![]() |
Particles and Nuclei Sub-Library: Version 0.910
|
Equation of state for a relativistic fermion. More...
#include <rel_fermion.h>
This class computes the thermodynamics of a relativistic fermion either as a function of the density or the chemical potential. It employs direct integration, using two different integrators for the degenerate and non-degenerate regimes. The default integrators are gsl_inte_qag (for degenerate fermions) and gsl_inte_qagiu (for non-degenerate fermions). For the functions calc_mu() and calc_density(), if the temperature argument is less than or equal to zero, the functions fermion_zerot::calc_mu_zerot() and fermion_zerot::calc_density_zerot() will be used to compute the result.
Define the degeneracy parameter
where is the effective chemical potential and
is the effective mass. For
greater than deg_limit (degenerate regime), a finite interval integrator is used and for
less than deg_limit (non-degenerate regime), an integrator over the interval from
is used. In the case where part::inc_rest_mass is false, the degeneracy parameter is
Integration limits:
The upper limit on the degenerate integration is given by
where and
is rel_fermion::upper_limit_fac . In the case where part::inc_rest_mass is false, the result is
The entropy is only significant at the Fermi surface, thus in the degenerate case, the lower limit of the entropy integral can be given be determined by the value of which solves
The solution is
but this solution is only valid if . In the case where part::inc_rest_mass is false, the result is
which is valid if .
Entropy integrand:
In the degenerate regime, the entropy integrand
where is the fermionic distribution function can lose precision when
is negative and sufficiently large in absolute magnitude. Thus when
where
is stored in deg_entropy_fac (default is -30), the integrand is written as
If is less than -1 times exp_limit (e.g. less than -200), then the entropy integrand is assumed to be zero.
Non-degenerate integrands:
The integrands in the non-degenerate regime are written in a dimensionless form, by defining ,
, and
. The density integrand is
the energy integrand is
and the entropy integrand is
where
The default settings for for this class give an accuracy of at least 1 part in (and frequently better than this).
When the integrators provide numerical uncertainties, these uncertainties are stored in unc. In the case of calc_density() and pair_density(), the uncertainty from the numerical accuracy of the solver is not included. (There is also a relatively small inaccuracy due to the mathematical evaluation of the integrands which is not included in unc.)
One way to improve the accuracy of the computation is just to decrease the tolerances on the default integration objects. This can be done, using, for example
rel_fermion rf(1.0,2.0); rf.def_dit.tolx/=1.0e2; rf.def_dit.tolf/=1.0e2; rf.def_nit.tolx/=1.0e2; rf.def_nit.tolf/=1.0e2;
which decreases the both the relative and absolute tolerances for both the degenerate and non-degenerate integrators. If one is using either the calc_density() or pair_density() functions, one may also have to improve the accuracy of the solver which determines the chemical potential from the density. For the default solver, this could be done with
rf.def_density_root.tolx/=1.0e2; rf.def_density_root.tolf/=1.0e2;
Of course if these tolerances are too small, the calculation may fail.
ret
is finite.ll
) makes bm_part2.cpp worse.Definition at line 215 of file rel_fermion.h.
Public Member Functions | |
rel_fermion () | |
Create a fermion with mass m and degeneracy g . | |
virtual void | calc_mu (fermion &f, double temper) |
Calculate properties as function of chemical potential. | |
virtual void | calc_density (fermion &f, double temper) |
Calculate properties as function of density. | |
virtual void | pair_mu (fermion &f, double temper) |
Calculate properties with antiparticles as function of chemical potential. | |
virtual void | pair_density (fermion &f, double temper) |
Calculate properties with antiparticles as function of density. | |
virtual void | nu_from_n (fermion &f, double temper) |
Calculate effective chemical potential from density. | |
int | set_inte (inte< funct > &non_it, inte< funct > °_it) |
Set integrators. | |
int | set_density_root (root< funct > &rp) |
Set the solver for use in calculating the chemical potential from the density. | |
virtual const char * | type () |
Return string denoting type ("rel_fermion") | |
Data Fields | |
fermion | unc |
Storage for the uncertainty. | |
cern_mroot_root< funct > | def_density_root |
The default solver for calc_density(). | |
gsl_inte_qag< funct > | def_dit |
The default integrator for degenerate fermions. | |
gsl_inte_qagiu< funct > | def_nit |
The default integrator for non-degenerate fermions. | |
Numerical parameters | |
double | deg_limit |
The critical degeneracy at which to switch integration techniques (default 2) | |
double | exp_limit |
The limit for exponentials to ensure integrals are finite (default 200) | |
double | upper_limit_fac |
The factor for the degenerate upper limits (default 20) | |
double | deg_entropy_fac |
A factor for the degenerate entropy integration (default 30) | |
Protected Member Functions | |
double | density_fun (double u) |
The integrand for the density for non-degenerate fermions. | |
double | energy_fun (double u) |
The integrand for the energy density for non-degenerate fermions. | |
double | entropy_fun (double u) |
The integrand for the entropy density for non-degenerate fermions. | |
double | deg_density_fun (double u) |
The integrand for the density for degenerate fermions. | |
double | deg_energy_fun (double u) |
The integrand for the energy density for degenerate fermions. | |
double | deg_entropy_fun (double u) |
The integrand for the entropy density for degenerate fermions. | |
double | solve_fun (double x) |
Solve for the chemical potential given the density. | |
double | pair_fun (double x) |
Solve for the chemical potential given the density with antiparticles. | |
Protected Attributes | |
inte< funct > * | nit |
The non-degenerate integrator. | |
inte< funct > * | dit |
The degenerate integrator. | |
root< funct > * | density_root |
The solver for calc_density() | |
double | T |
Temperature. | |
fermion * | fp |
Current fermion pointer. |
virtual void rel_fermion::calc_density | ( | fermion & | f, |
double | temper | ||
) | [virtual] |
This function uses the current value of nu
(or mu
if the particle is non interacting) for an initial guess to solve for the chemical potential. If this guess is too small, then this function may fail.
Implements fermion_eval_thermo.
Documentation generated with Doxygen. Provided under the GNU Free Documentation License (see License Information).