Physics and Astronomy Publications
Document Type
Article
Publication Date
10-8-2009
Journal
Journal of Mathematical Physics
Volume
50
Issue
10
First Page
102103-1
Last Page
102103-11
URL with Digital Object Identifier
http://dx.doi.org/10.1063/1.3230482
Abstract
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.
Notes
Copyright (2009) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.
This article appeared in (S. R. Valluri et al., J. Math. Phys. 50, 102103 (2009)) and may be found at S. R. Valluri et al., J. Math. Phys. 50, 102103 (2009).