sn_nr_fermion Class Reference

#include <sn_nr_fermion.h>

Inheritance diagram for sn_nr_fermion:

deriv_fermion fermion deriv_part part

Detailed Description

Equation of state for a nonrelativistic fermion.

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 221 of file sn_nr_fermion.h.


Public Member Functions

 sn_nr_fermion (double m=0.0, double g=0.0)
 Create a fermion with mass m and degeneracy g.
virtual int calc_mu (const double temper)
 Calculate properties as function of chemical potential.
virtual int calc_density (const double temper)
 Calculate properties as function of density.
virtual int pair_mu (const double temper)
 Calculate properties with antiparticles as function of chemical potential.
virtual int pair_density (const double temper)
 Calculate properties with antiparticles as function of density.
virtual int nu_from_n (const double temper)
 Calculate effective chemical potential from density.
int set_density_root (root< void *, funct< void * > > &rp)
 Set the solver for use in calculating the chemical potential from the density.
virtual const char * type ()
 Return string denoting type ("sn_nr_fermion").

Data Fields

double flimit
 The limit for the Fermi functions (default 20.0).
deriv_fermion 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.
cern_mroot_root
< void *, funct
< void * > > 
def_density_root
 The default solver for npen_density() and pair_density().

Protected Member Functions

int solve_fun (double x, double &yy, void *&pa)
 Function to compute chemical potential from density.
int pair_fun (double x, double &yy, void *&pa)
 Function to compute chemical potential from density when antiparticles are included.

Protected Attributes

root< void *, funct
< void * > > * 
density_root
 Solver to compute chemical potential from density.

Field Documentation

double flimit

The limit for the Fermi functions (default 20.0).

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

Definition at line 235 of file sn_nr_fermion.h.


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

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