On the Distribution of Integer Random Variables Related by a Certain Linear Inequality. I

We consider the mathematical problem of the allocation of indistinguishable particles to integer energy levels under the condition that the number of particles can be arbitrary and the total energy of the system is bounded above. Systems of integer as well as fractional dimension are considered. The occupation numbers can either be arbitrary nonnegative integers (the case of “Bose particles”) or lie in a finite set $\{0,1,dots,R\}$ (the case of so-called parastatistics; for example, $R=1$ corresponds to the Fermi–Dirac statistics). Assuming that all allocations satisfying the given constraints are equiprobable, we study the phenomenon whereby, for large energies, most of the allocations tend to concentrate near the limit distribution corresponding to the given parastatistics.