STATISTICAL-MECHANICAL DISTRIBUTIONS FOR SMALL NUMBERS OF SYSTEMS

We consider the problem of deriving the appropriate modification of th e standard Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein distribut ions for the situation where the number of systems is not necessarily very large and thus the usual derivations break down. We obtain for ea ch case a modified formula which yields only integer values for the re levant occupation number and which reduces to the standard result in t he limit of large numbers.