On the Stationary Motion of a System of Equal Elastic Spheres of Finite Diameter
Samuel Hawksley Burbury · Proceedings of the London Mathematical Society · 1896
1.The object of this paper is to prove that in such a system in stationary motion the velocities of spheres near to one another are correlated.That is, that the chance that n spheres forming a group together in, space shall simultaneously have component velocities is of the form Ae'^dti^ ... dw n , and Q is not merely the sum of the squares, as in Maxwell's system, but a quadratic function, of the velocities, namely,the second term containing the products of every pair of «'s, &c, but no products uv, uw, or vw, and b being a function of the distance at the instant between the two molecules whose velocities are u, &G., and n', &c, which becomes evanescent as that distance increases.The sufficiency of Maxwell's distribution of velocities, according to which Q contains only the squares of the velocities, has been proved by several writers, but always on a certain fundamental assumption, which is true only for infinitely small densities-The assumption is in effect this-let fdxdydzdudvdw be the number per unit volume of molecules Avhose coordinates of position, x, y, z, and momenta w, v, w, are in a certain state A, or, as we may express it, fdx ... dw is the chance that a molecule shall be in that state.Similarly, f'dx ... dw' is the chance that another molecule shall be in the state A'.Then it is always assumed that the chance of two molecules being in the states A, A' respectively is ff'dn ... dw'.That is, it is assumed that the chances are under all circumstances, i.e., even when the two molecules are on the point of collision, independent.*• See especially Boltzmann's Vorhstuigen tiler Gas Theorie, 1895, p. 22, where the assumption is made emphatically.