Fermi-Dirac Functions


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Contact: F.X.Timmes
my one page vitae,
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research statement, and
teaching statement.

The monikers "Fermi-Dirac", "generalized Fermi-Dirac", and "Fermi" function haven't received uniform usage in the literature. I'll use the term "Fermi-Dirac" for the two parameter integral: \begin{equation} F_{k}(\eta,\theta) = \int\limits_{0}^{\infty} \ {x^{k} \ (1 + 0.5 \ \theta \ x)^{1/2} \over \exp(x - \eta) + 1} \ dx \label{eq1} \tag{1} \end{equation} where $k$ is the order of the function, $\theta = k_B T / (mc^2)$ is the relativity parameter, and $\eta = \mu/(k_B T)$ is the normalized chemical potential energy $\mu$, which is sometimes called the degeneracy parameter. I'll use the term "Fermi" function as the $\theta=0$ special case of the Fermi-Dirac function: \begin{equation} F_{k}(\eta) = \int\limits_{0}^{\infty} \ {x^{k} \ \over \exp(x - \eta) + 1} \ dx \label{eq2} \tag{2} \end{equation} The Fermi function can be obtained from some remarkable rational function fits from H.M. Antia (ApJS 84, 101, 1993 and private communication). These rational function approximations are compared to the answers given by the Fermi-Dirac quadratures below in this bzip2 tarball fermi.tbz.

The Fermi-Dirac function are solved by two methods. The first uses an exact-as-you-like direct simpson integration on nested grids in tandem with some integral transformations. I've called this Cloutman's method (ApJS 71, 677, 1989). Some of my trivial additions were to generalize from the Fermi functions to the Fermi-Dirac functions in the code contained in this bzip2 tarball fermi_dirac.tbz. The second method solves the Fermi-Dirac functions by quadrature summations. You may be pleasantly suprised at how efficiently it runs for the accuracy achieved. The original code comes from Josep Aparicio (ApJS 117, 627, 1998). My relatively trivial contributions in this bzip2 tarball fermi_dirac_with_derivatives.tbz have been to add the first and second derivatives, and gather the various quadrature accuracies under one roof. One may also benefit from the function and derivative quadratures offered in Gong et al. 2001.

First derivatives with respect to η and θ
* *
Second derivatives with respect to η and θ
* * *

If you want to see how these Fermi-Dirac functions are used in a bare knuckle stellar equation of state, peek at the Timmes eos routine.

Please cite the relevant references if you publish a piece of work that use these codes, pieces of these codes, or modified versions of them. Offer co-authorship as appropriate.