Editing Apps/SMS (section)

==Entropy of Radiation Field==
===Planck's Law of Black-Body Radiation===
Drawing from the [https://en.wikipedia.org/wiki/Black-body_radiation#Equations Wikipedia discussion of black-body radiation], Planck's law states that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>B_\nu(T)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/kT} - 1} \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
[[Appendix/References#H87|[<b><font color="red">H87</font></b>]]], &sect;12.1 (p. 280), Eq. (12.14)<br />
[[Appendix/References#KW94|[<b><font color="red">KW94</font></b>]]], &sect;5.1 (p. 30), Eq. (5.15)<br />
[[Appendix/References#HK94|[<b><font color="red">HK94</font></b>]]], &sect;4.1 (p. 152), Eq. (4.7)<br />
[[Appendix/References#BLRY07|[<b><font color="red">BLRY07</font></b>]]], &sect;1.6.2 (p. 23), Eq. (1.103)
</td></tr>
</table>
Letting,
<div align="center"><math>x \equiv \frac{h\nu}{kT} \, ,</math></div>
and integrating over the entire frequency spectrum, we have,

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

<tr>
  <td align="right">
<math>\int_0^\infty B_\nu(T)d\nu</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{2h}{c^2}\biggl(\frac{kT}{h} \biggr)^4 \int_0^\infty \frac{x^3 dx}{e^{x} - 1} 
=
\frac{2k^4}{h^3c^2} \biggl(\frac{\pi^4}{15} \biggr)T^4
= \frac{\sigma_\mathrm{SB} T^4}{\pi} \, ,
</math>
  </td>
</tr>
</table>
where [with units of mass &times; (time)<sup>-3</sup> K<sup>-4</sup>],

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

<tr>
  <td align="right">
<math>\sigma_\mathrm{SB} </math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math> 
\frac{2\pi^5}{15} \frac{k^4}{h^3c^2} \, .
</math>
  </td>
</tr>
</table>
In discussions of the thermodynamic properties of (photon) radiation fields, the radiation [density] constant, {{ Template:Math/C_RadiationConstant }}, appears in many physical relations.  It is related to the Steffan-Boltzmann constant via the simple expression,
<div align="center"><math>a_\mathrm{rad} = \frac{4 \sigma_\mathrm{SB}}{c} \, ,</math></div>
and has units of energy per unit volume &times; K<sup>-4</sup>.

===Photon Entropy===
Here are expressions from the published literature that &#8212; to within an additive constant &#8212; give the entropy of the (photon) radiation field.  We begin with [<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], who states that, "&hellip; the entropy per gram of the photon [field]" is,
<div align="center">
<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{4a_\mathrm{rad}T^3}{3\rho} \, . 
</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#Clayton68|<font color="red">Clayton68</font>]]</b>], Chapter 2 (p. 121), Eq. (2-136)<br />
[[Appendix/References#Shu92|[<b><font color="red">Shu92</font></b>]]], Vol. I, Chapter 9 (p. 82), Eq. (9.22)
</div>
Providing the same expression, [[Appendix/References#Shu92|[<b><font color="red">Shu92</font></b>]]] states that it  is a measure of "&hellip; the entropy of blackbody radiation per unit mass of fluid."  From the group of terms on the right-hand side of this expression, we conclude that <math>s_\mathrm{rad}</math> has units of specific energy per Kelvin.  This is consistent with the units that are traditionally associated with entropy; see, for example, our [[PGE/FirstLawOfThermodynamics#VariableDimensions|separate discussion of the dimensions of various thermodynamic variables]].

According to [[Appendix/References#ST83|[<b><font color="red">ST83</font></b>]]], "&hellip; the photon entropy per baryon" is,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>s_r = \frac{4}{3} \frac{a_\mathrm{rad}T^3}{n}</math>
  </td>
  <td align="center">=</td>
  <td align="left">
<math>\frac{4m_H a_\mathrm{rad} T^3}{3\rho} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[[Appendix/References#ST83|[<b><font color="red">ST83</font></b>]]], &sect;17.2, Eq. (17.2.2) &amp;  &sect;17.3, Eq. (17.3.7)
  </td>
</tr>
</table>
where, <span title="Hydrogen mass"><math>m_H</math></span> is the mass of a hydrogen atom, and <math>n = \rho/m_H</math> is the baryon number density.  We conclude, therefore, that <math>s_\mathrm{rad} = s_r/m_H</math>.

