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Public Attributes |
Protected Member Functions |
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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:

## 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 | |

root * | density_root |

Solver to compute chemical potential from density. | |

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

where is the effective chemical potential, is the rest mass, and is the effective mass. For later use, we define .

Uncertainties are given in unc.

**Evaluation** **of** **the** **derivatives**

The relevant derivatives of the distribution function are

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

where the entropy density is

The derivatives can be integrated directly or they may be converted to integrals over the distribution function through an integration by parts

using the distribution function for and 0 and as the limits, we have

as long as vanishes at . Rewriting,

as long as vanishes at .

**Explicit** **forms**

1) The derivative of the density wrt the chemical potential

Using we get

2) The derivative of the density wrt the temperature

Using we get

3) The derivative of the entropy wrt the chemical potential

This verifies the Maxwell relation

4) The derivative of the entropy wrt the temperature

Using

5) The derivative of the density wrt the effective mass

Using we get

** New section**

and , so

or

1) The derivative of the density wrt the chemical potential

2) The derivative of the density wrt the temperature

4) The derivative of the entropy wrt the temperature

5) The derivative of the density wrt the effective mass

Definition at line 225 of file fermion_deriv_nr.h.

double o2scl::fermion_deriv_nr::flimit |

fermion_deriv_nr will ignore corrections smaller than about .

Definition at line 239 of file fermion_deriv_nr.h.

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

- /home/awsteiner/wcs/o2scl/src/part/fermion_deriv_nr.h

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