# Particles¶

The classes described below calculate the thermodynamic properties of interacting and non-interacting quantum and classical particles.

## Particles contents¶

## Particle data classes¶

The class `o2scl::part_tl`

is the basic structure for
a particle. The classes with a `_tl`

suffix are templates which
can handle a variety of floating point types. Typedefs without
a `_tl`

suffix specialize these templates for the floating
point type `double`

.

`o2scl::part_tl::m`

- mass (i.e. rest mass), \(m\)`o2scl::part_tl::g`

- degeneracy factor (e.g. \(g=2j+1\))`o2scl::part_tl::n`

- number density, \(n\)`o2scl::part_tl::ed`

- energy density, \(\varepsilon\)`o2scl::part_tl::pr`

- pressure, \(P\)`o2scl::part_tl::en`

- entropy density, \(s\)`o2scl::part_tl::ms`

- effective mass, \(m^{*}\)`o2scl::part_tl::mu`

- chemical potential, \(\mu\)`o2scl::part_tl::nu`

- effective chemical potential, \(\nu\)`o2scl::part_tl::inc_rest_mass`

- True if the rest mass is included (default true)`o2scl::part_tl::non_interacting`

- False if the particle includes interactions (default true)

The data members `o2scl::part_tl::ms`

and
`o2scl::part_tl::nu`

allow one to specify modifications to
the mass and the chemical potential due to interactions. This allows
one to calculate the properties of particle due to interactions so
long as the basic form of the free-particle dispersion relation is
unchanged, i.e.

If the particle is non-interacting, then `o2scl::part_tl::nu`

and
`o2scl::part_tl::ms`

are sometimes used by O₂scl_part
functions for temporary storage.

If `o2scl::part_tl::inc_rest_mass`

is c true (this is the
default in all of the classes except `o2scl::nucleus`

),
then all functions include the rest mass (stored in
`o2scl::part_tl::m`

) energy density in the energy density,
the “mu” functions expect that the rest mass is included in
`o2scl::part_tl::mu`

or
`o2scl::part_tl::nu`

as input and the “density” functions
output `o2scl::part_tl::mu`

or
`o2scl::part_tl::nu`

including the rest mass. Note that it
is assumed that `o2scl::part_tl::m`

contains the rest mass
even if the particle is interacting and an effective mass is stored in
`o2scl::part_tl::ms`

.

When `o2scl::part_tl::inc_rest_mass`

is true, antiparticles are
implemented by choosing the antiparticle chemical potential to be
\(- \mu\). When `o2scl::part_tl::inc_rest_mass`

is false,
there is an ambiguity in the relative definitions of the rest mass
contribution for the antiparticles and the combination of both
particles and antiparticles. Define energy density for particles
including the rest mass contribution as \(\varepsilon_+\), and
the energy density without the rest mass contribution as
\(\tilde{\varepsilon}_{+} = \varepsilon_{+} - n_{+} m\) .
Similarly, for antiparticles, we have \(\tilde{\varepsilon}_- =
\varepsilon_- - n_- m\). The total energy density including the
rest mass contribution is then \(\varepsilon = \varepsilon_{+} +
\varepsilon_-\) and without the rest mass contribution
\(\tilde{\varepsilon} \equiv \varepsilon - (n_{+}-n_-) m\). Then,

Similarly, for the chemical potentials, we have

thus \(\tilde{\mu}_- = - \tilde{\mu}_+ - 2 m\) . This bookkeeping
is handled by `o2scl::part_tl::anti()`

, the
`o2scl::fermion_thermo_tl::pair_mu()`

, and the
`o2scl::fermion_thermo_tl::pair_density()`

, functions.

The thermodynamic identity used to compute the pressure for interacting particles is

where `o2scl::part_tl::nu`

is used. This way, the particle
class doesn’t need to know about the structure of the interactions to
ensure that the thermodynamic identity is satisfied. Note that in the
O₂scl_eos library, where in the equations of state the
normal thermodynamic identity is used

Frequently, the interactions which create an effective chemical
potential which is different than `o2scl::part_tl::mu`

thus create
extra terms in the pressure and the energy density for the given
equation of state.

