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

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

#include <fermion_deriv_rel.h>

Inheritance diagram for o2scl::fermion_deriv_rel:

Public Member Functions

 fermion_deriv_rel ()
 Create a fermion with mass m and degeneracy g.
virtual int calc_mu (fermion_deriv &f, double temper)
 Calculate properties as function of chemical potential.
virtual int calc_density (fermion_deriv &f, double temper)
 Calculate properties as function of density.
virtual int pair_mu (fermion_deriv &f, double temper)
 Calculate properties with antiparticles as function of chemical potential.
virtual int pair_density (fermion_deriv &f, double temper)
 Calculate properties with antiparticles as function of density.
virtual int nu_from_n (fermion_deriv &f, double temper)
 Calculate effective chemical potential from density.
void set_inte (inte<> &unit, inte<> &udit)
 Set inte objects. More...
void set_density_root (root<> &rp)
 Set the solver for use in calculating the chemical potential from the density.
virtual const char * type ()
 Return string denoting type ("fermion_deriv_rel")
double deriv_calibrate (fermion_deriv &f, int verbose, std::string fname="")
 Calibrate with more accurate tabulated results.
- Public Member Functions inherited from o2scl::fermion_deriv_thermo
virtual bool calc_mu_deg (fermion_deriv &f, double temper, double prec)
 Calculate properties as a function of chemical potential using a degenerate expansion.

Public Attributes

double exp_limit
 Limit of arguments of exponentials for Fermi functions (default 200.0)
double deg_limit
 The critical degeneracy at which to switch integration techniques (default 2.0)
double upper_limit_fac
 The limit for the Fermi functions (default 20.0) More...
fermion_deriv unc
 Storage for the most recently calculated uncertainties.
bool err_nonconv
 If true, call the error handler when convergence fails (default true)
inte_qagiu_gsl def_nit
 The default integrator for the non-degenerate regime.
inte_qag_gsl def_dit
 The default integrator for the degenerate regime.
root_cern def_density_root
 The default solver for npen_density() and pair_density()

Protected Member Functions

double solve_fun (double x, fermion_deriv &f, double T)
 Solve for the chemical potential from the density for calc_density()
double pair_fun (double x, fermion_deriv &f, double T)
 Solve for the chemical potential from the density for pair_density()
The integrands, as a function of \f$ u=k/T \f$, for

non-degenerate integrals

double density_fun (double u, fermion_deriv &f, double T)
double energy_fun (double u, fermion_deriv &f, double T)
double entropy_fun (double u, fermion_deriv &f, double T)
double density_T_fun (double k, fermion_deriv &f, double T)
double density_mu_fun (double k, fermion_deriv &f, double T)
double entropy_T_fun (double k, fermion_deriv &f, double T)
double density_ms_fun (double k, fermion_deriv &f, double T)
The integrands, as a function of momentum, for the

degenerate integrals

double deg_density_fun (double u, fermion_deriv &f, double T)
double deg_energy_fun (double u, fermion_deriv &f, double T)
double deg_entropy_fun (double u, fermion_deriv &f, double T)
double deg_density_T_fun (double k, fermion_deriv &f, double T)
double deg_density_mu_fun (double k, fermion_deriv &f, double T)
double deg_entropy_T_fun (double k, fermion_deriv &f, double T)
double deg_density_ms_fun (double k, fermion_deriv &f, double T)

Protected Attributes

int intl_method
 The internal integration method.
 The integrator for non-degenerate fermions.
 The integrator for degenerate fermions.
 The solver for calc_density() and pair_density()

Method of computing derivatives

int method
 Method (default is automatic)
static const int automatic =0
 Automatically choose method.
static const int direct =1
 In the form containing $ f(1-f) $ .
static const int by_parts =2
 Integrate by parts.

Detailed Description

This class only has preliminary support for inc_rest_mass=true (more testing should be done, particularly for the "pair" functions)

This implements an equation of state for a relativistic fermion using direct integration. After subtracting the rest mass from the chemical potentials, the distribution function is

\[ \left\{1+\exp[(\sqrt{k^2+m^{* 2}}-m-\nu)/T]\right\}^{-1} \]

where $ k $ is the momentum, $ \nu $ is the effective chemical potential, $ m $ is the rest mass, and $ m^{*} $ is the effective mass. For later use, we define $ E^{*} = \sqrt{k^2 + m^{*2}} $ . The degeneracy parameter is

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

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 the value of the momentum $ k $ which is the solution of

\[ (\sqrt{k^2+m^{*,2}}-m-\nu)/T=\mathrm{f{l}imit} \]

which is

\[ \sqrt{(m+{\cal L})^2-m^{*2}} \]

where $ {\cal L}\equiv\mathrm{f{l}imit}\times T+\nu $ .

For the entropy integration, we set the lower limit to

\[ 2 \sqrt{\nu^2+2 \nu m} - \mathrm{upper~limit} \]

since the only contribution to the entropy is at the Fermi surface.

In the non-degenerate regime, we make the substitution $ u=k/T $ to help ensure that the variable of integration scales properly.

Uncertainties are given in unc.


This needs to be corrected to calculate $ \sqrt{k^2+m^{* 2}}-m $ gracefully when $ m^{*}\approx m \ll k $ .

Call error handler if inc_rest_mass is true or update to properly treat the case when inc_rest_mass is true.

