00001 /* 00002 ------------------------------------------------------------------- 00003 00004 Copyright (C) 2006, 2007, 2008, 2009, 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/boson.h> 00031 #include <o2scl/fermion.h> 00032 #include <o2scl/classical.h> 00033 00034 #ifndef DOXYGENP 00035 namespace o2scl { 00036 #endif 00037 00038 /** 00039 \brief Storage for deriviatives wrt \f$ \mu \f$ and T. 00040 00041 The variables \c dndmu, \c dndT, and \c dsdT correspond 00042 to 00043 \f[ 00044 \left(\frac{d n}{d \mu}\right)_{T}, \quad 00045 \left(\frac{d n}{d T}\right)_{\mu}, \quad \mathrm{and} \quad 00046 \left(\frac{d s}{d T}\right)_{\mu} 00047 \f] 00048 respectively. 00049 00050 All other derivatives can be expressed simply in terms of these 00051 three. 00052 00053 \hline 00054 00055 <b>Derivatives wrt to chemical potential and temperature:</b> 00056 00057 There is a Maxwell relation 00058 \f[ 00059 \left(\frac{d s}{d \mu}\right)_T = 00060 \left(\frac{d n}{d T}\right)_{\mu} 00061 \f] 00062 The pressure derivatives are trivial 00063 \f[ 00064 \left(\frac{d P}{d \mu}\right)_{T}=n, \quad 00065 \left(\frac{d P}{d T}\right)_{\mu}=s 00066 \f] 00067 The energy density derivatives are related through the 00068 thermodynamic identity: 00069 \f[ 00070 \left(\frac{d \varepsilon}{d \mu}\right)_{T}= 00071 \mu \left(\frac{d n}{d \mu}\right)_{T}+ 00072 T \left(\frac{d s}{d \mu}\right)_{T} 00073 \f] 00074 \f[ 00075 \left(\frac{d \varepsilon}{d T}\right)_{\mu}= 00076 \mu \left(\frac{d n}{d T}\right)_{\mu}+ 00077 T \left(\frac{d s}{d T}\right)_{\mu} 00078 \f] 00079 00080 \hline 00081 00082 <b>Other derivatives:</b> 00083 00084 Note that the derivative of the entropy with respect to the 00085 temperature above is not the specific heat, \f$ c_V \f$. 00086 The specific heat is 00087 \f[ 00088 C_V = \frac{T}{N} \left( \frac{\partial S}{\partial T} \right)_{V,N} 00089 = \frac{T}{n} \left( \frac{\partial s}{\partial T} \right)_{V,n} 00090 \f] 00091 To compute the specific heat in terms of the derivatives above, 00092 note that the descendants of deriv_part provide all of the 00093 thermodynamic functions in terms of \f$ \mu, V \f$ and \f$ T 00094 \f$, so we have 00095 \f[ 00096 s=s(\mu,V,T) \quad \mathrm{and} \quad n=n(\mu,V,T) \, . 00097 \f] 00098 We can then construct a function 00099 \f[ 00100 s=s[\mu(n,V,T),V,T] 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 Note that this is the specific heat per particle, and 00128 has no units. If specific heat per unit volume is required, 00129 you must multiply by the number density. 00130 00131 No derivative with respect to the bare mass is given, since 00132 classes cannot know how to relate the effective mass to the 00133 bare mass. 00134 00135 */ 00136 class deriv_part { 00137 00138 public: 00139 00140 /// Derivative of number density with respect to chemical potential 00141 double dndmu; 00142 00143 /// Derivative of number density with respect to temperature 00144 double dndT; 00145 00146 /// Derivative of entropy density with respect to temperature 00147 double dsdT; 00148 00149 /// Derivative of number density with respect to the effective mass 00150 double dndm; 00151 00152 }; 00153 00154 #ifndef DOXYGENP 00155 } 00156 #endif 00157 00158 #endif
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