Class fermion_rel_tl (o2scl)¶
-
template<class
fermion_t
= fermion_tl<double>, classfd_inte_t
= class o2scl::fermi_dirac_integ_gsl, classbe_inte_t
= o2scl::bessel_K_exp_integ_gsl, classnit_t
= inte_qagiu_gsl<>, classdit_t
= inte_qag_gsl<>, classdensity_root_t
= root_cern<>, classroot_t
= root_cern<>, classfunc_t
= funct, classfp_t
= double>
classo2scl
::
fermion_rel_tl
: public o2scl::fermion_thermo_tl<fermion_tl<double>, class o2scl::fermi_dirac_integ_gsl, o2scl::bessel_K_exp_integ_gsl, root_cern<>, funct, double>¶ Equation of state for a relativistic fermion.
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_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 \( \eta \equiv m^{*}/T \). Then, \( p/T=\sqrt{u^2+2 \eta u} \) \( E/T = \mathrm{mx+u} \) and \( p/T^2 dp = 2(\eta+u) du \) The density integrand is
\[ \left(\eta+u\right) \sqrt{u^2+2 (\eta) u} \left(\frac{e^{y}}{e^{\eta+u}+e^{y}}\right) \, , \]the energy integrand is\[ \left(\eta+u\right)^2 \sqrt{u^2+2 (\eta) u} \left(\frac{e^{y}}{e^{\eta+u}+e^{y}}\right) \, , \]and the entropy integrand is\[ \left(\eta+u\right) \sqrt{u^2+2 (\eta) u} \left(t_1+t_2\right) \, , \]where\[\begin{split}\begin{eqnarray*} t_1 &=& \log \left(1+e^{y-\eta-u}\right)/ \left(1+e^{y-\eta-u}\right) \nonumber \\ t_2 &=& \log \left(1+e^{\eta+u-y}\right)/ \left(1+e^{\eta+u-y}\right) \, . \end{eqnarray*}\end{split}\]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 can improve the accuracy to within 1 part in \( 10^{10} \) using
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.fermion_rel rf(1.0,2.0); rf.upper_limit_fac=40.0; rf.dit->tol_abs=1.0e-13; rf.dit->tol_rel=1.0e-13; rf.nit->tol_abs=1.0e-13; rf.nit->tol_rel=1.0e-13; rf.density_root->tol_rel=1.0e-10;
Todos:
- Idea for Future:
I had to remove the shared_ptr stuff because the default algorithm types don’t support multiprecision, but it might be nice to restore the shared_ptr mechanism somehow.
- 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 ifret
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.
Numerical parameters
-
bool
err_nonconv
¶ If true, call the error handler when convergence fails (default true)
-
int
verbose
¶ Verbosity parameter (default 0)
-
bool
use_expansions
¶ If true, use expansions for extreme conditions (default true)
-
bool
verify_ti
¶ If true, verify the thermodynamic identity (default false)
-
root<func_t, func_t, fp_t> *
density_root
¶ The solver for calc_density()
-
density_root_t
def_density_root
¶ The default solver for the chemical potential given the density.
-
int
last_method
¶ An integer indicating the last numerical method used.
In all functions
0: no previous calculation or last calculation failed
In nu_from_n():
1: default solver
2: default solver with smaller tolerances
3: bracketing solver
In calc_mu():
4: non-degenerate expansion
5: degenerate expansion
6: exact integration, non-degenerate integrands
7: exact integration, degenerate integrands, lower limit on entropy integration
8: exact integration, degenerate integrands, full entropy integration
9: T=0 result
In calc_density(), the integer is a two-digit number. The first digit (1 to 3) is the method used by nu_from_n() and the second digit is one of
1: nondegenerate expansion
2: degenerate expansion
3: exact integration, non-degenerate integrands
4: exact integration, degenerate integrands, lower limit on entropy integration
5: exact integration, degenerate integrands, full entropy integration If calc_density() uses the T=0 code, then last_method is 40.
In pair_mu(), the integer is a three-digit number. The third digit is always 0 (to ensure a value of last_method which is unique from the other values reported from other functions as described above). The first digit is the method used for particles from calc_mu() above and the second digit is the method used for antiparticles.
In pair_density(), the integer is a four-digit number. The first digit is from the list below and the remaining three digits, if nonzero, are from pair_mu().
1: T=0 result
2: default solver
3: bracketing solver
4: default solver with smaller tolerances
5: default solver with smaller tolerances in log units
6: bracketing in log units
-
fermion_rel_tl
()¶ Create a fermion with mass
m
and degeneracyg
.
-
~fermion_rel_tl
()¶
-
const char *
type
()¶ Return string denoting type (“fermion_rel”)
Template versions of base functions
-
int
nu_from_n
(fermion_t &f, fp_t temper)¶ Calculate the chemical potential from the density (template version)
-
int
calc_density
(fermion_t &f, fp_t temper)¶ Calculate properties as function of density.
This function uses the current value of
nu
(ormu
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.- Idea for Future:
There is still quite a bit of code duplication between this function and calc_mu() .
-
void
pair_mu
(fermion_t &f, fp_t temper)¶ Calculate properties with antiparticles as function of chemical potential.
-
int
pair_density
(fermion_t &f, fp_t temper)¶ Calculate thermodynamic properties with antiparticles from the density.
-
fp_t
density_fun
(fp_t u, fermion_t &f, fp_t T)¶ The integrand for the density for non-degenerate fermions.
-
fp_t
pressure_fun
(fp_t u, fermion_t &f, fp_t T)¶ The integrand for the pressure for non-degenerate fermions.
-
fp_t
energy_fun
(fp_t u, fermion_t &f, fp_t T)¶ The integrand for the energy density for non-degenerate fermions.
-
fp_t
entropy_fun
(fp_t u, fermion_t &f, fp_t T)¶ The integrand for the entropy density for non-degenerate fermions.
-
fp_t
deg_density_fun
(fp_t k, fermion_t &f, fp_t T)¶ The integrand for the density for degenerate fermions.
-
fp_t
deg_energy_fun
(fp_t k, fermion_t &f, fp_t T)¶ The integrand for the energy density for degenerate fermions.
-
fp_t
deg_pressure_fun
(fp_t k, fermion_t &f, fp_t T)¶ The integrand for the energy density for degenerate fermions.
-
fp_t
deg_entropy_fun
(fp_t k, fermion_t &f, fp_t T)¶ The integrand for the entropy density for degenerate fermions.
-
fp_t
pair_fun
(fp_t x, fermion_t &f, fp_t T, bool log_mode)¶ Solve for the chemical potential given the density with antiparticles.
- 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.