Editing SSC/FreeEnergy/PolytropesEmbedded/Pt3C (section)

====Graphical Depiction of Free-Energy Surface====

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Figure 1: &nbsp; Free-Energy Surface for <math>~(n_c,n_e) = (5,1)</math> and <math>~\frac{\mu_e}{\mu_c} = 1</math></b></font></td></tr>
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[[File:FreeEnergy51Surface2.png|center|300px|Free-Energy surface for 5_1 bipolytrope]]
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[[File:Bipolytrope51Muratio1.gif|center|439px|Free-Energy surface for 5_1 bipolytrope]]
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  <td align="left" colspan="2">
''Left Panel:''  The free energy (vertical, blue axis) is plotted as a function of the radial interface location, <math>~\xi_i</math> (red axis), and the normalized configuration radius, <math>~\Chi \equiv \chi/\chi_\mathrm{eq}</math> (green axis).  ''Right Panel:'' Same as the left panel, but animated in order to highlight undulations of the surface.  The value of the free energy is indicated by color as well as by the height of the warped surface &#8212; red identifies the lowest depicted energies while blue identifies the highest depicted energies; these same colors have been projected down onto the <math>~z = 0</math> plane to present a two-dimensional, color-contour plot.  A multi-colored line segment drawn parallel to the <math>~\xi_i</math> axis at the value, <math>~\Chi = 1</math>, identifies the configuration's ''equilibrium'' radius for each value of the interface location.  Equilibrium configurations marked in white lie at the bottom of the principal free-energy "valley" and are stable, while configurations marked in blue lie at the top of a free-energy "hill," indicating that they are unstable; the red dot identifies the location along this equilibrium sequence where the transition from stable to unstable configurations occurs.
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For purposes of reproducibility, it is incumbent upon us to clarify how the values of the free energy were normalized in order to produce the free-energy surface displayed in Figure 1.  The normalization steps are explicitly detailed within the [[#Fortran_Code|fortran program, below]], that generated the data for plotting purposes; here we provide a brief summary.  We evaluated the normalized free energy, <math>~\mathfrak{G}^*_{51}</math>, across a <math>~200 \times 200</math> zone grid of uniform spacing, covering the following <math>~(x,y) = (\ell_i,\Chi)</math> domain:
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<math>~\frac{1}{\sqrt{3}}</math>
  </td>
  <td align="center">
<math>~\le \ell_i \le</math>
  </td>
  <td align="left">
<math>~\frac{3}{\sqrt{3}}</math>
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  <td align="right">
<math>~0.469230769</math>
  </td>
  <td align="center">
<math>~\le \Chi\le</math>
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  <td align="left">
<math>~2.0</math>
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(With this specific definition of the ''y''-coordinate grid, <math>~\Chi = 1</math> is associated with zone 70.)  After this evaluation, a constant, <math>~E_\mathrm{fudge} = -10</math>, was added to <math>~\mathfrak{G}^*</math> in order to ensure that the free energy was negative across the entire domain.  Then (inorm = 1), for each specified interface location, <math>~x = \ell_i</math>, employing the ''equilibrium'' value of the free energy,
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<math>~E_0 = \mathfrak{G}^*_{51}(\ell_i, \Chi = 1) + E_\mathrm{fudge} \, ,</math>
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the free energy was normalized across all values of <math>~y = \Chi</math> via the expression,
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<math>~\mathrm{fe} = \frac{(\mathfrak{G}^*_{51} + E_\mathrm{fudge}) - (E_0)_i}{|E_0|_i} \, .</math>
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Finally, for plotting purposes, at each grid cell vertex ("vertex") &#8212; as well as at each grid cell center ("cell") &#8212; the value of the free energy, <math>~\mathrm{fe}</math>, was renormalized as follows,
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<math>~\mathrm{vertex} =  \frac{\mathrm{fe} - \mathrm{min}(\mathrm{fe})}{ \mathrm{max}(\mathrm{fe}) - \mathrm{min}(\mathrm{fe})} \, .</math>
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Via this last step, the minimum "vertex" energy across the entire domain was 0.0 while the maximum "vertex" energy was 1.0.


