Exact Solution of Particle's Occupation Number in the Lambert Boltzmann Distribution using Lambert W Function

Abstract

By applying the exact Stirling formula and the exact function , the  occupation number of particles was calculated based on the statistics of Boltzmann distribution for a finite number of particles n. The exact analytical expression of occupation number of particles (ni) is found in terms of the Lambert W function and is more general than that usually calculated by the standard Boltzmann distribution based on the Stirling approximation. The new expression in the exact and algebraic closed form eliminates the need for the complex iterative computation. Its high accuracy is proved by a comparison of calculating occupation number of particles (ni) with respective numerical solution.