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

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

#include <fermion_deriv_nr.h>

Inheritance diagram for o2scl::fermion_deriv_nr:
o2scl::fermion_deriv_thermo

Public Member Functions

 fermion_deriv_nr ()
 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_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_nr")
 
- 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 flimit
 The limit for the Fermi functions (default 20.0) More...
 
fermion_deriv unc
 Storage for the most recently calculated uncertainties.
 
bool guess_from_nu
 If true, use the present value of the chemical potential as a guess for the new chemical potential.
 
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)
 Function to compute chemical potential from density.
 
double pair_fun (double x, fermion_deriv &f, double T)
 Function to compute chemical potential from density when antiparticles are included.
 

Protected Attributes

rootdensity_root
 Solver to compute chemical potential from density.
 

Detailed Description

This does not include the rest mass energy in the chemical potential or the rest mass energy density in the energy density to alleviate numerical precision problems at low densities

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

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

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

Uncertainties are given in unc.

Evaluation of the derivatives

The relevant derivatives of the distribution function are

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

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

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

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

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

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

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 or they may be converted to integrals over the distribution function through an integration 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,

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

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

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

2) The derivative of the density wrt the temperature

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

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

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

3) The derivative of the entropy wrt the chemical potential

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

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

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

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

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

5) The derivative of the density wrt the effective mass

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

Using $ h(k)=k^4/(2 m^{* 2} T) $ we get

\[ \left(\frac{d n}{d m^{*}}\right)_{T,\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} f \frac{3 k^2}{2 m^{*}} d k \]

New section

$ u = k^2/2/m^{*}/T $ and $ y=\mu/T $, so

\[ k d k = m^{*} T d u \]

or

\[ d k = \frac{m^{*} T}{\sqrt{2 m^{*} T u}} d u = \sqrt{\frac{m^{*} T}{2 u}} d u \]

1) The derivative of the density wrt the chemical potential

\[ \left(\frac{d n}{d \mu}\right)_T = \frac{g m^{* 3/2} \sqrt{T}}{2^{3/2} \pi^2} \int_0^{\infty} u^{-1/2} f d u \]

2) The derivative of the density wrt the temperature

\[ \left(\frac{d n}{d T}\right)_{\mu} = \frac{g m^{* 3/2} \sqrt{T}} {2^{3/2} \pi^2} \int_0^{\infty} f d u \left[ 3 u^{1/2} - y u^{-1/2}\right] \]

4) The derivative of the entropy wrt the temperature

\[ \left(\frac{d s}{d T}\right)_{\mu} = \frac{g m^{* 3/2} T^{1/2}}{2^{3/2} \pi^2} \int_0^{\infty} f \left[ 5 u^{3/2} - 6 y u^{1/2} + y^2 u^{-1/2}\right] d u \]

5) The derivative of the density wrt the effective mass

\[ \left(\frac{d n}{d m^{*}}\right)_{T,\mu} = \frac{3 g m{* 1/2} T^{3/2}}{2^{3/2} \pi^2} \int_0^{\infty} u^{1/2} f d u \]

Definition at line 225 of file fermion_deriv_nr.h.

Member Data Documentation

◆ flimit

double o2scl::fermion_deriv_nr::flimit

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

Definition at line 239 of file fermion_deriv_nr.h.


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