We have also found it useful to examine the expression for the entropy of a radiation field used by {{ FWW86full }} &#8212; hereafter {{ FWW86hereafter }} &#8212; primarily because the authors provide, for comparison, a numerical evaluation of the expression's leading coefficient.

<table border="1" width="80%" align="center" cellpadding="8">
<tr><td align="left">
In terms of the variables, {{ Template:Math/VAR_Temperature01 }} and {{ Template:Math/VAR_Density01 }}, and the four physical constants, {{ Template:Math/C_PlanckConstant }}, {{ Template:Math/C_SpeedOfLight }}, {{ Template:Math/C_BoltzmannConstant }}, {{ Template:Math/C_AvogadroConstant }}, {{ FWW86hereafter }} state that the "&hellip; entropy per baryon &hellip; for photons is,"
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>s_\gamma</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{4\pi^2}{45}\biggl[ \frac{kT }{\hbar c}\biggr]^3 \frac{1}{\rho N_A}. 
</math>
  </td>
</tr>
<tr>
<td align="center" colspan="3">
{{ FWW86hereafter }} &sect;III (p. 678), Eq. (11)
</td>
</tr>
</table>

From the numerical values of the physical constants &#8212; which can be obtained by scrolling your cursor over each representative letter in our text (see also our [[Appendix/VariablesTemplates#Physical_Constants|accompanying table of physical constants]]) &#8212; and acknowledging that <math>\hbar \equiv h/(2\pi)</math>, we deduce that, to six significant digits,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>hc</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
1.986446 \times 10^{-16} ~\mathrm{erg} \cdot \mathrm{cm}
=
1.239841 \times 10^{-10} ~\mathrm{MeV} \cdot \mathrm{cm}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~s_\gamma</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl[ \frac{(2\pi)^5}{45 N_A(h c)^3}\biggr] \frac{(kT)^3}{\rho} 
= (4.610053 \times 10^{25}~\mathrm{erg}^{-3} \cdot \mathrm{cm}^{-3}) \frac{(kT)^3}{\rho}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
(1.895998 \times 10^{8}~\mathrm{MeV}^{-3} \cdot \mathrm{cm}^{-3}) \frac{(kT)^3}{\rho} \,.
</math>
  </td>
</tr>
</table>
This is consistent with the statement in {{ FWW86hereafter }} that, when energy is measured in MeV, the leading coefficient is, <math>1.905 \times 10^8</math>.

Notice that, because the quantity, {{ Template:Math/C_BoltzmannConstant }}{{ Template:Math/VAR_Temperature01 }}, has units of energy, the dimension of <math>s_\gamma</math> is inverse mass.   In contrast to this, as has been highlighted in our [[PGE/FirstLawOfThermodynamics#VariableDimensions|separate discussion of the physical units of various thermodynamic variables]], we expect the units of specific entropy <math>(s_\mathrm{rad})</math> to be specific energy per Kelvin.  We suspect that, for convenience, {{ FWW86hereafter }} have dropped a leading factor of {{ Template:Math/C_BoltzmannConstant }} on the RHS of their expression for <math>s_\gamma</math> and that it is appropriate to draw from their presentation that the specific 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> 
k s_\gamma \, . 
</math>
  </td>
</tr>
</table>
</td></tr>
</table>


[[Image:CommentButton02.png|right|100px|Note that in the {{ BAC84hereafter }} publication, the Boltzmann constant, k, does not appear in this expression because "T is the temperature in energy units (Boltzmann constant = 1)."]]Similarly, from {{ BAC84hereafter }} we find that "&hellip; the photon entropy [is],"

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

<tr>
  <td align="right">
<math>s_\gamma</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{4}{3} \biggl[ \frac{\pi^2 }{15(\hbar c)^3}\biggr] \frac{(kT)^3}{n_B} \, , 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">{{ BAC84hereafter }}, &sect;2b (p. 827), Eq. (5)</td>
</tr>
</table>
where <math>n_B</math> is the "baryon density."  


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