= Availability curvature and equilibrium fluctuations
{title2=$\langle(x-x_0)^2\rangle\simeq k_BT_R/A''(x_0)$}
After integrating out other variables, let $A(x)$ include the constrained <entropy> and the <measure> appropriate to the chosen coordinate $x$. Its equilibrium <probability density function> is proportional to $e^{-A(x)/(k_BT_R)}$. Near a stable interior minimum, $A(x)=A(x_0)+A''(x_0)(x-x_0)^2/2+\cdots$ with $A''(x_0)>0$. The resulting <normal distribution> has <variance> $k_BT_R/A''(x_0)$. This is the leading <Gaussian fluctuation approximation>; a coordinate-dependent <measure> contributes to the effective $A$, and higher-order terms matter near a vanishing curvature.
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