Title

D-dimensional Bose Gases and the Lambert W Function

Document Type

Article

Publication Date

12-2010

Journal

Journal of Mathematical Physics

Volume

12

Issue

51

First Page

123303-1

Last Page

123303-14

URL with Digital Object Identifier

http://dx.doi.org/10.1063/1.3496906

Abstract

The applications of the Lambert W function (also known as the W function) to D-dimensional Bose gases are presented. We introduce two sets of families of logarithmic transcendental equations that occur frequently in thermodynamics and statistical mechanics and present their solution in terms of the W function. The low temperature T behavior of free ideal Bose gases is considered in three and four dimensions. It is shown that near condensation in four dimensions, the chemical potential μ and pressure P can be expressed in terms of T through the W function. The low T behavior of one- and two-dimensional ideal Bose gases in a harmonic trap is studied. In 1D, the W function is used to express the condensate temperature, TC, in terms of the number of particles N; in 2D, it is used to express μ in terms of T. In the low T limit of the 1D hard-core and the 3D Bose gas, T can be expressed in terms of P and μ through the W function. Our analysis allows for the possibility to consider μ, T, and P as complex variables. The importance of the underlying logarithmic structure in ideal quantum gases is seen in the polylogarithmic and W function expressions relating thermodynamic variables such as μ, T, and P.

Notes

This paper was presented at the Quantum Probability and Related Topics section of the International Congress of Mathematics in August 2010 at Bangalore, India.