Public Member Functions | Public Attributes | Protected Member Functions | List of all members
o2scl::fermion_rel Class Reference

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

#include <fermion_rel.h>

Inheritance diagram for o2scl::fermion_rel:
o2scl::fermion_eval_thermo o2scl::fermion_zerot

Public Member Functions

 fermion_rel ()
 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 int calc_density (fermion &f, double temper)
 Calculate properties as function of density. More...
virtual void pair_mu (fermion &f, double temper)
 Calculate properties with antiparticles as function of chemical potential.
virtual int pair_density (fermion &f, double temper)
 Calculate properties with antiparticles as function of density.
virtual int nu_from_n (fermion &f, double temper)
 Calculate effective chemical potential from density. More...
virtual const char * type ()
 Return string denoting type ("fermion_rel")
- Public Member Functions inherited from o2scl::fermion_eval_thermo
virtual bool calc_mu_ndeg (fermion &f, double temper, double prec=1.0e-18, bool inc_antip=false)
 Non-degenerate expansion for fermions. More...
virtual bool calc_mu_deg (fermion &f, double temper, double prec=1.0e-18)
 Degenerate expansion for fermions. More...
void set_massless_root (root<> &rp)
 Set the solver for use in massless_calc_density()
virtual double calibrate (fermion &f, int verbose=0, std::string fname="")
 Test the thermodynamics of calc_density() and calc_mu() More...
virtual void massless_calc_mu (fermion &f, double temper)
 Finite temperature massless fermions.
virtual void massless_calc_density (fermion &f, double temper)
 Finite temperature massless fermions.
virtual void massless_pair_mu (fermion &f, double temper)
 Finite temperature massless fermions and antifermions.
virtual void massless_pair_density (fermion &f, double temper)
 Finite temperature massless fermions and antifermions. More...
- Public Member Functions inherited from o2scl::fermion_zerot
void kf_from_density (fermion &f)
 Calculate the Fermi momentum from the density. More...
void energy_density_zerot (fermion &f)
 Energy density at T=0 from fermion::kf and part::ms. More...
void pressure_zerot (fermion &f)
 Pressure at T=0 from fermion::kf and part::ms. More...
virtual void calc_mu_zerot (fermion &f)
 Zero temperature fermions from part::mu or part::nu and part::ms.
virtual void calc_density_zerot (fermion &f)
 Zero temperature fermions from part::n and part::ms.

Public Attributes

fermion unc
 Storage for the uncertainty.
bool use_expansions
 If true, use expansions for extreme conditions (default true)
std::shared_ptr< inte<> > nit
 The non-degenerate integrator.
std::shared_ptr< inte<> > dit
 The degenerate integrator.
std::shared_ptr< root<> > density_root
 The solver for calc_density()
Numerical parameters
bool err_nonconv
 If true, call the error handler when convergence fails (default true)
double min_psi
 The smallest value of $ (\mu-m)/T $ for which integration is used.
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)
- Public Attributes inherited from o2scl::fermion_eval_thermo
root_cern def_massless_root
 The default solver for massless_calc_density() More...

Protected Member Functions

double density_fun (double u, fermion &f, double T)
 The integrand for the density for non-degenerate fermions.
double energy_fun (double u, fermion &f, double T)
 The integrand for the energy density for non-degenerate fermions.
double entropy_fun (double u, fermion &f, double T)
 The integrand for the entropy density for non-degenerate fermions.
double deg_density_fun (double u, fermion &f, double T)
 The integrand for the density for degenerate fermions.
double deg_energy_fun (double u, fermion &f, double T)
 The integrand for the energy density for degenerate fermions.
double deg_entropy_fun (double u, fermion &f, double T)
 The integrand for the entropy density for degenerate fermions.
double solve_fun (double x, fermion &f, double T)
 Solve for the chemical potential given the density.
double pair_fun (double x, fermion &f, double T, bool log_mode)
 Solve for the chemical potential given the density with antiparticles. More...
- Protected Member Functions inherited from o2scl::fermion_eval_thermo
double massless_solve_fun (double x, fermion &f, double temper)
 Solve for the chemical potential for massless fermions.

