User:Tohline/StabilityVariationalPrincipal

Free-Energy Stability Analysis

Most General Case

Consider a free-energy function of the form,

<math>~\mathcal{G}</math>

<math>~=</math>

<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{-3/j} + \mathcal{G}_0 \, ,</math>

where, <math>~a, b, c,</math> and <math>~\mathcal{G}_0</math> are constants, and the dimensionless configuration radius,

<math>~\chi \equiv \frac{R}{R_0} \, ,</math>

is defined in terms of a characteristic length, <math>~R_0</math>, which is likely to be different for each type of problem.

Virial Equilibrium

The first variation (first derivative) of this function with respect to the configuration's radius is,

<math>~\frac{d\mathcal{G}}{d\chi}</math>

<math>~=</math>

<math>~a\chi^{-2} - \biggl(\frac{3b}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 c}{j}\biggr) \chi^{-3/j -1} \, .</math>

According to the virial theorem, the radius of an equilibrium configuration is obtained by setting <math>~d\mathcal{G}/d\chi = 0</math> and identifying the roots of the resulting equation. For example, identifying roots of the polynomial expression,

<math>~0</math>

<math>~=</math>

<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} - \biggl(\frac{1}{j}\biggr) \chi_\mathrm{eq}^{(j-3)/j } \, .</math>

Stability

Let's rewrite the first variation of the free-energy function in terms of three coefficients <math>~(e,f,g)</math> which, in general, we will permit to have different values from the original three <math>~(a,b,c)</math>,

<math>~\mathcal{G}^'</math>

<math>~=</math>

<math>~e\chi^{-2} - \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-1} - \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -1} \, .</math>

The first variation (first derivative) of this function with respect to the configuration's radius — which, in effect, represents the second variation of the free-energy function — gives,

<math>~\frac{d\mathcal{G}^'}{d\chi}</math>

<math>~=</math>

<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + \biggl(\frac{3}{j} + 1\biggr) \biggl(\frac{3 g}{j}\biggr) \chi^{-3/j -2} \, .</math>

If we evaluate this function by setting <math>~\chi = \chi_\mathrm{eq}</math>, the sign of the resulting expression should indicate stability (positive) or dynamical instability (negative); and the marginally unstable configuration is identified by the value of <math>~\chi_\mathrm{eq}</math> for which <math>~d\mathcal{G}^'/d\chi = 0</math>.

Pressure-Truncated Configurations

For pressure-truncated polytropes, we set <math>~j = -1</math> and let <math>~n</math> represent the chosen polytropic index. In this situation, then, we have,

Free-energy expression:		<math>~\mathcal{G}</math>	<math>~=</math>	<math>~- a\chi^{-1} + b \chi^{-3/n} + c \chi^{3} + \mathcal{G}_0 \, ;</math>
Virial equlibrium:		<math>~0</math>	<math>~=</math>	<math>~\frac{a}{3c} - \biggl(\frac{b}{nc}\biggr) \chi_\mathrm{eq}^{(n-3)/n} + \chi_\mathrm{eq}^{4 } \, ;</math>
Stability indicator:		<math>~\frac{d\mathcal{G}^'}{d\chi}</math>	<math>~=</math>	<math>~-2e\chi^{-3} + \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi^{-3/n-2} + 6g \chi \, .</math>

Hence, the (critical) equilibrium radius of the marginally unstable configuration is given by the expression,

<math>~6g \chi_\mathrm{eq}^4 </math>	<math>~=</math>	<math>~2e - \biggl(\frac{3}{n} + 1\biggr) \biggl(\frac{3f}{n}\biggr) \chi_\mathrm{eq}^{(n-3)/n}</math>
	<math>~=</math>	<math>~2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} + \chi_\mathrm{eq}^4 \biggr]</math>
<math>~\Rightarrow ~~~ 6g \chi_\mathrm{eq}^4 +\biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\chi_\mathrm{eq}^4 </math>	<math>~=</math>	<math>~ 2e - \biggl[\frac{3f(n+3)}{n^2} \biggr] \biggl(\frac{nc}{b} \biggr)\biggl[\frac{a}{3c} \biggr] </math>
<math>~\Rightarrow ~~~ \biggl[6g + \frac{3cf(n+3)}{nb} \biggr]\chi_\mathrm{eq}^4 </math>	<math>~=</math>	<math>~ 2e - \biggl[\frac{af(n+3)}{nb} \biggr] </math>
<math>~\Rightarrow ~~~ \chi_\mathrm{eq}^4\biggr|_\mathrm{crit} </math>	<math>~=</math>	<math>~ \biggl[\frac{2nbe -af(n+3)}{6nbg +3cf(n+3)} \biggr] \, . </math>

Notice that, if <math>~(e,f,g) \rightarrow (a,b,c)</math>, this gives,

<math>~ \chi_\mathrm{eq}^4\biggr|_\mathrm{crit} </math>	<math>~=</math>	<math>~ \biggl[\frac{2nba -ab(n+3)}{6nbc +3cb(n+3)} \biggr] </math>
	<math>~=</math>	<math>~ \frac{a}{3^2c}\biggl[\frac{n-3}{n+1} \biggr] \, . </math>

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation