Editing SSC/FreeEnergy/PolytropesEmbedded/Pt3A (section)

====Out of Equilibrium====
And now, in combination ''out'' of equilibrium,
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<math>~\frac{\mathfrak{G}}{E_\mathrm{norm}} </math>
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<math>~=</math>
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<math>~ 
\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-1} \biggl\{ \biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}  
+ \biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}\biggr\}
+\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3/5} \biggl(\frac{2n_c}{3}\biggr) \biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}
+\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3} \biggl(\frac{2n_e}{3}\biggr)\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}
\, .
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Hence, quite generally ''out'' of equilibrium,
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<math>~\frac{\partial}{\partial \chi} \biggl[ \frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr] </math>
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<math>~=</math>
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<math>~ 
-\chi^{-1}\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-1} \biggl\{ \biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}  
+ \biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}\biggr\}
-\frac{3}{5}\chi^{-1}\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3/5} \biggl(\frac{10}{3}\biggr) \biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}
-3\chi^{-1}\biggl(\frac{\chi}{\chi_\mathrm{eq}}\biggr)^{-3} \biggl(\frac{2}{3}\biggr)\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} \, .
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Let's see what the value of this derivative is if the dimensionless radius, <math>~\chi</math>, is set to the value that has been determined, via a detailed force-balanced analysis, to be the equilibrium radius, namely, <math>~\chi = \chi_\mathrm{eq}</math>.  In this case, we have,
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<math>~\biggl\{\frac{\partial}{\partial \chi} \biggl[ \frac{\mathfrak{G}}{E_\mathrm{norm}} \biggr] \biggr\}_\mathrm{\chi \rightarrow \chi_\mathrm{eq}}</math>
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<math>~=</math>
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<math>~ -\chi_\mathrm{eq}^{-1}\biggl\{
\biggl[ \frac{W_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}  
+ \biggl[\frac{W_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}
+2\biggl[ \frac{S_\mathrm{core}}{E_\mathrm{norm}}\biggr]_\mathrm{eq}
+2\biggl[\frac{S_\mathrm{env}}{E_\mathrm{norm}}\biggr]_\mathrm{eq} \biggr\} \, .
</math>
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But, according to the virial theorem &#8212; and, as we have just demonstrated &#8212; the four terms inside the curly braces sum to zero.  So this demonstrates that the derivative of our out-of-equilibrium free-energy expression does go to zero at the equilibrium radius, as it should!