Additional Inherited Members

- Protected Attributes inherited from o2scl::fermion_eval_thermo
 A pointer to the solver for massless fermions.

Detailed Description

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 inte_qag_gsl (for degenerate fermions) and inte_qagiu_gsl (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.

Degeneracy parameter:

Define the degeneracy parameter

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

where $ \nu $ is the effective chemical potential (including the rest mass) and $ m^{*} $ is the effective mass. For $ \psi $ smaller than min_psi, the non-degenerate expansion in fermion_eval_thermo::calc_mu_ndeg() is attempted first. If that fails, then integration is used. 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. In the case where part::inc_rest_mass is false, the degeneracy parameter is

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

Integration limits:

The upper limit on the degenerate integration is given by

\[ \mathrm{upper~limit} = \sqrt{{\cal L}^2-m^{*,2}} \]

where $ {\cal L}\equiv u T+\nu $ and $ u $ is fermion_rel::upper_limit_fac . In the case where part::inc_rest_mass is false, the result is

\[ \mathrm{upper~limit} = \sqrt{(m+{\cal L})^2-m^{*2}} \]

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 $ k $ which solves

\[ - u = \frac{\sqrt{k^2+m^{* 2}}-\nu}{T} \]

The solution is

\[ \mathrm{lower~limit} = \sqrt{(-u T+{\nu})^2-m^{*,2}} \]

but this solution is only valid if $ (m^{*}-\nu)/T < -u $. In the case where part::inc_rest_mass is false, the result is

\[ \mathrm{lower~limit} = \sqrt{(-u T + m +\nu)^2-m^{*,2}} \]

which is valid if $ (m^{*}-\nu - m)/T < -u $.

Entropy integrand:

In the degenerate regime, the entropy integrand

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

where $ f $ is the fermionic distribution function 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.

Non-degenerate integrands:

The integrands in the non-degenerate regime are written in a dimensionless form, by defining $ u $ with the relation $ p = \sqrt{\left(T u + m^{*}\right)^2-m^{* 2}} $, $ 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

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


\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 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 can improve the accuracy to within 1 part in $ 10^{10} $ using

fermion_rel rf(1.0,2.0);

which decreases the both the relative and absolute tolerances for both the degenerate and non-degenerate integrators and improves the accuracy of the solver which determines the chemical potential from the density. Of course if these tolerances are too small, the calculation may fail.


Idea for Future:
The expressions which appear in in the integrand functions density_fun(), etc. could likely be improved, especially in the case where o2scl::part::inc_rest_mass is false. There should not be a need to check if ret is finite.
Idea for Future:
It appears this class doesn't compute the uncertainty in the chemical potential or density with calc_density(). This could be fixed.
Idea for Future:
I'd like to change the lower limit on the entropy integration, but the value in the code at the moment (stored in ll) makes bm_part2.cpp worse.
Idea for Future:
The function pair_mu() should set the antiparticle integrators as done in fermion_deriv_rel.

Definition at line 216 of file fermion_rel.h.

Member Function Documentation

◆ calc_density()

virtual int o2scl::fermion_rel::calc_density ( fermion f,
double  temper 

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 o2scl::fermion_eval_thermo.

◆ nu_from_n()

virtual int o2scl::fermion_rel::nu_from_n ( fermion f,
double  temper 
Idea for Future:
This function might be improved by generating a bracket for a bracketing solver, rather than o2scl::root_cern which is the default for o2scl::fermion_rel::density_root.

◆ pair_fun()

double o2scl::fermion_rel::pair_fun ( double  x,
fermion f,
double  T,
bool  log_mode 
Idea for Future:
Particles and antiparticles have different degeneracy factors, so we separately use the expansions one at a time. It is probably better to separately generate a new expansion function which automatically handles the sum of particles and antiparticles.

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

Documentation generated with Doxygen. Provided under the GNU Free Documentation License (see License Information).