Cumulant expansion of a coarse-grained free energy

A cumulant expansion expresses the free energy obtained by integrating out fast Fourier modes in terms of their connected fluctuations. Write the physical free energy as $F_0+V$, where the quadratic reference part defines a normalized Gaussian measure for the fast modes. At fixed slow field, the interaction contribution isHere $T$ is temperature, $k_B$ is the Boltzmann constant and $⟨\cdot {⟩}_0$ averages only the fast modes. Expanding the logarithm givesEvery term has units of energy. Equivalently, define the dimensionless interaction $\mathcal{V}=\beta V$ and dimensionless free energy $\mathcal{F}=\beta F$; the expansion then has coefficients $1,-1/2,1/6$ without explicit $\beta$ factors. This is the convention commonly used in statistical field theory. ^[ref]

The subtraction in the second cumulant removes products of disconnected averages. For a Gaussian measure, Wick theorem turns the remaining terms into connected Feynman diagrams. Truncating this formal expansion is a perturbation theory approximation, whose accuracy depends on the interaction and the fluctuation scales.