On the Distribution of Integer Random Variables Related by Two Linear Inequalities: I

The authors' two preceding papers deal with the problem on the allocation of indistinguishable particles to positive integer energy levels under the condition that the total energy of the system is bounded above by some constant $M$. The estimates proved there imply that, for large $M$, most of the allocations concentrate near a limit distribution (which is the Bose–Einstein distribution, provided that the particles obey the corresponding statistics). The present paper continues this trend of research by considering the case in which not only the total energy is constrained but also the overall number of particles is specified. We study both the Bose and the Gibbs distribution and analyze the phenomenon whereby the Bose distribution passes into the Gibbs distribution in the limit as the number of particles is relatively small.