Editing SSC/Virial/PolytropesSummary (section)

====Curiosity====

[[File:PiVersusPa.png|thumb|300px|Pressure vs. pressure plot]]
The figure displayed here, on the right, is a magnification of a segment of the <math>~\Pi(p_a)</math> curve (light blue diamonds) shown in the lefthand panel of the preceding figure, although here we have used a linear, rather than a log, scale on both axes.  The quantity being plotted along both axes is the external pressure, but normalized in different ways.  The quantity, <math>~p_a</math> (horizontal axis), provides a direct measure of the physical external (hence, also, surface) pressure, while the quantity, <math>~\Pi</math> (vertical axis), is the external pressure ''renormalized'' by a specific combination of the free-energy coefficients.  Our stability analysis has been conducted assuming that the free-energy coefficients &#8212; which are expressible in terms of structural form factors &#8212; are constants, that is, they do not vary with the size of the configuration.  Hence, it is the limiting value of <math>~\Pi_\mathrm{ad}</math>, specifically,
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<math>~\Pi_\mathrm{max}\biggr|_{n=4}</math>
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<math>~=</math>
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<math>~\frac{15^{15}}{16^{16}} = 0.02373828 \, ,</math>
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that identifies the demarcation between stable and unstable states.  This limiting value is identified by the horizontal red-dashed line in the figure; and the relevant demarcation point appears where this tangent line touches the curve.  According to our stability analysis, equilibrium configurations to the left of this demarcation point are stable while configurations to the right are unstable.

In the context of our discussion of the lefthand diagram in the preceding figure &#8212; see especially the relevant figure caption &#8212; we claimed that, for each physically allowed value of the external pressure, <math>~p_a</math>, the parameter, <math>~\Pi</math>, was double-valued and that configurations along the ''upper'' segment of its curve were stable.  After studying a magnification of this parameter curve near its turning point, a bit of clarification is required.  It appears as though equilibrium models lying along the short ''upper'' segment of the curve that falls between the demarcation/tangent point at <math>~\Pi_\mathrm{max}</math> and the maximum value of <math>~p_a</math> are unstable.  This means that, even though two equilibrium configurations can be constructed at each value of <math>~p_a</math> in this region near and including the turning point, ''both'' configurations are dynamically unstable.  We conclude, therefore, that stable configurations only exist for values of <math>~p_a</math> that are less than the value associated with <math>~\Pi_\mathrm{max}</math>.