Cococubed.com Fermi-Dirac Functions
Home Astronomy research Software instruments    Stellar equation of states    EOS with ionization    EOS for supernovae    Chemical potentials    Stellar atmospheres    Voigt Function    Jeans escape    Polytropic stars    Cold white dwarfs    Adiabatic white dwarfs    Cold neutron stars    Stellar opacities    Neutrino energy loss rates    Ephemeris routines    Fermi-Dirac functions    Polyhedra volume    Plane - cube intersection    Coating an ellipsoid    Nuclear reaction networks    Nuclear statistical equilibrium    Laminar deflagrations    CJ detonations    ZND detonations    Fitting to conic sections    Unusual linear algebra    Derivatives on uneven grids    Pentadiagonal solver    Quadratics, Cubics, Quartics    Supernova light curves    Exact Riemann solutions    1D PPM hydrodynamics    Hydrodynamic test cases    Galactic chemical evolution    Universal two-body problem    Circular and elliptical 3 body    The pendulum    Phyllotaxis    MESA    MESA-Web    FLASH    Zingale's software    Brown's dStar    GR1D code    Iliadis' STARLIB database    Herwig's NuGRID    Meyer's NetNuc Presentations Illustrations cococubed YouTube Bicycle adventures Public Outreach Education materials 2023 ASU Solar Systems Astronomy 2023 ASU Energy in Everyday Life AAS Journals AAS YouTube 2023 MESA VI 2023 MESA Marketplace 2023 MESA Classroom 2022 Earendel, A Highly Magnified Star 2022 White Dwarfs & 12C(α,γ)16O 2022 Black Hole Mass Spectrum 2022 MESA in Don't Look Up 2021 Bill Paxton, Tinsley Prize 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. Fk(η,θ) First derivatives with respect to η and θ Second derivatives with respect to η and θ
	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.