The Lambert W function and quantum statistics

We present some applications of the Lambert $W$ function ($W$ function) to the formalism of quantum statistics (QS). We consider the problem of finding extrema in terms of energy for a general QS distribution, which involves the solution of a transcendental equation in terms of the $W$ function. We then present some applications of this formula including Bose–Einstein systems in $d$ dimensions, Maxwell–Boltzmann systems, and black body radiation. We also show that for the appropriate parameter values, this formula reduces to an analytic expression in connection with Wien’s displacement law that was found in a previous study. In addition, we show that for Maxwell–Boltzmann and Bose–Einstein systems, the $W$ function allows us to express the temperature of the system as a function of the thermodynamically relevant chemical potential, the particle density, and other parameters. Finally, we explore an indirect relationship of the $W$ function to the polylogarithm function and to the Lambert transform.