Class fermion_deriv_rel_tl (o2scl)¶
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template<class fermion_deriv_t = fermion_deriv_tl<double>, class fermion_rel_t = fermion_rel_tl<fermion_deriv_t>, class nit_t = inte_qagiu_gsl<>, class dit_t = inte_qag_gsl<>, class fp_t = double>
class fermion_deriv_rel_tl : public o2scl::fermion_deriv_thermo_tl<fermion_deriv_tl<double>, double>, public o2scl::fermion_deriv_rel_integ<double>¶ Equation of state for a relativistic fermion.
This implements an equation of state for a relativistic fermion using direct integration. After subtracting the rest mass from the chemical potentials, the distribution function is
\[ \left\{1+\exp[(\sqrt{k^2+m^{* 2}}-m-\nu)/T]\right\}^{-1} \]where \( k \) is the momentum, \( \nu \) is the effective chemical potential, \( m \) is the rest mass, and \( m^{*} \) is the effective mass. For later use, we define \( E^{*} = \sqrt{k^2 + m^{*2}} \) . The degeneracy parameter is\[ \psi=(\nu+(m-m^{*}))/T \]For \( \psi \) greater than deg_limit (degenerate regime), a finite interval integrator is used and for \( \psi \) less than deg_limit (non-degenerate regime), an integrator over the interval from \( [0,\infty) \) is used. The upper limit on the degenerate integration is given by the value of the momentum \( k \) which is the solution of\[ (\sqrt{k^2+m^{*,2}}-m-\nu)/T=\mathrm{f{l}imit} \]which is\[ \sqrt{(m+{\cal L})^2-m^{*2}} \]where \( {\cal L}\equiv\mathrm{f{l}imit}\times T+\nu \) .For the entropy integration, we set the lower limit to
\[ 2 \sqrt{\nu^2+2 \nu m} - \mathrm{upper~limit} \]since the only contribution to the entropy is at the Fermi surface.In the non-degenerate regime, we make the substitution \( u=k/T \) to help ensure that the variable of integration scales properly.
Uncertainties are given in unc.
Evaluation of the derivatives
The relevant derivatives of the distribution function are
\[ \frac{\partial f}{\partial T}= f(1-f)\frac{E^{*}-m-\nu}{T^2} \]\[ \frac{\partial f}{\partial \nu}= f(1-f)\frac{1}{T} \]\[ \frac{\partial f}{\partial k}= -f(1-f)\frac{k}{E^{*} T} \]\[ \frac{\partial f}{\partial m^{*}}= -f(1-f)\frac{m^{*}}{E^{*} T} \]We also need the derivative of the entropy integrand w.r.t. the distribution function, which is
\[ {\cal S}\equiv f \ln f +(1-f) \ln (1-f) \qquad \frac{\partial {\cal S}}{\partial f} = \ln \left(\frac{f}{1-f}\right) = \left(\frac{\nu-E^{*}+m}{T}\right) \]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 (method = direct) or they may be converted to integrals over the distribution function through an integration by parts (method = 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,\[ \frac{g}{2 \pi^2} \int_0^{\infty} h(k) f (1-f) dk = \frac{g}{2 \pi^2} \int_0^{\infty} f \frac{T}{k} \left[ h^{\prime} E^{*}-\frac{h E^{*}}{k}+\frac{h k}{E^{*}} \right] dk \]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\[ \left(\frac{d n}{d \mu}\right)_T = \frac{g}{2 \pi^2} \int_0^{\infty} \left(\frac{k^2+E^{*2}}{E^{*}}\right) f dk \]2) The derivative of the density wrt the temperature
\[ \left(\frac{d n}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{k^2(E^{*}-m-\nu)}{T^2} f (1-f) dk \]Using \( h(k)=k^2(E^{*}-\nu)/T^2 \) we get\[ \left(\frac{d n}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{f}{T} \left[2 k^2+E^{*2}-E^{*}\left(\nu+m\right)- k^2 \left(\frac{\nu+m}{E^{*}}\right)\right] dk \]3) The derivative of the entropy wrt the chemical potential
\[ \left(\frac{d s}{d \mu}\right)_T = \frac{g}{2 \pi^2} \int_0^{\infty} k^2 f (1-f) \frac{(E^{*}-m-\nu)}{T^2} dk \]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
\[ \left(\frac{d s}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} k^2 f (1-f) \frac{(E^{*}-m-\nu)^2}{T^3} dk \]Using \( h(k)=k^2 (E^{*}-\nu)^2/T^3 \)\[ \left(\frac{d s}{d T}\right)_{\mu} = \frac{g}{2 \pi^2} \int_0^{\infty} \frac{f(E^{*}-m-\nu)}{E^{*}T^2} \left[E^{* 3}+3 E^{*} k^2- (E^{* 2}+k^2)(\nu+m)\right] d k \]5) The derivative of the density wrt the effective mass
\[ \left(\frac{d n}{d m^{*}}\right)_{T,\mu} = -\frac{g}{2 \pi^2} \int_0^{\infty} \frac{k^2 m^{*}}{E^{*} T} f (1-f) dk \]Using \( h(k)=-(k^2 m^{*})/(E^{*} T) \) we get\[ \left(\frac{d n}{d m^{*}}\right)_{T,\mu} = -\frac{g}{2 \pi^2} \int_0^{\infty} m^{*} f dk \]Todo
In class fermion_deriv_rel_tl:
Future: The option err_nonconv=false is not really implemented yet.