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<tr>
  <th align="center" colspan="3">FORTRAN Program Documentation</th>
  <th align="center" colspan="3">Example Evaluations<p></p>(See also [[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|associated Table 1]])</th>
</tr>
<tr>
  <th align="center" colspan="1">Coord. Axis</th>
  <th align="center" colspan="1">Coord. Name</th>
  <th align="left">Associated Physical Quantity</th>
  <td align="center"><math>~\frac{\mu_e}{\mu_c} = 1</math></td>
  <td align="center" colspan="2"><math>~\frac{\mu_e}{\mu_c} = 0.305</math></td>
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<tr>
  <td align="center">x-axis</td>
  <td align="center">bsize</td>
  <td align="left"><math>~\ell_i \equiv \frac{\xi_i}{\sqrt{3}}</math></td>
  <td align="center"><math>~\frac{2.416}{\sqrt{3}} = 1.395</math></td>
  <td align="center"><math>~\frac{8.1938}{\sqrt{3}} = 4.7307</math></td>
  <td align="center"><math>~\frac{14.389}{\sqrt{3}} = 8.3076</math></td>
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  <td align="center">y-axis</td>
  <td align="center">csize</td>
  <td align="left"><math>~\Chi \equiv \frac{\chi}{\chi_\mathrm{eq}}</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~1</math></td>
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  <th align="center" colspan="3">Relevant Lines of Code</th>
  <td colspan="3" rowspan="3">&nbsp;</td>
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  <td align="left" colspan="3">
<pre>
      eta = 3.0d0*muratio*bsize/(1.0d0+bsize**2)
      Gami = 1.0d0/eta-bsize
      frakL = (bsize**4-1.0d0)/bsize**2 + &
     &        DATAN(bsize)*((1.0d0+bsize**2)/bsize)**3
      frakK = Gami/eta + ((1.0d0+Gami**2)/eta)*(pii/2.0d0+DATAN(Gami))
      E0    = ((5.0d0*frakL) + (4.0d0*frakK)&
     &        - (3.0d0*frakL+12.0d0*frakK))/bsize**2+Efudge
          fescalar(j,k) = (csize**(-0.6d0)*(5.0d0*frakL)&
     &        + csize**(-3.0d0)*(4.0d0*frakK)&
     &   - (3.0d0*frakL+12.0d0*frakK)/csize)/bsize**2 + Efudge
          if(inorm.eq.1)fescalar(j,k)=fescalar(j,k)/DABS(E0) &
     &                  - E0/DABS(E0)
</pre>
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<tr>
  <th align="center" colspan="1">Variable</th>
  <th align="center" colspan="1">Represents</th>
  <th align="left">Value calculated via the expression &hellip;</th>
</tr>
<tr>
  <td align="center">eta</td>
  <td align="center"><math>~\eta_i</math></td>
  <td align="left">
<math>~3 \biggl(\frac{\mu_e}{\mu_c}\biggr)\biggl[ \frac{\ell_i}{(1+\ell_i^2)} \biggr]</math>
  </td>
  <td align="center"><math>~1.421</math></td>
  <td align="center"><math>~0.1851</math></td>
  <td align="center"><math>~0.1086</math></td>
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  <td align="center">Gami</td>
  <td align="center"><math>~\Lambda_i</math></td>
  <td align="left"><math>~\frac{1}{\eta_i} - \ell_i</math></td>
  <td align="center"><math>~-0.691</math></td>
  <td align="center"><math>~0.6705</math></td>
  <td align="center"><math>~0.9033</math></td>
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  <td align="center">frakL</td>
  <td align="center"><math>~\mathfrak{L}_i</math></td>
  <td align="left"><math>~
\frac{(\ell_i^4 - 1)}{\ell_i^2} + \biggl[ \frac{1+\ell_i^2}{\ell_i} \biggr]^3 \tan^{-1}\ell_i
</math></td>
  <td align="center"><math>~10.37</math></td>
  <td align="center"><math>~186.80</math></td>
  <td align="center"><math>~937.64</math></td>
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  <td align="center">frakK</td>
  <td align="center"><math>~\mathfrak{K}_i</math></td>
