Editing PGE/FirstLawOfThermodynamics (section)

===Example C===
In astrophysical discussions of the time-rate-of-change of the fluid entropy, it is not unusual to include a scalar function, <math>\Gamma</math>, that accounts in a generic manner for ''volumetric gains'' of energy due to local sources, and another scalar function, <math>\Lambda</math>, that accounts in a generic manner for ''volumetric loses'' of energy due to local sinks.  In place of the above "Example A" right-hand-side expression, then, we would expect to see,
<div align="center">
<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>~
- \nabla\cdot \vec{F}_\mathrm{cond} + \Psi + \Gamma - \Lambda \, .
</math>
  </td>
</tr>
</table>
</div>

When, for example, the fluid (gas) is exposed to photon radiation, heating of the fluid by the radiation is handled by setting,
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<math>~\Gamma = c\kappa_E E_\mathrm{rad} \, ,</math>
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and the fluid cools &#8212; returning energy to the radiation field &#8212; according to the reciprocating expression,
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<math>~\Lambda = 4\pi \kappa_p B_p = 4 \kappa_p \sigma T^4 \, ,</math>
</div>
where, <math>\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math> is the Stefan-Boltzmann constant.  In such a case, the right-hand-side of the equation describing the corresponding time-rate-of-change of the entropy of the ''radiation'' field, <math>s_\mathrm{rad}</math>, would necessarily contain the same two terms, but in both cases with ''opposite signs.''  That is, the entropy of the radiation field sees <math>\Lambda</math> as a ''source'' while it sees <math>\Gamma</math> as a "sink."

[In addition to <math>~\Gamma</math> and <math>~\Lambda</math>, other terms involving spatial variations in the velocity field and in the radiation energy density also appear on the right-hand-side of the expression for <math>~ds_\mathrm{rad}/dt</math>.  For simplicity, and because these other terms are not relevant to the principal point we are making, we have opted not to detail the entire expression for <math>~ds_\mathrm{rad}/dt</math> here.  The additional terms and details can be found in, for example, {{ ZEUS-MP2006 }} or {{ MT2012 }}.]

====First Elaboration====

When the expressions for <math>ds_\mathrm{fluid}/dt</math> and <math>ds_\mathrm{rad}/dt</math> are added together to obtain a prescription for the time-rate-of-change of <math>s_\mathrm{tot}</math> &#8212; see, for example, "Example B" above &#8212; neither of the functions, <math>\Gamma</math> or <math>\Lambda</math>, will appear explicitly because they have opposite signs in the two separate expressions.  This will be the case whether the environment is ''optically thin'' or ''optically thick.''

====Second Elaboration====
In an ''optically thick'' environment where local thermodynamic equilibrium has been achieved, <math>E_\mathrm{rad} = a_\mathrm{rad}T^4</math>, so,
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<math>~\Gamma = c\kappa_E a_\mathrm{rad}T^4 = \biggl( \frac{\kappa_E}{\kappa_p} \biggr) \Lambda \, .</math>
</div>
In such an environment, we also expect <math>~\kappa_E \leftrightarrow \kappa_p</math>, so the heating and cooling terms will cancel out each other.  As a result, the quantity <math>~(\Lambda - \Gamma) </math> will disappear from the ''separate'' expressions for <math>~ds_\mathrm{fluid}/dt</math> and <math>~ds_\mathrm{rad}/dt</math>.


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