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Particles and Nuclei 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_DERIV_PART_H 00024 #define O2SCL_DERIV_PART_H 00025 00026 #include <string> 00027 #include <iostream> 00028 #include <fstream> 00029 #include <cmath> 00030 #include <o2scl/part.h> 00031 #include <o2scl/fermion.h> 00032 00033 #ifndef DOXYGENP 00034 namespace o2scl { 00035 #endif 00036 00037 /** \brief Storage for deriviatives wrt \f$ \mu \f$ and T. 00038 00039 The variables \c dndmu, \c dndT, and \c dsdT correspond 00040 to 00041 \f[ 00042 \left(\frac{d n}{d \mu}\right)_{T,V}, \quad 00043 \left(\frac{d n}{d T}\right)_{\mu,V}, \quad \mathrm{and} \quad 00044 \left(\frac{d s}{d T}\right)_{\mu,V} 00045 \f] 00046 respectively. All other derivatives can be expressed simply in 00047 terms of these three. Note that volume derivatives are trivial, 00048 since both the entropy and number scale linearly with the 00049 volume. 00050 00051 \hline 00052 00053 <b>Derivatives wrt to chemical potential and temperature:</b> 00054 00055 There is a Maxwell relation 00056 \f[ 00057 \left(\frac{d s}{d \mu}\right)_{T,V} = 00058 \left(\frac{d n}{d T}\right)_{\mu,V} 00059 \f] 00060 The pressure derivatives are trivial 00061 \f[ 00062 \left(\frac{d P}{d \mu}\right)_{T,V}=n, \quad 00063 \left(\frac{d P}{d T}\right)_{\mu,V}=s 00064 \f] 00065 The energy density derivatives are related through the 00066 thermodynamic identity: 00067 \f[ 00068 \left(\frac{d \varepsilon}{d \mu}\right)_{T,V}= 00069 \mu \left(\frac{d n}{d \mu}\right)_{T,V}+ 00070 T \left(\frac{d s}{d \mu}\right)_{T,V} 00071 \f] 00072 \f[ 00073 \left(\frac{d \varepsilon}{d T}\right)_{\mu,V}= 00074 \mu \left(\frac{d n}{d T}\right)_{\mu,V}+ 00075 T \left(\frac{d s}{d T}\right)_{\mu,V} 00076 \f] 00077 00078 \hline 00079 00080 <b>Other derivatives:</b> 00081 00082 Note that the derivative of the entropy with respect to the 00083 temperature above is not the specific heat per particle, \f$ c_V \f$. 00084 The specific heat per particle is 00085 \f[ 00086 c_V = \frac{T}{N} \left( \frac{\partial S}{\partial T} \right)_{V,N} 00087 = \frac{T}{n} \left( \frac{\partial s}{\partial T} \right)_{V,n} 00088 \f] 00089 As noted in \ref part_section in the User's Guide for \o2p, we 00090 work in units so that \f$ \hbar = c = k_B = 1 \f$. In this case, 00091 \f$ c_V \f$ is unitless as defined here. To compute \f$ c_V \f$ 00092 in terms of the derivatives above, note that the 00093 descendants of deriv_part provide all of the thermodynamic 00094 functions in terms of \f$ \mu, V \f$ and \f$ T \f$, so we have 00095 \f[ 00096 s=s(\mu,T,V) \quad \mathrm{and} \quad n=n(\mu,T,V) \, . 00097 \f] 00098 We can then construct a function 00099 \f[ 00100 s=s[\mu(n,T,V),T,V] 00101 \f] 00102 and then write the required derivative directly 00103 \f[ 00104 \left(\frac{\partial s}{\partial T}\right)_{n,V} = 00105 \left(\frac{\partial s}{\partial \mu}\right)_{T,V} 00106 \left(\frac{\partial \mu}{\partial T}\right)_{n,V} + 00107 \left(\frac{\partial s}{\partial T}\right)_{\mu,V} \, . 00108 \f] 00109 Now we use the identity 00110 \f[ 00111 \left(\frac{\partial \mu}{\partial T}\right)_{n,V} = - 00112 \left(\frac{\partial n}{\partial T}\right)_{\mu,V} 00113 \left(\frac{\partial n}{\partial \mu}\right)_{T,V}^{-1} \, , 00114 \f] 00115 and the Maxwell relation above to give 00116 \f[ 00117 C_V = \frac{T}{n} 00118 \left[ 00119 \left(\frac{\partial s}{\partial T}\right)_{\mu,V} 00120 -\left(\frac{\partial n}{\partial T}\right)_{\mu,V}^2 00121 \left(\frac{\partial n}{\partial \mu}\right)_{T,V}^{-1} 00122 \right] 00123 \f] 00124 which expresses the specific heat in terms of the three 00125 derivatives which are given. 