Abstract: Based on high-precision Stirling＇s formula,we present a new form of Boltzmann distribution as well as its rigorous derivation,utilizing Taylor＇s theorem for multivariate functions and Lambert W function.Also,this method can be applied for rigorous derivation of Bose distribution and Dirac distribution.

Key words: Boltzmann distribution, Lagrange multipliers method, Stirling＇s formula, Taylor＇s expression, Lambert W function