= 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 is
$$
\Delta F=-\frac1\beta\log\left\langle e^{-\beta V}\right\rangle_0,\qquad\beta=\frac1{k_BT}.
$$
Here $T$ is <temperature>, $k_B$ is the <Boltzmann constant> and $\langle\cdot\rangle_0$ averages only the fast modes. Expanding the logarithm gives
$$
\Delta F=\langle V\rangle_0-\frac\beta2\left(\langle V^2\rangle_0-\langle V\rangle_0^2\right)+\frac{\beta^2}{6}\langle(V-\langle V\rangle_0)^3\rangle_0+\cdots.
$$
Every 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>. <Statistical Field Theory by David Tong>[Chapter 2, opening discussion: physical thermodynamic free energy versus the dimensionless logarithm of the partition function]{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.
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