00126 00127 For, \f$ c_P \f$, defined as 00128 \f[ 00129 c_P = \frac{T}{N} \left( \frac{\partial S}{\partial T} 00130 \right)_{N,P} 00131 \f] 00132 (which is also unitless) we can write functions 00133 \f[ 00134 S=S(N,T,V) \qquad \mathrm{and} \qquad V=V(N,P,T) 00135 \f] 00136 which imply 00137 \f[ 00138 \left( \frac{\partial S}{\partial T} \right)_{N,P} = 00139 \left( \frac{\partial S}{\partial T} \right)_{N,V} + 00140 \left( \frac{\partial S}{\partial V} \right)_{N,T} 00141 \left( \frac{\partial V}{\partial T} \right)_{N,P} \, . 00142 \f] 00143 Thus we require the derivatives 00144 \f[ 00145 \left( \frac{\partial S}{\partial T} \right)_{N,V} , 00146 \left( \frac{\partial S}{\partial V} \right)_{N,T} , 00147 \qquad\mathrm{and}\qquad 00148 \left( \frac{\partial V}{\partial T} \right)_{N,P} 00149 \, . 00150 \f] 00151 00152 To compute the new entropy derivatives, we can write 00153 \f[ 00154 S=S(\mu(N,T,V),T,V) 00155 \f] 00156 to get 00157 \f[ 00158 \left( \frac{\partial S}{\partial T} \right)_{N,V} = 00159 \left( \frac{\partial S}{\partial \mu} \right)_{T,V} 00160 \left( \frac{\partial \mu}{\partial T} \right)_{N,V} + 00161 \left( \frac{\partial S}{\partial T} \right)_{\mu,V} \, , 00162 \f] 00163 and 00164 \f[ 00165 \left( \frac{\partial S}{\partial V} \right)_{N,T} = 00166 \left( \frac{\partial S}{\partial \mu} \right)_{T,V} 00167 \left( \frac{\partial \mu}{\partial V} \right)_{N,T} + 00168 \left( \frac{\partial S}{\partial V} \right)_{\mu,T} \, . 00169 \f] 00170 These require the chemical potential derivatives which have 00171 associated Maxwell relations 00172 \f[ 00173 \left( \frac{\partial \mu}{\partial T} \right)_{N,V} = 00174 -\left( \frac{\partial S}{\partial N} \right)_{T,V} 00175 \qquad\mathrm{and}\qquad 00176 \left( \frac{\partial \mu}{\partial V} \right)_{N,T} = 00177 -\left( \frac{\partial P}{\partial N} \right)_{T,V} \, . 00178 \f] 00179 Finally, we can rewrite the derivatives on the right hand sides 00180 in terms of derivatives of functions of \f$ \mu, V \f$ and 00181 \f$ T \f$, 00182 \f[ 00183 \left( \frac{\partial S}{\partial N} \right)_{T,V} = 00184 \left( \frac{\partial S}{\partial \mu} \right)_{T,V} 00185 \left( \frac{\partial N}{\partial \mu} \right)_{T,V}^{-1} \, , 00186 \f] 00187 and 00188 \f[ 00189 \left( \frac{\partial P}{\partial N} \right)_{T,V} = 00190 \left( \frac{\partial P}{\partial \mu} \right)_{T,V} 00191 \left( \frac{\partial N}{\partial \mu} \right)_{T,V}^{-1} \, . 00192 \f] 00193 00194 The volume derivative, 00195 \f[ 00196 \left( \frac{\partial V}{\partial T} \right)_{N,P} \, , 00197 \f] 00198 is related to the coefficient of thermal expansion, sometimes 00199 called \f$ \alpha \f$, 00200 \f[ 00201 \alpha \equiv \frac{1}{V} 00202 \left( \frac{\partial V}{\partial T} \right)_{N,P} \, . 00203 \f] 00204 We can rewrite the derivative 00205 \f[ 00206 \left( \frac{\partial V}{\partial T} \right)_{N,P} = 00207 -\left( \frac{\partial P}{\partial T} \right)_{N,V} 00208 \left( \frac{\partial P}{\partial V} \right)_{N,T}^{-1} \, . 