= Volume fluctuations of a closed gas bubble
{title2=$\langle\Delta V^2\rangle\simeq k_BTV/[\kappa_T^{-1}-(2/9)\Gamma\alpha V^{-1/3}]$}
For fixed <particle number> and <temperature>, a bubble with <surface tension> $\Gamma$ and <area> $\alpha V^{2/3}$ has effective <availability> $A(V)=F_{\mathrm{gas}}(T,V,N)+\Gamma\alpha V^{2/3}+p_RV$. Its stationary condition is $p_{\mathrm{gas}}-p_R=(2/3)\Gamma\alpha V^{-1/3}$. Its curvature is $A''=V^{-1}[\kappa_T^{-1}-(2/9)\Gamma\alpha V^{-1/3}]$. Positive curvature is required for stable <Gaussian fluctuation approximation>. The <variance> follows from <availability curvature and equilibrium fluctuations>. For an <ideal gas>, the restoring denominator becomes $p_R+(4/9)\Gamma\alpha V^{-1/3}>0$ for positive $p_R$. Treating the internal <pressure> as externally fixed instead of fixing the gas <particle number> changes the stability problem.
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