James Clerk Maxwell and Modern Physics

by Richard Glazebrook

In the introduction to this paper Maxwell points out, while there was then no means of measuring the quantities which occurred in Clausius’ expression for the mean free path, “the phenomena of the internal friction of gases, the conduction of heat through a gas, and the diffusion of one gas through another, seem to indicate the possibility of determining accurately the mean length of path which a particle describes between two collisions. In order, therefore, to lay the foundation of such investigations on strict mechanical principles,” he continues, “I shall demonstrate the laws of motion of an indefinite number of small, hard and perfectly elastic spheres acting on one another only during impact.”

Maxwell then proceeds to consider in the first case the impact of two spheres.

But a gas consists of an indefinite number of molecules. Now it is impossible to deal with each molecule individually, to trace its history and follow its path. In order, therefore, to avoid this difficulty Maxwell introduced the statistical method of dealing with such problems, and this introduction is the first great step in molecular theory with which his name is connected.

He was led to this method by his investigation into the theory of Saturn’s rings, which had been completed in 1856, and in which he had shown that the conditions of stability required the supposition that the rings are composed of an indefinite number of free particles revolving round the planet, with velocities depending on their distances from the centre. These particles may either be arranged in separate rings, or their motion may be such that they are continually coming into collision with each other.

As an example of the statistical method, let us consider a crowd of people moving along a street. Taken as a whole the crowd moves steadily forwards. Any individual in the crowd, however, is jostled backwards and forwards and from side to side; if a line were drawn across the street we should find people crossing it in both directions. In a considerable interval more people would cross it, going in the direction in which the crowd is moving, than in the other, and the velocity of the crowd might be estimated by counting the number which crossed the line in a given interval. This velocity so found would differ greatly from the velocity of any individual, which might have any value within limits, and which is continually changing. If we knew the velocity of each individual and the number of individuals we could calculate the average velocity, and this would agree with the value found by counting the resultant number of people who cross the line in a given interval.

Again, the people in the crowd will naturally fall into groups according to their velocities. At any moment there will be a certain number of people whose velocities are all practically equal, or, to be more accurate, do not differ among themselves by more than some small quantity. The number of people at any moment in each of these groups will be very different. The number in any group, which has a velocity not differing greatly from the mean velocity of the whole, will be large; comparatively few will have either a very large or a very small velocity.

Again, at any moment, individuals are changing from one group to another; a man is brought to a stop by some obstruction, and his velocity is considerably altered—he passes from one group to a different one; but while this is so, if the mean velocity remains constant, and the size of the crowd be very great, the number of people at any moment in a given group remains unchanged. People pass from that group into others, but during any interval the same number pass back again into that group.

It is clear that if this condition is satisfied the distribution is a steady one, and the crowd will continue to move on with the same uniform mean velocity.

Now, Maxwell applies these considerations to a crowd of perfectly elastic spheres, moving anyhow in a closed space, acting upon each other only when in contact. He shows that they may be divided into groups according to their velocities, and that, when the steady state is reached, the number in each group will remain the same, although the individuals change. Moreover, it is shown that, if A and B represent any two groups, the state will only be steady when the numbers which pass from the group A to the group B are equal to the numbers which pass back from the group B to the group A. This condition, combined with the fact that the total kinetic energy of the motion remains unchanged, enables him to calculate the number of particles in any group in terms of the whole number of particles, the mean velocity, and the actual velocity of the group.

From this an accurate expression can be found for the pressure of the gas, and it is proved that the value found by others, on the assumption that all the particles were moving with a common velocity, is correct. Previous to this paper of Maxwell’s it had been realised that the velocities could not be uniform throughout. There had been no attempt to determine the distribution of velocity, or to submit the problem to calculation, making allowance for the variations in velocity.