The evaluation of Gibbs' phase-integral for imperfect gases

Statistical mechanics is concerned primarily with what are known as “normal properties” of assemblies. The underlying idea is that of the generalised phase-space. The configuration of an assembly is determined (on classical mechanics) by a certain number of pairs of Hamiltonian canonical coordinates p, q , which are the coordinates of the phase-space referred to. Liouville's theorem leads us to take the element of volume d τ=Π dp dq as giving the correct element of a priori probability. Any isolated assembly is confined to a surface in the phase-space, for its energy at least is constant; when there are no other uniform integrals of the equations of motion, the actual probability of a given aggregate of states of the proper energy, i.e., of a given portion of the surface, varies as the volume, in the neighbourhood of points of this portion, included between two neighbouring surfaces of constant energies E, E + dE ; it therefore varies as the integral of (∂ E /∂ n ) −1 taken over the portion. If I be the measure of the total phase-space available, interpreted in this way, and i that of the portion in which some particular condition is satisfied, then i / I is the probability of that condition being satisfied.