Availability curvature and equilibrium fluctuations ($⟨(x-x_0{)}^2⟩\simeq k_B{T}_R/A^{\prime \prime }(x_0)$)

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 $e^{-A(x)/(k_B{T}_R)}$. Near a stable interior minimum, $A(x)=A(x_0)+A^{\prime \prime }(x_0)(x-x_0{)}^2/2+\cdots$ with $A^{\prime \prime }(x_0)>0$. The resulting normal distribution has variance $k_B{T}_R/A^{\prime \prime }(x_0)$. 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.

For a material parcel of fixed amount inside a much larger thermal and mechanical environment, availability is $A(V)=F(V,T)+p_{\mathrm{bath}}V$. Since $F_V=-p$, its curvature is $A_{VV}=-p_V=1/(V{\kappa }_T)$. The Gaussian fluctuation approximation gives $⟨\delta V^2⟩=k_{B}T{\kappa }_T\bar{V}$. This describes parcel-volume fluctuations, not the geometric volume of a fixed observation box or a globally rigid vessel. A fixed spatial box instead has number fluctuations related to the same compressibility.