Fermi-Dirac Functions


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

The terms "Fermi-Dirac", "generalized Fermi-Dirac", and "Fermi" function haven't received uniform usage in the literature. I'll use "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 "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 functions can be obtained from some remarkable rational function approximations. The Fermi-Dirac function are solved by two methods. The first uses simpson integration on nested grids in tandem with integral transformations. The second method uses quadrature summations (Also see this article). The answers these methods produce are compared in fermi_dirac.tbz. My contributions to fermi_dirac.tbz include adding the first and second partial derivatives to the quadrature method, and gathering the various quadrature accuracies under one roof.

To see how these Fermi-Dirac functions are used in a bare knuckle stellar equation of state, peek at the Timmes eos instrument.


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