  <td align="left"><math>~
\frac{\Lambda_i}{\eta_i} + \frac{(1 + \Lambda_i^2)}{\eta_i} \biggl[ \frac{\pi}{2} + \tan^{-1}\Lambda_i \biggr]
</math></td>
  <td align="center"><math>~0.518</math></td>
  <td align="center"><math>~20.544</math></td>
  <td align="center"><math>~46.882</math></td>
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  <td align="center">&nbsp;</td>
  <td align="center"><math>~\frac{\mathfrak{L}_i}{\mathfrak{K}_i}</math></td>
  <td align="left">&nbsp;</td>
  <td align="center"><math>~20</math></td>
  <td align="center"><math>~9.093</math></td>
  <td align="center"><math>~20</math></td>
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<!-- OMIT
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  <td align="center">fescalar(x,y) - Efudge</td>
  <td align="center"><math>~\mathfrak{G}^*_{51}(\ell_i,\Chi)</math></td>
  <td align="left">
<math>~ \frac{1}{\ell_i^2} \biggl[
\Chi^{-3/5} (5 \mathfrak{L}_i)
+\Chi^{-3} (4\mathfrak{K}_i)
-\Chi^{-1} (3\mathfrak{L}_i  +12\mathfrak{K}_i ) \biggr]
</math>
</td>
  <td align="center"><math>~8.525</math></td>
  <td align="center">&hellip;</td>
  <td align="center">&hellip;</td>
</tr>
END OMIT-->
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  <td align="center">E0 - Efudge</td>
  <td align="center"><math>~\mathfrak{G}^*_{51}(\ell_i,\Chi=1)</math></td>
  <td align="left">
<math>~ 
\frac{1}{\ell_i^2} \biggl[ 5 \mathfrak{L}_i + 4\mathfrak{K}_i -  (3\mathfrak{L}_i  +12\mathfrak{K}_i ) \biggr]
= \frac{2(\mathfrak{L}_i - 4\mathfrak{K}_i)}{\ell_i^2} 
</math>
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  <td align="center"><math>~8.525</math></td>
  <td align="center"><math>~9.3496</math></td>
  <td align="center"><math>~21.737</math></td>
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<table border="0" cellpadding="8" width="880px">
<tr><td align="center"><font size="+1"><b>
Figure 2: &nbsp; Free-Energy Surface for <math>~(n_c,n_e) = (5,1)</math> and <math>~\frac{\mu_e}{\mu_c} = 0.305</math></b></font></td></tr>
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<td align="center" bgcolor="#CCFFFF">
[[File:FEsurface01.png|center|400px|Free-Energy surface for 5_1 bipolytrope]]
</td>
</tr>
<tr>
  <td align="left" colspan="1">
''Left Panel:''  The free energy (vertical, blue axis) is plotted as a function of the radial interface location, <math>~\xi_i</math> (red axis), and the normalized configuration radius, <math>~\Chi \equiv \chi/\chi_\mathrm{eq}</math> (green axis).  ''Right Panel:'' Same as the left panel, but animated in order to highlight undulations of the surface.  The value of the free energy is indicated by color as well as by the height of the warped surface &#8212; red identifies the lowest depicted energies while blue identifies the highest depicted energies; these same colors have been projected down onto the <math>~z = 0</math> plane to present a two-dimensional, color-contour plot.  A multi-colored line segment drawn parallel to the <math>~\xi_i</math> axis at the value, <math>~\Chi = 1</math>, identifies the configuration's ''equilibrium'' radius for each value of the interface location.  Equilibrium configurations marked in white lie at the bottom of the principal free-energy "valley" and are stable, while configurations marked in blue lie at the top of a free-energy "hill," indicating that they are unstable; the red dot identifies the location along this equilibrium sequence where the transition from stable to unstable configurations occurs.
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