A New Proof of the H Theorem

Boltzmann's $H$ theorem is here given a new proof based upon the properties of the probability coefficients that determine the transitions between different states of the total system or of any part of the system. The customary assumption is made that the frequency of these transitions is proportional to the population of the initial state, and the solutions of the equations determining statistical equilibrium are then fully discussed. These solutions either agree with those obtained directly with the aid of the principle of microscopic reversibility, or they divide into mutually independent and noncombining groups each of which in itself satisfies the principle of microscopic reversibility. The proof applies equally to Boltzmann, Bose and Fermi statistics. It involves a theorem concerning the rank of a matrix formed from coefficients appearing in the equations of equilibrium.