Editing PGE/FirstLawOfThermodynamics (section)

==Left-Hand Side (LHS)==
There are several potentially useful expressions for the time-rate of change of fluid entropy.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho T \frac{ds_\mathrm{fluid}}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho\frac{d\epsilon}{dt} + \rho P \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho\frac{d\epsilon}{dt} + P\nabla\cdot \vec{v}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\gamma_g \tau^{\gamma_g-1}\biggl[ \frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho\epsilon \biggl[
\frac{d\ln(\tau/\rho)^{\gamma_g}}{dt}
\biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{P}{(\gamma_g - 1)} ~\frac{d}{dt}\biggl[
\ln\biggl( \frac{P}{\rho^{\gamma_g}}\biggr)
\biggr] \, .
</math>
  </td>
</tr>
</table>

Clearly to within an additive constant, an expression for the fluid entropy, itself, is

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~s</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ c_P \ln\biggl( \frac{\tau}{\rho} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_V \ln \biggl(\frac{P}{\rho^{\gamma_g}} \biggr) \, .</math>
  </td>
</tr>
</table>

In [[Appendix/Ramblings/Radiation/RadHydro#Optically_Thick_Regime|optically thick environments]] where the radiation field is intermixed and in equilibrium with the fluid (gas), the time-rate-of-change in the entropy of the radiation field is characterized by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho T \frac{ds_\mathrm{rad}}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho\frac{d}{dt}\biggl(\frac{E_\mathrm{rad}}{\rho}\biggr) + \rho P_\mathrm{rad} \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4a_\mathrm{rad} \rho T}{3} \frac{d}{dt} \biggl( \frac{T^3}{\rho}\biggr)
</math>
  </td>
</tr>
</table>

Hence, to within an additive constant, an expression for the entropy of the radiation field is,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~s_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{4 a_\mathrm{rad} T^3}{3\rho} \, .</math>
  </td>
</table>
In a [[Apps/SMS#Entropy_of_Radiation_Field|separate discussion]] we identify other references where this expression for <math>s_\mathrm{rad}</math> can be found.