Future: The ref pair_density() function is a bit slow because it computes the non-derivative thermodynamic quantities twice, and this could be improved.
Note
This class only has preliminary support for inc_rest_mass=true (more testing should be done, particularly for the “pair” functions)
Note
The dsdT integration may fail if the system is very degenerate. When method is byparts, the integral involves a large cancellation between the regions from \( k \in (0, \mathrm{ulimit/2}) \) and \( k \in (\mathrm{ulimit/2}, \mathrm{ulimit}) \). Switching to method=direct and setting the lower limit to \( \mathrm{llimit} \), may help, but recent testing on this gave negative values for dsdT. For very degenerate systems, an expansion may be better than trying to perform the integration. The value of the integrand at k=0 also looks like it might be causing difficulties.
Integration objects
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inte_double_exp_boost it_multip¶
Adaptive ultiprecision integrator.
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inline void set_inte(inte<std::function<fp_t(fp_t)>, fp_t> &unit, inte<std::function<fp_t(fp_t)>, fp_t> &udit)¶
Set inte objects.
The first integrator is used for non-degenerate integration and should integrate from 0 to \( \infty \) (like o2scl::inte_qagiu_gsl). The second integrator is for the degenerate case, and should integrate between two finite values.
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inline virtual const char *type()¶
Return string denoting type (“fermion_deriv_rel”)
Public Functions
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inline fermion_deriv_rel_tl()¶
Create a fermion with mass
mand degeneracyg.
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inline virtual ~fermion_deriv_rel_tl()¶
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inline virtual int calc_mu(fermion_deriv_t &f, fp_t temper)¶
Calculate properties as function of chemical potential.
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inline virtual int calc_density(fermion_deriv_t &f, fp_t temper)¶
Calculate properties as function of density.
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inline virtual int pair_mu(fermion_deriv_t &f, fp_t temper)¶
Calculate properties with antiparticles as function of chemical potential.
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inline virtual int pair_density(fermion_deriv_t &f, fp_t temper)¶
Calculate properties with antiparticles as function of density.
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inline virtual int nu_from_n(fermion_deriv_t &f, fp_t temper)¶
Calculate effective chemical potential from density.
Public Members
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fp_t upper_limit_fac¶
The limit for the Fermi functions (default 20)
fermion_deriv_rel will ignore corrections smaller than about \( \exp(-\mathrm{f{l}imit}) \) .
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fermion_deriv_t unc¶
Storage for the most recently calculated uncertainties.
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fermion_rel_t fr¶
Object for computing non-derivative quantities.
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int verbose¶
Verbosity parameter (default 0)
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bool multip¶
If true, use multiprecision to improve the integrations.
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bool verify_ti¶
If true, verify the thermodynamic identity (default false)
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int last_method¶
An integer indicating the last numerical method used.
The function calc_mu() sets this integer to a two-digit number. It is equal to 10 times the value reported by o2scl::fermion_rel::calc_mu() plus a value from the list below corresponding to the method used for the derivatives
1: nondegenerate expansion
2: degenerate expansion
4: nondegenerate integrand, using user-specified value for method
5: degenerate integrand, using direct
6: degenerate integrand, using by_parts
7: degenerate integrand, using user-specified value for method
The function nu_from_n() sets this value equal to 100 times the value reported by o2scl::fermion_rel_tl::nu_from_n() .
The function calc_density() sets this value equal to the value from o2scl::fermion_deriv_rel_tl::nu_from_n() plus the value from o2scl::fermion_deriv_rel_tl::calc_mu() .
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std::string last_method_s¶
String detailing last method used.
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bool err_nonconv¶
If true, call the error handler when convergence fails (default true)