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.