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Equation of State Sub-Library: Version 0.910
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00001 /* 00002 ------------------------------------------------------------------- 00003 00004 Copyright (C) 2006-2012, Andrew W. Steiner 00005 00006 This file is part of O2scl. 00007 00008 O2scl is free software; you can redistribute it and/or modify 00009 it under the terms of the GNU General Public License as published by 00010 the Free Software Foundation; either version 3 of the License, or 00011 (at your option) any later version. 00012 00013 O2scl is distributed in the hope that it will be useful, 00014 but WITHOUT ANY WARRANTY; without even the implied warranty of 00015 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 00016 GNU General Public License for more details. 00017 00018 You should have received a copy of the GNU General Public License 00019 along with O2scl. If not, see <http://www.gnu.org/licenses/>. 00020 00021 ------------------------------------------------------------------- 00022 */ 00023 #ifndef O2SCL_GEN_POTENTIAL_EOS_H 00024 #define O2SCL_GEN_POTENTIAL_EOS_H 00025 00026 #include <iostream> 00027 #include <string> 00028 #include <cmath> 00029 #include <o2scl/constants.h> 00030 #include <o2scl/mroot.h> 00031 #include <o2scl/hadronic_eos.h> 00032 #include <o2scl/part.h> 00033 #include <o2scl/gsl_deriv.h> 00034 #include <o2scl/nonrel_fermion.h> 00035 #include <cstdlib> 00036 00037 #ifndef DOXYGENP 00038 namespace o2scl { 00039 #endif 00040 00041 /** \brief Generalized potential model equation of state 00042 00043 The single particle energy is defined by the functional derivative 00044 of the energy density with respect to the distribution function 00045 \f[ 00046 e_{\tau} = \frac{\delta \varepsilon}{\delta f_{\tau}} 00047 \f] 00048 00049 The effective mass is defined by 00050 \f[ 00051 \frac{m^{*}}{m} = \left( \frac{m}{k} 00052 \frac{d e_{\tau}}{d k} 00053 \right)^{-1}_{k=k_F} 00054 \f] 00055 00056 In all of the models, the kinetic energy density is 00057 \f$\tau_n+\tau_p\f$ where 00058 \f[ 00059 \tau_i = \frac{2}{(2 \pi)^3} \int d^3 k~ 00060 \left(\frac{k^2}{2 m}\right)f_i(k,T) 00061 \f] 00062 and the number density is 00063 \f[ 00064 \rho_i = \frac{2}{(2 \pi)^3} \int d^3 k~f_i(k,T) 00065 \f] 00066 00067 When \ref form == \ref mdi_form or 00068 \ref gbd_form, the potential energy 00069 density is given by \ref Das03 : 00070 \f[ 00071 V(\rho,\delta) = \frac{Au}{\rho_0} \rho_n \rho_p + 00072 \frac{A_l}{2 \rho_0} \left(\rho_n^2+\rho_p^2\right)+ 00073 \frac{B}{\sigma+1} \frac{\rho^{\sigma+1}}{\rho_0^{\sigma}} 00074 \left(1-x \delta^2\right)+V_{mom}(\rho,\delta) 00075 \f] 00076 where \f$\delta=1-2 \rho_p/(\rho_n+\rho_p)\f$. 00077 If \ref form == \ref mdi_form, then 00078 \f[ 00079 V_{mom}(\rho,\delta)= 00080 \frac{1}{\rho_0} 00081 \sum_{\tau,\tau^{\prime}} C_{\tau,\tau^{\prime}} 00082 \int \int 00083 d^3 k d^3 k^{\prime} 00084 \frac{f_{\tau}(\vec{k}) f_{{\tau}^{\prime}} (\vec{k}^{\prime})} 00085 {1-(\vec{k}-\vec{k}^{\prime})^2/\Lambda^2} 00086 \f] 00087 where \f$C_{1/2,1/2}=C_{-1/2,-1/2}=C_{\ell}\f$ and 00088 \f$C_{1/2,-1/2}=C_{-1/2,1/2}=C_{u}\f$. 00089 Otherwise if \ref form == \ref gbd_form, then 00090 \f[ 00091 V_{mom}(\rho,\delta)= 00092 \frac{1}{\rho_0}\left[ 00093 C_{\ell} \left( \rho_n g_n + \rho_p g_p \right)+ 00094 C_u \left( \rho_n g_p + \rho_p g_n \right) 00095 \right] 00096 \f] 00097 where 00098 \f[ 00099 g_i=\frac{\Lambda^2}{\pi^2}\left[ k_{F,i}-\Lambda 00100 \mathrm{tan}^{-1} \left(k_{F,i}/\Lambda\right) \right] 00101 \f] 00102 00103 Otherwise, if \ref form == \ref bgbd_form, \ref bpalb_form 00104 or \ref sl_form, then the potential energy density is 00105 given by \ref Bombaci01 : 00106 \f[ 00107 V(\rho,\delta) = V_A+V_B+V_C 00108 \f] 00109 \f[ 00110 V_A = \frac{2 A}{3 \rho_0} 00111 \left[ \left(1+\frac{x_0}{2}\right)\rho^2- 00112 \left(\frac{1}{2}+x_0\right)\left(\rho_n^2+\rho_p^2\right)\right] 00113 \f] 00114 \f[ 00115 