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J.D. Maldonado
F.X.Timmes, my vitae
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" function for the three parameter integral:
where k is the order of the function, beta = mc2/kT is the relativity parameter, and eta = mu/kT is the normalized chemical potential energy mu, which is sometimes called the degeneracy parameter. I'll reserve the term "Fermi" function as a special case, beta=0, of the Fermi-Dirac function:
The Fermi-Dirac function is 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) in the code below. Some of my trivial additions were to generalize the code from the Fermi functions to the Fermi-Dirac functions.
The second method solves the Fermi-Dirac functions by an as-exact-as-you-like quadrature method. The first derivatives with respect to eta and beta are also returned. You may be very pleasantly suprised at how efficiently it runs given that it a direct integration. The original idea and code comes from Josep Aparicio (ApJS 117, 627, 1998). My trivial contributions have been to add the first derivatives, gather the various quadrature accuracies under one roof, and make a machine readable code more generally available.
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Fermi functions: test_fermi.f plus these include files: implno.dek const.dek funct_fermi1.f funct_fermi45.f interp_spline1.f gauss_fermi.f |
Fermi-Dirac functions: test_clout.f plus these include files: implno.dek const.dek cloutman.f gauss_fermi.f |
Fermi-Dirac functions and their derivatives: test_fermi_dirac.f plus these include files: implno.dek const.dek gauss_fermi.f |
If you want to see how the Fermi-Dirac functions are used in a bare knuckle stellar equation of state, take a look at the Timmes eos routine.