A Model for Fermi-Dirac Integrals

Fermi–Dirac integrals arise in calculating pressure and density in degenerate matter, such as neutron stars; they also occur in the electronic density of semiconductors. A (semi-)closed form was not known until 1995, when Howard Lee noticed the application of the integral form of polylogarithms. We developed an alternative expression, (plotted here), using an exponential model that is accurate for (arXiv:0909.3653v5 [math.GM]). Here we compare the model, in green, with the polylogarithm expression, in blue, for Fermi–Dirac integrals half-integer order , with (orders commonly used in astrophysics and semiconductors).


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Briefly, the model is based on the idea of rewriting as a product of
and a function that is a basic exponential model; for more details, see [1]. The dependence on the factor explains the restriction to values .
Written in terms of polylogarithms, the (normalized) Fermi–Dirac integral is [2] .
Meanwhile, the model is found to be ,
where is a function written in terms of the Lambert function (see [1] and references therein).
[1] M. H. Lee, "Polylogarithmic Analysis of Chemical Potential and Fluctuations in a d-Dimensional Free Fermi Gas at Low Temperatures," Journal of Mathematical Physics, 36, 1995 pp. 1217–1231.
[2] M. Morales, "Fermi-Dirac Integrals in Terms of Zeta Functions," arXiv:0909.3653v5 [math.GM].
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