Thermal fluctuation ($⟨(\delta A{)}^2⟩=k_{B}T{\partial }_h⟨A⟩$)

A thermal fluctuation is the randomness of an observable in an equilibrium Boltzmann distribution. If the energy contains $-hA$, differentiation of its partition function gives ${\partial }_h⟨A⟩=\beta (⟨A^2⟩-⟨A{⟩}^2)$. Thus the fluctuation variance is related to the response to a conjugate source. In a field theory, the two-point version is the connected correlation function.

For fixed particle number and temperature, a bubble with surface tension $\Gamma$ and area $\alpha V^{2/3}$ has effective availability $A(V)=F_{\mathrm{gas}}(T,V,N)+\Gamma \alpha V^{2/3}+p_{R}V$. Its stationary condition is $p_{\mathrm{gas}}-p_R=(2/3)\Gamma \alpha V^{-1/3}$. Its curvature is $A^{\prime \prime }=V^{-1}[{\kappa }_T^{-1}-(2/9)\Gamma \alpha V^{-1/3}]$. 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.