X. On the collision of elastic bodies
Samuel Hawksley Burbury · Philosophical Transactions of the Royal Society of London (A ) · 1892
Abstract In a paper read before the Society on June 11, Sir William Thomson expressed a doubt as to the general truth of the Maxwell-Boltzmann doctrine concerning the distribution of energy among a great number of mutually acting bodies, and suggested that certain test cases should be investigated. The test that he proposed on that occasion was a number of hollow elastic spheres, each of mass M, and each containing a smaller elastic sphere of mass m, free to move within a larger one. This pair he calls a doublet. This case is within the general proof of the doctrine given below. It is, however, I think amenable to a simpler treatment, which has been applied to the case of elastic spheres external to one another. 1. Every doublet has a centre of inertia of the sphere M and its imprisoned m. Let V be the velocity of that centre of inertia, R the relative velocity of M and m. If V and R be given in magnitude, R given in direction, V may have any direction, and in Maxwell’s distribution, for given direction of R, all directions of V are equally probable. Conversely, if, whatever be the values of V and R, for given direction of R all directions of V are equally probable, Maxwell’s law prevails. Now consider a very great number of doublets, all having their relative velocity and the velocity of centre of inertia within limits R, R + dR, and V, V + dV. Consider them before and after collisions between M and m. Nothing is changed by collision except the direction of R, and that change of direction is independent of the direction of V. Therefore after collision for given direction of R all directions of V are equally probable, and therefore Maxwell’s distribution prevails after as well as before collision, and is therefore not affected by collisions.