*
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

AAS Journals
AAS Youtube
2020 Celebration of Margaret Burbidge
2020 Digital Infrastructure
2021 MESA Marketplace
2021 MESA Summer School
2021 ASU Solar Systems
2021 ASU Energy in Everyday Life


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.