and if we review all of these numbers, that gives the statistical distribution of velocities within the given gas.

Maxwell found that this velocity distribution is exactly like the distribution of errors when we make many measurements of any physical quantity. Such distributions are called "normal distribution," and you know that they have a beautiful bell-shape, if you draw them in two-dimensional graphs. And Maxwell was able to determine relevant mean values of mechanical quantities, which are necessary for deriving various properties of a gas. But we need not go into details.

What is important here is that we have a strange mixture of mechanics and statistics. Notice that thermodynamic properties are connected with statistical properties of a set of large number of molecules. In order to derive, in the kinetic theory, the well-known formulas for gases (such as Boyle's or Gay-Lussac's law), we start from Newtonian mechanics for molecules; but somewhere in the derivation, we have to introduce such extra-assumptions as that the gas is in equilibrium, or that molecules are distributed uniformly within the container, or that each molecule has an equal probability for going into any direction, etc. Only with such assumptions we can derive the statistical distribution of velocities within the gas. And you can see, by the way, that the derivation of gas properties cannot be called a reduction in the strict sense, if we need such extra-assumptions in addition to Newtonian mechanics.

Notice that, these assumptions are used not as a mathematical means for calculation but as a factual or physical assumption about gas molecules. Thus the question of consistency or justifiablity becomes really important. The kinetic theory is not only an extended application of Newtonian mechanics to gases, but also an attempt to bring in new concepts and assumptions into Newtonian mechanics. How can we be sure that such an attempt is consistent, or justifiable on the basis of Newtonian mechanics?

One way to answer these questions is to investigate the process through which a gas reaches equilibrium. That is, starting from an arbitrary initial state of a gas, we inquire by what process the gas changes its state; presumably, in most cases the gas finally reaches equilibrium or the state with highest entropy. This way was pursued by Maxwell and, more energetically, by Boltzmann.

ｪｬ 5. Mechanics and Irreversibility

ｪｩ 3. A Quick Review of Thermodynamics

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March 1, 1999. (c) Soshichi Uchii