There are a few distinctions between how relativistic and nonrelativistic fermions are handled in O2scl which are worth noting. For an interacting relativistic fermion, the effective mass, , and the effective chemical potential,
are defined so that the energy density is
for a relativistic fermion, where we define the chemical potential without the rest mass with . When o2scl::part_tl::inc_rest_mass is true, o2scl::part::nu is equal to
and when o2scl::part_tl::inc_rest_mass is false, o2scl::part_tl::nu is equal to
. If we define
,
,
then the energy density is
This expression is used for o2scl::part_calibrate_class::part_calibrate() because depends only on
and
. For a nonrelativistic fermion,
where . Note that the rest mass energy density is
(not
) in both cases, but it is included in
while it is not included in
. Taking the nonrelativsitic limit of the relativistic energy density shows that
. Thus the class o2scl::fermion_nonrel_tl uses the value stored in o2scl::part_tl::nu slightly differently than does o2scl::fermion_rel_tl and o2scl::fermion_eff . The Fermi momentum is also handled slightly differently,
and
.
Now if we define and
then the argument of the exponential is
which is inconvenient because then is no longer a function of
and
alone. Thus we define
and then the energy density is
which is now a function of and
alone. This is the form used to compute the energy density in o2scl::fermion_nonrel_tl and the definition of
used for nonrelativistic fermions in o2scl::part_calibrate_class::part_calibrate() .
(Slowly moving the documentation from fermion_rel and fermion_deriv_rel to this location.)
In many cases, the non-interacting expressions for fermion thermodynamics can be used in interacting systems as long as one replaces the mass with an effective mass, and the chemical potential with an effective chemical potential,
. In the case where
includes the rest mass (still denoted
), the fermionic distribution function is
where the left expression is used when the chemical potential includes the rest mass and the energy density includes the rest mass energy density, (o2scl::part_tl::inc_rest_mass is true
) and the right expression is used when the rest mass is not included (o2scl::part_tl::inc_rest_mass is false
). For convenience, we define .
The fermionic integrands vanish when the argument of the exponential becomes large compared to a positive number . This condition is
Thus solving for the momentum, an upper limit, is
The entropy is only significant at the Fermi surface, thus in the degenerate case, the lower limit of the entropy integral can be given be determined by the value of which solves
The solution is
which is a valid lower limit only if the argument under the square root is positive.
The energy density is
the number density is
and the entropy density is
where
The derivative can also be written
In the degenerate regime, , can lose precision when
is negative and sufficiently large in absolute magnitude. Thus when
(for
) an alternative expression
can be used.
Non-degenerate integrands:
The integrands in the non-degenerate regime are written in a dimensionless form, by defining (this choice ensures
corresponds to
),
(or
if the chemical potential does not include the mass), and
. Then
,
, and
. The density is
the energy density is
and the entropy density is
Evaluation of the derivatives
The relevant derivatives of the distribution function are
The derivatives can be integrated directly direct) 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 using
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
Presuming the chemical potential includes the rest mass, and , the pressure for non-interacting fermions with degeneracy
is
where the second form is obtained with an integration by parts. We use units where . The variable substitutions from Johns96 are
,
, and
. (Presumably this choice of variables gives better results for non-relativistic fermions because the mass is separated from the chemical potential in the definition of
, but I haven't checked this.) These replacements give
where . Re-expressing in terms of
, one obtains
The pressure is
where .
The Sommerfeld expansion for is
This is an asymptotic expansion, and must thus be used with care. Define . The first term in the Sommerfeld expansion for
depends only on
alone:
where . This expression cannot be used when
is small, but a Taylor series expansion can be used instead. A few terms are
The number density is
Note that because the density is a derivative, it is possible that the terms in the density fail before the terms in the pressure, thus we should use one less term for the density when using the expansion. The entropy is
The derivative of the number density with respect to the chemical potential is
The derivative of the number density with respect to the temperature is
and the derivative of the entropy density with respect to the temperature is
Finally, the derivative of the number density with respect to the mass is more involved because of the mass-dependent prefactor.
These expansions are used in o2scl::fermion_thermo_tl::calc_mu_deg() and o2scl::fermion_deriv_thermo_tl::calc_mu_deg() .
There is a useful identity (Chandrasekhar10 and Tooper69)
which works well when . This result directly gives the sum in Johns96
The function is implemented in GSL as
gsl_sf_bessel_Kn_scaled
. In the case that one wants to include antiparticles, the result is similar
where the scaled Bessel function has been separated out. Similarly defining
the terms in the expansion for the density (without and with antiparticles) are
The entropy terms (with and without antiparticles) are
included. For the derivatives, no additional Bessel functions are required
These expansions are used in o2scl::fermion_thermo_tl::calc_mu_ndeg() .
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