Class fermion_deriv_rel_tl (o2scl)

O2scl_part : Class List

template<class fermion_deriv_t = fermion_deriv_tl<double>, class fp_t = double>
class o2scl::fermion_deriv_rel_tl : public o2scl::fermion_deriv_thermo_tl<double>

Equation of state for a relativistic fermion.

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 \( {\cal L}\equiv\mathrm{f{l}imit}\times T+\nu \) .

Note

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

Note

The testing of this class is apparently sensitive to the difference between gsl_hypot and std::hypot in o2hypot in misc.cpp. Further testing needs to be done to verify which is more accurate. This further testing will probably need to wait until the full multiprecision fermion classes are done.

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.

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

Todo

In class fermion_deriv_rel_tl:

  • Future: The option err_nonconv=false is not really implemented yet.

  • Future: The ref pair_density() function is a bit slow because it computes the non-derivative thermodynamic quantities twice, and this could be improved.

Note

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.

Method of computing derivatives

int method

Method (default is automatic)

int last_method

An integer indicating the last numerical method used.

The function calc_mu() sets this integer to a two-digit number. It is equal to 10 times the value reported by o2scl::fermion_rel::calc_mu() plus a value from the list below corresponding to the method used for the derivatives

  • 1: nondegenerate expansion

  • 2: degenerate expansion

  • 3: nondegenerate integrand, using by_parts for method

  • 4: nondegenerate integrand, using user-specified value for method

  • 5: degenerate integrand, using direct

  • 6: degenerate integrand, using by_parts

  • 7: degenerate integrand, using user-specified value for method

The function nu_from_n() sets this value equal to 100 times the value reported by o2scl::fermion_rel_tl::nu_from_n() .

The function calc_density() sets this value equal to the value from o2scl::fermion_deriv_rel_tl::nu_from_n() plus the value from o2scl::fermion_deriv_rel_tl::calc_mu() .

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()

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.

int intl_method

The internal integration method.

inte *nit

The integrator for non-degenerate fermions.

inte *dit

The integrator for degenerate fermions.

root *density_root

The solver for calc_density() and pair_density()

inline virtual int calc_mu(fermion_deriv_t &f, fp_t temper)

Calculate properties as function of chemical potential.

inline virtual int calc_density(fermion_deriv_t &f, fp_t temper)

Calculate properties as function of density.

inline virtual int pair_mu(fermion_deriv_t &f, fp_t temper)

Calculate properties with antiparticles as function of chemical potential.

inline virtual int pair_density(fermion_deriv_t &f, fp_t temper)

Calculate properties with antiparticles as function of density.

inline virtual int nu_from_n(fermion_deriv_t &f, fp_t temper)

Calculate effective chemical potential from density.

inline void set_inte(inte<> &unit, inte<> &udit)

Set inte objects.

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.

inline void set_density_root(root<> &rp)

Set the solver for use in calculating the chemical potential from the density.

inline virtual const char *type()

Return string denoting type (“fermion_deriv_rel”)

The integrands, as a function of \f$ u=k/T \f$, for

non-degenerate integrals

inline fp_t density_T_fun(fp_t u, fermion_deriv_t &f, fp_t T)

Integrand for derivative of density with respect to temperature for non-degenerate particles.

inline fp_t density_mu_fun(fp_t u, fermion_deriv_t &f, fp_t T)

Integrand for derivative of density with respect to chemical potential for non-degenerate particles.

inline fp_t entropy_T_fun(fp_t u, fermion_deriv_t &f, fp_t T)

Integrand for derivative of entropy with respect to temperature for non-degenerate particles.

inline fp_t density_ms_fun(fp_t u, fermion_deriv_t &f, fp_t T)

Integrand for derivative of density with respect to effective mass for non-degenerate particles.

The integrands, as a function of momentum, for the

degenerate integrals

inline fp_t deg_density_T_fun(fp_t k, fermion_deriv_t &f, fp_t T)

Integrand for derivative of density with respect to temperature for degenerate particles.

inline fp_t deg_density_mu_fun(fp_t k, fermion_deriv_t &f, fp_t T)

Integrand for derivative of density with respect to chemical potential for degenerate particles.

inline fp_t deg_entropy_T_fun(fp_t k, fermion_deriv_t &f, fp_t T)

Integrand for derivative of entropy with respect to temperature for degenerate particles.

inline fp_t deg_density_ms_fun(fp_t k, fermion_deriv_t &f, fp_t T)

Integrand for derivative of density with respect to effective mass for degenerate particles.

Public Functions

inline fermion_deriv_rel_tl()

Create a fermion with mass m and degeneracy g.

inline virtual ~fermion_deriv_rel_tl()

Public Members

fp_t exp_limit

Limit of arguments of exponentials for Fermi functions (default 200.0)

fp_t deg_limit

The critical degeneracy at which to switch integration techniques (default 2.0)

fp_t upper_limit_fac

The limit for the Fermi functions (default 20.0)

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

fermion_deriv_t unc

Storage for the most recently calculated uncertainties.

fermion_rel_tl<fermion_deriv_t> fr

Object for computing non-derivative quantities.