Class fermion_thermo_tl (o2scl)¶
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template<class fermion_t = fermion, class fd_inte_t = fermi_dirac_integ_gsl, class be_inte_t = bessel_K_exp_integ_gsl, class root_t = root_cern<>, class func_t = funct, class fp_t = double>
class fermion_thermo_tl : public o2scl::fermion_zerot_tl<double>¶ Fermion with finite-temperature thermodynamics [abstract base].
This is an abstract base for the computation of finite-temperature fermionic statistics. Different children (e.g. fermion_rel_tl) use different techniques to computing the momentum integrations.
Because massless fermions at finite temperature are much simpler, there are separate member functions included in this class to handle them. The functions massless_calc_density() and massless_calc_mu() compute the thermodynamics of massless fermions at finite temperature given the density or the chemical potentials. The functions massless_pair_density() and massless_pair_mu() perform the same task, but automatically include antiparticles.
The function massless_calc_density() uses a root object to solve for the chemical potential as a function of the density. The default is an object of type root_cern. The function massless_pair_density() does not need to use the root object because of the simplification afforded by the inclusion of antiparticles.
In class fermion_thermo_tl:
Future: Create a Chebyshev approximation for inverting the
the Fermi functions for massless_calc_density() functions?
Template types:
fermion_t: the type of particle object which will be passed to the methods defined by this class
fd_inte_t: the integration object for massless fermions
be_inte_t: the integration object for the nondegenerate limit
root_t: the solver for massless fermions
func_t: the function object type
fp_t: the floating point type
Subclassed by o2scl::fermion_rel_tl< fermion_deriv_tl< double > >, o2scl::fermion_rel_tl< fermion_tl< cpp_dec_float_25 >, fermi_dirac_integ_direct< cpp_dec_float_25, funct_cdf35, cpp_dec_float_35 >, bessel_K_exp_integ_boost< cpp_dec_float_25, cpp_dec_float_35 >, inte_double_exp_boost<>, inte_double_exp_boost<>, root_cern< funct_cdf25, cpp_dec_float_25 >, funct_cdf25, cpp_dec_float_25 >, o2scl::fermion_rel_tl< fermion_tl< long double >, fermi_dirac_integ_direct< long double, funct_cdf25, cpp_dec_float_25 >, bessel_K_exp_integ_boost< long double, cpp_dec_float_25 >, inte_double_exp_boost<>, inte_double_exp_boost<>, root_cern< funct_ld, long double >, funct_ld, long double >, o2scl::fermion_rel_tl< fermion_deriv >, o2scl::fermion_nonrel_tl< fermion_t, fd_inte_t, be_inte_t, root_t, func_t, fp_t >, o2scl::fermion_rel_tl< fermion_t, fd_inte_t, be_inte_t, nit_t, dit_t, root_t, func_t, fp_t >
Massless fermions
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root_t def_massless_root¶
The default solver for massless_calc_density()
We default to a solver of type root_cern here since we don’t have a bracket or a derivative.
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inline void massless_pair_mu(fermion_t &f, fp_t temper)¶
Finite temperature massless fermions and antifermions.
When \( \mu=0 \) (e.g. for photons), the density is zero, the energy density is
\[ \varepsilon = \frac{7g T^4}{120 \pi^2} \]and the pressure is \( P = \varepsilon/3 \). When the chemical potential is finite, we have\[\begin{eqnarray*} n &=& \frac{g \mu^3}{6 \pi^2} \left[ 1 + \left( \frac{\pi T}{\mu} \right)^2 \right] \varepsilon &=& \frac{g \mu^4}{8 \pi^2} \left[ 1 + 2 \left( \frac{\pi T}{\mu} \right)^2 + \frac{7}{15} \left( \frac{\pi T}{\mu} \right)^4 \right] \end{eqnarray*}\]and the pressure is \( P = \varepsilon/3 \).
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inline void massless_pair_density(fermion_t &f, fp_t temper)¶
Finite temperature massless fermions and antifermions.
In the cases \( n^3 \gg T \) and \( T \gg n^3 \) , expansions are used instead of the exact formulas to avoid loss of precision.
In particular, using the parameter
\[ \alpha = \frac{g^2 \pi^2 T^6}{243 n^2} \]and defining the expression\[ \mathrm{cbt} = \alpha^{-1/6} \left( -1 + \sqrt{1+\alpha}\right)^{1/3} \]we can write the chemical potential as\[ \mu = \frac{\pi T}{\sqrt{3}} \left(\frac{1}{\mathrm{cbt}} - \mathrm{cbt} \right) \]These expressions, however, do not work well when \( \alpha \) is very large or very small, so series expansions are used whenever \( \alpha > 10^{4} \) or \( \alpha < 3 \times 10^{-4} \). For small \( \alpha \),
\[ \left(\frac{1}{\mathrm{cbt}} - \mathrm{cbt} \right) \approx \frac{2^{1/3}}{\alpha^{1/6}} - \frac{\alpha^{1/6}}{2^{1/3}} + \frac{\alpha^{5/6}}{6{\cdot}2^{2/3}} + \frac{\alpha^{7/6}}{12{\cdot}2^{1/3}} - \frac{\alpha^{11/6}}{18{\cdot}2^{2/3}} - \frac{5 \alpha^{13/6}}{144{\cdot}2^{1/3}} + \frac{77 \alpha^{17/6}}{2592{\cdot}2^{2/3}} \]and for large \( \alpha \),\[ \left(\frac{1}{\mathrm{cbt}} - \mathrm{cbt} \right) \approx \frac{2}{3} \sqrt{\frac{1}{\alpha}} - \frac{8}{81} \left(\frac{1}{\alpha}\right)^{3/2} + \frac{32}{729} \left(\frac{1}{\alpha}\right)^{5/2} \]This approach works to within about 1 part in \( 10^{12} \), and is tested in
fermion_ts.cpp.In function massless_pair_density()
Future: This could be improved by including more terms
in the expansions.
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inline void set_massless_root(root_t &rp)¶
Set the solver for use in massless_calc_density()
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inline virtual const char *type()¶
Return string denoting type (“fermion_thermo”)
Public Functions
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inline fermion_thermo_tl()¶
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inline virtual ~fermion_thermo_tl()¶
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inline bool pair_den_ndeg(fermion_t &f, fp_t temper, fp_t prec = 1.0e-18)¶
Compute the net density including antiparticles in the nondegenerate approximation.
This function is experimental.
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virtual int calc_mu(fermion_t &f, fp_t temper) = 0¶
Calculate properties as function of chemical potential.
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virtual int calc_density(fermion_t &f, fp_t temper) = 0¶
Calculate properties as function of density.
Note
This function returns an integer value, in contrast to calc_mu(), because of the potential for non-convergence.
Public Members
Protected Attributes
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o2scl::cubic_real_coeff_gsl2<fp_t> crcg2¶
Cubic solver for massless fermions.