Evaluation of the derivatives

The relevant derivatives of the distribution function are

\[ \frac{\partial f}{\partial T}= f(1-f)\frac{E^{*}-m-\nu}{T^2} \]

\[ \frac{\partial f}{\partial \nu}= f(1-f)\frac{1}{T} \]

\[ \frac{\partial f}{\partial k}= -f(1-f)\frac{k}{E^{*} T} \]

\[ \frac{\partial f}{\partial m^{*}}= -f(1-f)\frac{m^{*}}{E^{*} T} \]

We also need the derivative of the entropy integrand w.r.t. the distribution function, which is

\[ {\cal S}\equiv f \ln f +(1-f) \ln (1-f) \qquad \frac{\partial {\cal S}}{\partial f} = \ln \left(\frac{f}{1-f}\right) = \left(\frac{\nu-E^{*}+m}{T}\right) \]

where the entropy density is

\[ s = - \frac{g}{2 \pi^2} \int_0^{\infty} {\cal S} k^2 d k \]

The derivatives can be integrated directly (method = direct) or they may be converted to integrals over the distribution function through an integration by parts (method = by_parts)

\[ \int_a^b f(k) \frac{d g(k)}{dk} dk = \left.f(k) g(k)\right|_{k=a}^{k=b} - \int_a^b g(k) \frac{d f(k)}{dk} dk \]

using the distribution function for $ f(k) $ and 0 and $ \infty $ as the limits, we have

\[ \frac{g}{2 \pi^2} \int_0^{\infty} \frac{d g(k)}{dk} f dk = \frac{g}{2 \pi^2} \int_0^{\infty} g(k) f (1-f) \frac{k}{E^{*} T} dk \]

as long as $ g(k) $ vanishes at $ k=0 $ . Rewriting,

\[ \frac{g}{2 \pi^2} \int_0^{\infty} h(k) f (1-f) dk = \frac{g}{2 \pi^2} \int_0^{\infty} f \frac{T}{k} \left[ h^{\prime} E^{*}-\frac{h E^{*}}{k}+\frac{h k}{E^{*}} \right] dk \]

as long as $ h(k)/k $ vanishes at $ k=0 $ .

Explicit forms

1) The derivative of the density wrt the chemical potential

\[ \left(\frac{d n}{d \mu}\right)_T = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{k^2}{T} f (1-f) dk \]

Using $ h(k)=k^2/T $ we get

\[ \left(\frac{d n}{d \mu}\right)_T = \frac{g}{2 \pi^2} \int_0^{\infty} \left(\frac{k^2+E^{*2}}{E^{*}}\right) f dk \]

2) The derivative of the density wrt the temperature

\[ \left(\frac{d n}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{k^2(E^{*}-m-\nu)}{T^2} f (1-f) dk \]

Using $ h(k)=k^2(E^{*}-\nu)/T^2 $ we get

\[ \left(\frac{d n}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{f}{T} \left[2 k^2+E^{*2}-E^{*}\left(\nu+m\right)- k^2 \left(\frac{\nu+m}{E^{*}}\right)\right] dk \]

3) The derivative of the entropy wrt the chemical potential

\[ \left(\frac{d s}{d \mu}\right)_T = \frac{g}{2 \pi^2} \int_0^{\infty} k^2 f (1-f) \frac{(E^{*}-m-\nu)}{T^2} dk \]

This verifies the Maxwell relation

\[ \left(\frac{d s}{d \mu}\right)_T = \left(\frac{d n}{d T}\right)_{\mu} \]

4) The derivative of the entropy wrt the temperature

\[ \left(\frac{d s}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} k^2 f (1-f) \frac{(E^{*}-m-\nu)^2}{T^3} dk \]

Using $ h(k)=k^2 (E^{*}-\nu)^2/T^3 $

\[ \left(\frac{d s}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{f(E^{*}-m-\nu)}{E^{*}T^2} \left[E^{* 3}+3 E^{*} k^2- (E^{* 2}+k^2)(\nu+m)\right] d k \]

5) The derivative of the density wrt the effective mass

\[ \left(\frac{d n}{d m^{*}}\right)_{T,\mu} = -\frac{g}{2 \pi^2} \int_0^{\infty} \frac{k^2 m^{*}}{E^{*} T} f (1-f) dk \]

Using $ h(k)=-(k^2 m^{*})/(E^{*} T) $ we get

\[ \left(\frac{d n}{d m^{*}}\right)_{T,\mu} = -\frac{g}{2 \pi^2} \int_0^{\infty} m^{*} f dk \]

The dsdT integration may fail if the system is very degenerate. When method is byparts, the integral involves a large cancellation between the regions from $ k \in (0, \mathrm{ulimit/2}) $ and $ k \in (\mathrm{ulimit/2}, \mathrm{ulimit}) $. Switching to method=direct and setting the lower limit to $ \mathrm{llimit} $, may help, but recent testing on this gave negative values for dsdT. For very degenerate systems, an expansion may be better than trying to perform the integration. The value of the integrand at k=0 also looks like it might be causing difficulties.
err_nonconv=false not implemented yet.
Idea for Future:
It might be worth coding up direct differentiation, or differentiating the eff results, as these may succeed more generally.
Idea for Future:
This class will have difficulty with extremely degenerate or extremely non-degnerate systems. Fix this.
Idea for Future:
Create a more intelligent method for dealing with bad initial guesses for the chemical potential in calc_density().

Definition at line 259 of file fermion_deriv_rel.h.

Member Function Documentation

◆ set_inte()

void o2scl::fermion_deriv_rel::set_inte ( inte<> &  unit,
inte<> &  udit 

The first integrator is used for non-degenerate integration and should integrate from 0 to $ \infty $ (like o2scl::inte_qagiu_gsl). The second integrator is for the degenerate case, and should integrate between two finite values.

Member Data Documentation

◆ upper_limit_fac

double o2scl::fermion_deriv_rel::upper_limit_fac

fermion_deriv_rel will ignore corrections smaller than about $ \exp(-\mathrm{f{l}imit}) $ .

Definition at line 283 of file fermion_deriv_rel.h.

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

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