The `o2scl::fermion_tl`

class is a child of
`o2scl::part_tl`

which contains data members for the Fermi
momentum and energy gap. The `o2scl::boson`

class contains
an extra data member for the condensate. The `o2scl::quark`

class is a descendant of the `o2scl::fermion_tl`

class
which contains extra data members for the quark condensate and the
contribution to the bag constant. Nuclei are represented by the
`o2scl::nucleus`

class and documented in nuclei_section.

## Units¶

Factors of \(\hbar, c\) and \(k_B\) have been removed everywhere, so that mass, energy, and temperature all have the same units. Number and entropy densities have units of mass cubed (or energy cubed). The particle classes can be used with any system of units which is based on powers of one unit, i.e. \([n] = [T]^3 = [m]^3 = [P]^{3/4} = [\varepsilon]^{3/4}\), etc. For O₂scl EOS classes, powers of \(\mathrm{fm}^{-1}\) are often chosen.

## Classes for particle thermodynamics¶

At zero temperature, the thermodynamic properties of fermions can
be computed using `o2scl::fermion_zerot_tl`

. The class
`o2scl::classical_thermo_tl`

computes the properties of particles
in the classical limit.

At finite temperature, there are different classes corresponding to
different approaches to computing the integrals over the distribution
functions. The approximation scheme from [Johns96] is used in
`o2scl::boson_eff`

and `o2scl::fermion_eff`

. An
exact method employing direct integration of the distribution
functions is used in `o2scl::boson_rel`

and
`o2scl::fermion_rel_tl`

, but these are necessarily quite a
bit slower. All of these classes use expansions to give ensure
comparably accurate results in the degenerate and non-degenerate
limits.

The class `o2scl::fermion_eff`

usually works to within about 1
part in \(10^4\), but can be as bad as 1 part in \(10^2\)
in some more extreme cases. The default settings for
`o2scl::fermion_rel_tl`

give an accuracy of at least 1 part in
\(10^6\) (and frequently better than this). For
`o2scl::fermion_rel_tl`

, the accuracy can be improved to 1 part in
\(10^{10}\) by decreasing the integration tolerances.

See Fermion Details for more discussion on the fermion integrals.

The class `o2scl::fermion_nonrel_tl`

assumes a non-relativistic
dispersion relation for fermions. It uses an exact method for both
zero and finite temperatures. The non-relativistic integrands are
much simpler and `o2scl::fermion_nonrel_tl`

uses the appropriate
GSL functions (which are nearly exact) to compute them.

## Thermodynamics with derivatives¶

Sometimes it is useful to know derivatives like \(ds/dT\) in
addition to the energy and pressure.
The class `o2scl::part_deriv_press_tl`

stores the three
derivatives which correspond to second derivatives
of the pressure

All other first derivatives of the thermodynamic functions can be written in terms of these three.

The new data classes are
`o2scl::part_deriv_tl`

and `o2scl::fermion_deriv_tl`

which store the basic particle thermodynamics described
above with these additional three derivatives.

There are three classes which compute these derivatives for
fermions and classical particles. The class
`o2scl::classical_deriv_thermo_tl`

handles the nondegenerate limit,
`o2scl::fermion_deriv_rel_tl`

handles fermions and
`o2scl::fermion_deriv_nr_tl`

handles nonrelativistic fermions.
The class `o2scl::fermion_deriv_thermo_tl`

is a base
class for `o2scl::fermion_deriv_rel_tl`

and uses
degenerate and nondegenerate expansions to evaluate
both the base thermodynamic quantities and the three
derivatives from `o2scl::part_deriv_press_tl`

.

The function `o2scl::part_deriv_tl::deriv_f()`

computes
the derivatives which are second derivatives of the
free energy from the three computed above.

## Other derivatives¶

For the derivative of the entropy with respect to the chemical potential, there is a Maxwell relation

The first derivatives of the energy density can be computed using the thermodynamic identity:

Most of the other common derivatives which are used are those which can be obtained by second derivatives of the Gibbs free energy, \(G = F + P V\).

Other common derivatives are the heat capacity per particle at
constant volume, \(c_V\), and the speed of sound, \(( d P / d
\varepsilon)_{\{N_i\},S}\). These derivatives are computed by functions
in `o2scl::deriv_thermo_base_tl`

from the three second
derivatives of the pressure stored in a
`o2scl::part_deriv_tl`

or
`o2scl::fermion_deriv_tl`

object.