Determine the equilibrium distribution by avoiding utilizing the Stirling formula.

The derivations of the most probable distribution are not self-consistent in most statistical physics textbooks since the Stirling's approximation adopted in the derivations is not valid for systems with very few particles. Any improvement by taking more higher terms of the Stirling series is insufficient due to erroneous divergence for small number. In this paper, we derive the equilibrium distribution by extrapolating the factorial function by the gamma function so that the Stirling's approximation or the Stirling formula is avoided. For the ideal Bose system in a one-dimensional harmonic trap, we find that the equilibrium distribution given by this paper coincide with the rigorous result perfectly even for very small system size. [ABSTRACT FROM AUTHOR]