V_B=\frac{4 B}{3 \rho_0^{\sigma}} \frac{T}{1+4 B^{\prime} T / 00116 \left(3 \rho_0^{\sigma-1} \rho^2\right)} 00117 \f] 00118 where 00119 \f[ 00120 T = \rho^{\sigma-1} \left[ \left( 1+\frac{x_3}{2} \right) \rho^2 - 00121 \left(\frac{1}{2}+x_3\right)\left(\rho_n^2+\rho_p^2\right)\right] 00122 \f] 00123 The term \f$V_C\f$ is: 00124 \f[ 00125 V_C=\sum_{i=1}^{i_{\mathrm{max}}} 00126 \frac{4}{5} \left(C_{i}+2 z_i\right) \rho 00127 (g_{n,i}+g_{p,i})+\frac{2}{5}\left(C_i -8 z_i\right) 00128 (\rho_n g_{n,i} + \rho_p g_{p,i}) 00129 \f] 00130 where 00131 \f[ 00132 g_{\tau,i} = \frac{2}{(2 \pi)^3} \int d^3 k f_{\tau}(k,T) 00133 g_i(k) 00134 \f] 00135 00136 For \ref form == \ref bgbd_form or \ref form == \ref bpalb_form, 00137 the form factor is given by 00138 \f[ 00139 g_i(k) = \left(1+\frac{k^2}{\Lambda_i^2}\right)^{-1} 00140 \f] 00141 while for \ref form == \ref sl_form, the form factor 00142 is given by 00143 \f[ 00144 g_i(k) = 1-\frac{k^2}{\Lambda_i^2} 00145 \f] 00146 where \f$\Lambda_1\f$ is specified in the parameter 00147 \c Lambda when necessary. 00148 00149 \htmlonly 00150 See Mathematica notebook at 00151 <a href="gen_potential_eos.nb"> 00152 gen_potential_eos.nb</a>, and 00153 <a href="gen_potential_eos.ps"> 00154 gen_potential_eos.ps</a>. 00155 \endhtmlonly 00156 \latexonly 00157 See Mathematica notebook at 00158 \begin{verbatim} 00159 doc/o2scl/extras/gen_potential_eos.nb 00160 doc/o2scl/extras/gen_potential_eos.ps 00161 \end{verbatim} 00162 \endlatexonly 00163 00164 \bug The BGBD EOS doesn't work and the 00165 effective mass for the GBD EOS doesn't work 00166 00167 \future Calculate the chemical potentials analytically 00168 00169 */ 00170 class gen_potential_eos : public hadronic_eos_eden { 00171 00172 public: 00173 00174 /// \name The parameters for the various interactions 00175 //@{ 00176 double x,Au,Al,rho0,B,sigma,Cl,Cu,Lambda; 00177 double A,x0,x3,Bp,C1,z1,Lambda2,C2,z2,bpal_esym; 00178 int sym_index; 00179 //@} 00180 00181 gen_potential_eos(); 00182 00183 /// Equation of state as a function of density. 00184 virtual int calc_e(fermion &ne, fermion &pr, thermo <); 00185 00186 /// Form of potential 00187 int form; 00188 00189 /// The "momentum-dependent-interaction" form 00190 static const int mdi_form=1; 00191 00192 /// The modifed GBD form 00193 static const int bgbd_form=2; 00194 00195 /// The form from \ref Prakash88 as formulated in \ref Bombaci01 00196 static const int bpalb_form=3; 00197 00198 /// The "SL" form. See \ref Bombaci01. 00199 static const int sl_form=4; 00200 00201 /// The Gale, Bertsch, Das Gupta from \ref Gale87. 00202 static const int gbd_form=5; 00203 00204 /// The form from \ref Prakash88. 00205 static const int bpal_form=6; 00206 00207 /** \brief Set the derivative object to calculate the 00208 chemical potentials 00209 */ 00210 int set_mu_deriv(deriv<funct> &de) { 00211 mu_deriv_set=true; 00212 mu_deriv_ptr=&de; 00213 return 0; 00214 } 00215 00216 /// The default derivative object for calculating chemical potentials 00217 gsl_deriv<funct> def_mu_deriv; 00218 00219 /// Return string denoting type ("gen_potential_eos") 00220 virtual const char *type() { return "gen_potential_eos"; } 00221 00222 protected: 00223 00224 #ifndef DOXYGEN_INTERNAL 00225 00226 /// Desc 00227 nonrel_fermion nrf; 00228 00229 /// True of the derivative object has been set 00230 bool mu_deriv_set; 00231 00232 /// The derivative object 00233 deriv<funct> *mu_deriv_ptr; 00234 00235 /// Compute the momentum integral for \ref mdi_form 00236 double mom_integral(double pft, double pftp); 00237 00238 /** \name The mode for the energy() function [protected] */ 00239 //@{ 00240 int mode; 00241 static const int nmode=1; 00242 static const int pmode=2; 00243 static const int normal=0; 00244 //@} 00245 00246 /// Compute the energy 00247 double energy(double x); 00248 00249 #endif 00250 00251 }; 00252 00253 #ifndef DOXYGENP 00254 } 00255 #endif 00256 00257 #endif
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