rel_fermion Class Reference

Equation of state for a relativistic fermion. More...

#include <rel_fermion.h>

Inheritance diagram for rel_fermion:

fermion_T fermion part

Detailed Description


Method:

This implements an equation of state for a relativistic fermion using direct integration. Define the degeneracy parameter

\[ \psi=(\nu-m^{*})/T \]

where $ \nu $ is the effective chemical potential and $ m^{*} $ is the effective mass. For $ \psi $ greater than deg_limit (degenerate regime), a finite interval integrator is used and for $ \psi $ less than deg_limit (non-degenerate regime), an integrator over the interval from $ [0,\infty) $ is used. The upper limit on the degenerate integration is given by

\[ \sqrt{(u T+{\nu})^2-m^{*,2}} \]

where $ u $ is rel_fermion::upper_limit_fac .

In the degenerate regime, the entropy integrand

\[ - k^2 \left[ n \log n + \left(1-n\right) \log \left(1-n \right) \right] \]

can lose precision when $ (E - \nu)/T $ is negative and sufficiently large in absolute magnitude. Thus when $ (E - \nu)/T < S $ where $ S $ is stored in deg_entropy_fac (default is -30), the integrand is written as

\[ -k^2 \left( E/T-\nu/T \right) e^{E/T-\nu/T} \, . \]

If $ (E - \nu)/T < S $ is less than -1 times exp_limit (e.g. less than -200), then the entropy integrand is assumed to be zero.

The integrands in the non-degenerate regime are written in a dimensionless form, by defining $ u \equiv p/T, y \equiv \nu/T, $ and $ \mathrm{mx} \equiv m^{*}/T $. The density integrand is

\[ \left(\mathrm{mx}+u\right) \sqrt{u^2+2 \mathrm{mx} u} \left(\frac{e^{y}}{e^{\mathrm{mx}+u}+e^{y}}\right) \, , \]

the energy integrand is

\[ \left(\mathrm{mx}+u\right)^2 \sqrt{u^2+2 \mathrm{mx} u} \left(\frac{e^{y}}{e^{\mathrm{mx}+u}+e^{y}}\right) \, , \]

and the entropy integrand is

\[ \left(\mathrm{mx}+u\right) \sqrt{u^2+2 \mathrm{mx} u} \left(t_1+t_2\right) \, , \]

where

\begin{eqnarray*} t_1 &=& \log \left(1+e^{y-\mathrm{mx}-u}\right)/ \left(1+e^{y-\mathrm{mx}-u}\right) \nonumber \\ t_2 &=& \log \left(1+e^{\mathrm{mx}+u-y}\right)/ \left(1+e^{\mathrm{mx}+u-y}\right) \, . \end{eqnarray*}

The default integrators are gsl_inte_qag (for degenerate particles) and gsl_inte_qagiu (for non-degenerate particles).


Accuracy:

The default settings for for this class give an accuracy of at least 1 part in $ 10^6 $ (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.


Todos:

Note:
This does not work with inc_rest_mass=false (3/30/09: There is a significant amount of testing code to ensure that it does work with inc_rest_mass=false, so this may have been fixed already.)
Idea for future:
Improve the asymptotics of the non-degenerate integrands.
Idea for future:
It appears this doesn't compute the uncertainty in the chemical potential or density with calc_density(). This could be fixed.

Definition at line 164 of file rel_fermion.h.


Public Member Functions

 rel_fermion (double m=0.0, double g=0.0)
 Create a fermion with mass m and degeneracy g.
virtual int calc_mu (double temper)
 Calculate properties as function of chemical potential.
virtual int calc_density (double temper)
 Calculate properties as function of density.
virtual int pair_mu (double temper)
 Calculate properties with antiparticles as function of chemical potential.
virtual int pair_density (double temper)
 Calculate properties with antiparticles as function of density.
virtual int nu_from_n (double temper)
 Calculate effective chemical potential from density.
int set_inte (inte< double, funct< double > > &non_it, inte< double, funct< double > > &deg_it)
 Set integrators.
int set_density_root (root< double, funct< double > > &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.
bool guess_from_nu
 If true, use the present value of the chemical potential as a guess for the new chemical potential.
cern_mroot_root< double, funct
< double > > 
def_density_root
 The default solver for calc_density().
gsl_inte_qag< double, funct
< double > > 
def_dit
 The default integrator for degenerate fermions.
gsl_inte_qagiu< double, funct
< double > > 
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, double &pa)
 The integrand for the density for non-degenerate fermions.
double energy_fun (double u, double &pa)
 The integrand for the energy density for non-degenerate fermions.
double entropy_fun (double u, double &pa)
 The integrand for the entropy density for non-degenerate fermions.
double deg_density_fun (double u, double &pa)
 The integrand for the density for degenerate fermions.
double deg_energy_fun (double u, double &pa)
 The integrand for the energy density for degenerate fermions.
double deg_entropy_fun (double u, double &pa)
 The integrand for the entropy density for degenerate fermions.
int solve_fun (double x, double &yy, double &pa)
 Solve for the chemical potential given the density.
int pair_fun (double x, double &yy, double &pa)
 Solve for the chemical potential given the density with antiparticles.

Protected Attributes

inte< double, funct< double > > * nit
 The non-degenerate integrator.
inte< double, funct< double > > * dit
 The degenerate integrator.
root< double, funct< double > > * density_root
 The solver for calc_density().

The documentation for this class was generated from the following file:

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