Cococubed.com Fermi-Dirac Functions

Home

Astronomy Research
2024 Neutrino Emission from Stars
2023 White Dwarfs & 12C(α,γ)16O
2023 MESA VI
2022 Earendel, A Highly Magnified Star
2022 Black Hole Mass Spectrum
2021 Skye Equation of State
2021 White Dwarf Pulsations & 22Ne
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

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

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
Herwig's NuGRID
Meyer's NetNuc

AAS Journals
2024 AAS Peer Review Workshops

2024 ASU Energy in Everyday Life
2024 MESA Classroom
Outreach and Education Materials

Other Stuff:
Illustrations
Presentations

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: $$F_{k}(\eta,\theta) = \int\limits_{0}^{\infty} \ {x^{k} \ (1 + 0.5 \ \theta \ x)^{1/2} \over \exp(x - \eta) + 1} \ dx \label{eq1} \tag{1}$$ 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: $$F_{k}(\eta) = \int\limits_{0}^{\infty} \ {x^{k} \ \over \exp(x - \eta) + 1} \ dx \label{eq2} \tag{2}$$ 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.