Class eos_had_potential (o2scl)¶
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class eos_had_potential : public o2scl::eos_had_eden_base¶
Generalized potential model equation of state.
The single particle energy is defined by the functional derivative of the energy density with respect to the distribution function
\[ e_{\tau} = \frac{\delta \varepsilon}{\delta f_{\tau}} \]The effective mass is defined by
\[ \frac{m^{*}}{m} = \left( \frac{m}{k} \frac{d e_{\tau}}{d k} \right)^{-1}_{k=k_F} \]In all of the models, the kinetic energy density is \(\tau_n+\tau_p\) where
\[ \tau_i = \frac{2}{(2 \pi)^3} \int d^3 k~ \left(\frac{k^2}{2 m}\right)f_i(k,T) \]and the number density is\[ \rho_i = \frac{2}{(2 \pi)^3} \int d^3 k~f_i(k,T) \]See the review in [Li01].
When form is equal to mdi_form or gbd_form, the potential energy density is given by
\[ V(\rho,\delta) = \frac{Au}{\rho_0} \rho_n \rho_p + \frac{A_l}{2 \rho_0} \left(\rho_n^2+\rho_p^2\right)+ \frac{B}{\sigma+1} \frac{\rho^{\sigma+1}}{\rho_0^{\sigma}} \left(1-x \delta^2\right)+V_{mom}(\rho,\delta) \]where \(\delta=1-2 \rho_p/(\rho_n+\rho_p)\).This potential energy is from [Das03].
If form is equal to mdi_form, then
\[ V_{mom}(\rho,\delta)= \frac{1}{\rho_0} \sum_{\tau,\tau^{\prime}} C_{\tau,\tau^{\prime}} \int \int d^3 k d^3 k^{\prime} \frac{f_{\tau}(\vec{k}) f_{{\tau}^{\prime}} (\vec{k}^{\prime})} {1-(\vec{k}-\vec{k}^{\prime})^2/\Lambda^2} \]where \(C_{1/2,1/2}=C_{-1/2,-1/2}=C_{\ell}\) and \(C_{1/2,-1/2}=C_{-1/2,1/2}=C_{u}\). Laterparameterizations in this form are given in [Chen05].
Otherwise if form is equal to gbd_form, then
\[ V_{mom}(\rho,\delta)= \frac{1}{\rho_0}\left[ C_{\ell} \left( \rho_n g_n + \rho_p g_p \right)+ C_u \left( \rho_n g_p + \rho_p g_n \right) \right] \]where\[ g_i=\frac{\Lambda^2}{\pi^2}\left[ k_{F,i}-\Lambda \mathrm{tan}^{-1} \left(k_{F,i}/\Lambda\right) \right] \]Otherwise, if form is equal to bgbd_form, bpal_form or sl_form, then the potential energy density is given by:
\[ V(\rho,\delta) = V_A+V_B+V_C \]\[ V_A = \frac{2 A}{3 \rho_0} \left[ \left(1+\frac{x_0}{2}\right)\rho^2- \left(\frac{1}{2}+x_0\right)\left(\rho_n^2+\rho_p^2\right)\right] \]\[ V_B=\frac{4 B}{3 \rho_0^{\sigma}} \frac{T}{1+4 B^{\prime} T / \left(3 \rho_0^{\sigma-1} \rho^2\right)} \]where\[ T = \rho^{\sigma-1} \left[ \left( 1+\frac{x_3}{2} \right) \rho^2 - \left(\frac{1}{2}+x_3\right)\left(\rho_n^2+\rho_p^2\right)\right] \]The term \(V_C\) is:\[ V_C=\sum_{i=1}^{i_{\mathrm{max}}} \frac{4}{5} \left(C_{i}+2 z_i\right) \rho (g_{n,i}+g_{p,i})+\frac{2}{5}\left(C_i -8 z_i\right) (\rho_n g_{n,i} + \rho_p g_{p,i}) \]where\[ g_{\tau,i} = \frac{2}{(2 \pi)^3} \int d^3 k f_{\tau}(k,T) g_i(k) \]This potential energy is from [Bombaci01].
For form is equal to bgbd_form or form is equal to bpal_form, the form factor is given by
\[ g_i(k) = \left(1+\frac{k^2}{\Lambda_i^2}\right)^{-1} \]while for form is equal to sl_form, the form factor is given by\[ g_i(k) = 1-\frac{k^2}{\Lambda_i^2} \]where \(\Lambda_1\) is specified in the parameterLambdawhen necessary.- Bug:
The BGBD and SL EOSs do not work.
- Idea for Future:
Calculate the chemical potentials analytically.
Subclassed by o2scl::eos_had_sym4_mdi
The parameters for the various interactions
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double x¶
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double Au¶
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double Al¶
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double rho0¶
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double B¶
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double sigma¶
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double Cl¶
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double Cu¶
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double Lambda¶
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double A¶
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double x0¶
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double x3¶
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double Bp¶
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double C1¶
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double z1¶
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double Lambda2¶
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double C2¶
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double z2¶
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double bpal_esym¶
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int sym_index¶
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int form¶
Form of potential.
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static const int mdi_form = 1¶
The “momentum-dependent-interaction” form from Das et al. (2003)
See [Das03].
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static const int bgbd_form = 2¶
The modifed GBD form.
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static const int bpal_form = 3¶
The form from Prakash et al. (1988) as formulated in Bombaci et al. (2001)
See [Prakash88] and [Bombaci01].
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static const int sl_form = 4¶
The “SL” form. See Bombaci et al. (2001)
See [Bombaci01].
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static const int pal_form = 6¶
The form from Prakash et al. 1988.
See [Prakash88].
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fermion_nonrel nrf¶
Non-relativistic fermion thermodyanmics.
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bool mu_deriv_set¶
True of the derivative object has been set.
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deriv_base *mu_deriv_ptr¶
The derivative object.
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eos_had_potential()¶
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inline virtual int calc_temp_f_gen(double nB, double nQ, double nS, double T, thermo &th)¶
Equation of state as a function of baryon, charge, and strangeness density at finite temperature.
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virtual int calc_e(fermion &ne, fermion &pr, thermo <)¶
Equation of state as a function of density.
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inline int set_mu_deriv(deriv_base<> &de)¶
Set the derivative object to calculate the chemical potentials.
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inline virtual const char *type()¶
Return string denoting type (“eos_had_potential”)