00209 \f] 00210 The first term can be computed from the Maxwell relation 00211 \f[ 00212 \left( \frac{\partial P}{\partial T} \right)_{N,V} = 00213 \left( \frac{\partial S}{\partial V} \right)_{N,T} \, , 00214 \f] 00215 where the entropy derivative was computed above. The second term 00216 (related to the inverse of the isothermal compressibility, \f$ 00217 \kappa_T \equiv (-1/V) (\partial V/\partial P)_{T,N} \f$ can be 00218 computed from the function \f$ P = P(\mu(N,V,T),V,T) \f$ 00219 \f[ 00220 \left( \frac{\partial P}{\partial V} \right)_{N,T} = 00221 \left( \frac{\partial P}{\partial \mu} \right)_{T,V} 00222 \left( \frac{\partial \mu}{\partial V} \right)_{N,T} + 00223 \left( \frac{\partial P}{\partial V} \right)_{\mu,T} 00224 \f] 00225 where the chemical potential derivative was computed above. 00226 00227 The results above can be collected to give 00228 \f[ 00229 \left( \frac{\partial S}{\partial T} \right)_{N,P} = 00230 \left( \frac{\partial S}{\partial T} \right)_{\mu,V} + 00231 \frac{S^2}{N^2} 00232 \left( \frac{\partial N}{\partial \mu} \right)_{T,V} - 00233 \frac{2 S}{N} 00234 \left( \frac{\partial N}{\partial T} \right)_{\mu,V} \, , 00235 \f] 00236 which implies 00237 \f[ 00238 c_P = 00239 \frac{T}{n} 00240 \left( \frac{\partial s}{\partial T} \right)_{\mu,V} + 00241 \frac{s^2 T}{n^3} 00242 \left( \frac{\partial n}{\partial \mu} \right)_{T,V} - 00243 \frac{2 s T}{n^2} 00244 \left( \frac{\partial n}{\partial T} \right)_{\mu,V} \, , 00245 \f] 00246 00247 This derivation also gives the well-known relationship between 00248 the specific heats at constant volume and constant pressure, 00249 \f[ 00250 c_P = c_V + \frac{T \alpha^2}{n \kappa_T} \, . 00251 \f] 00252 00253 No derivative with respect to the bare mass is given, since 00254 classes cannot know how to relate the effective mass to the 00255 bare mass. 00256 00257 */ 00258 class part_deriv : public part { 00259 00260 public: 00261 00262 /// Derivative of number density with respect to chemical potential 00263 double dndmu; 00264 00265 /// Derivative of number density with respect to temperature 00266 double dndT; 00267 00268 /// Derivative of entropy density with respect to temperature 00269 double dsdT; 00270 00271 /// Derivative of number density with respect to the effective mass 00272 double dndm; 00273 00274 part_deriv(double mass=0.0, double dof=0.0) : part(mass,dof) { 00275 } 00276 00277 }; 00278 00279 class fermion_deriv : public part_deriv { 00280 00281 public: 00282 00283 /// Fermi momentum 00284 double kf; 00285 00286 fermion_deriv(double mass=0.0, double dof=0.0) : part_deriv(mass,dof) { 00287 } 00288 00289 }; 00290 00291 class fermion_deriv_thermo { 00292 00293 public: 00294 00295 virtual ~fermion_deriv_thermo() { 00296 } 00297 00298 /** \brief Calculate properties as function of chemical potential 00299 */ 00300 virtual void calc_mu(fermion_deriv &f, double temper)=0; 00301 00302 /** \brief Calculate properties as function of density 00303 */ 00304 virtual void calc_density(fermion_deriv &f, double temper)=0; 00305 00306 /** \brief Calculate properties with antiparticles as function of 00307 chemical potential 00308 */ 00309 virtual void pair_mu(fermion_deriv &f, double temper)=0; 00310 00311 /** \brief Calculate properties with antiparticles as function of 00312 density 00313 */ 00314 virtual void pair_density(fermion_deriv &f, double temper)=0; 00315 00316 /// Calculate effective chemical potential from density 00317 virtual void nu_from_n(fermion_deriv &f, double temper)=0; 00318 00319 }; 00320 00321 00322 #ifndef DOXYGENP 00323 } 00324 #endif 00325 00326 #endif
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