On partitions of bipartite numbers

Abstract

1. In statistical mechanics, we usually deal with assemblies containing a fixed number of particles in which energy is the only conserved quantity. Recently, Fermi (1) has shown that the angular distribution of the pions produced in high-energy nuclear collisions can be explained if one takes into account the conservation of angular momentum in addition to the conservation of energy. We are, therefore, led to discuss the thermodynamical properties of assemblies characterized by the conservation of two or more parameters. The simplest assumption of this type that we can make is that there are two parameters, say E and P, which are conserved, and that each particle of the assembly can occupy the levels (r, s) (r, s are non-negative integers), where the contribution of the level (r, s) to E is rε0 and to P is sη0. In order to find the entropy, and hence other thermodynamical properties of the system, we have to enumerate the distinct number of ways, p(m, n), in which an assembly of particles corresponding to given values of E = mε0 and P = nη0 can be realized. In this paper we find asymptotic expressions for p(m, n) in the following cases: (a) m is a fixed number, (b) m and n are of the same order. It is assumed here that the number of particles is greater than m and n. We deal with the case (a) in §2 and the case (b) in §4. §3 deals with the asymptotic expansions of the generating function for p(m, n) which are used in §4.