[Mayer expansions and the Hamilton–Jacobi equation](https://link.springer.com/article/10.1007/BF01010398) by D. Brydges and T. Kennedy begins mentioning that many problems in statistical mechanics and QFT center on the analysis of integrals of the form:
\begin{equation}
\int d\mu\_{C}(\varphi)e^{-V(\varphi+\varphi')} \tag{1} \label{1}
\end{equation}
where $d\mu\_{C}$ is a Gaussian measure with mean zero and covariance $C$. I'm primarily interested in statistical mechanics so I'd like to understand these integrals in this context. I don't know much about QFT but (correct me if I'm wrong) a common object of study which satisfies (\ref{1}) is the so-called effective action $\mathcal{G}\_{\lambda, C}$. But this effective action is also useful in statistical mechanics. To be more specific, we define it as:
\begin{equation}
\mathcal{G}\_{\lambda,C}(\psi) \mathrel{:=} \log \int d\mu\_{C}(\varphi)e^{-\lambda V(\psi+\varphi)} \tag{2}\label{2}
\end{equation}
where $V$ is an interaction function (I'm assuming here that (\ref{2}) is well-defined). Note that (\ref{2}) has the exact same form of (\ref{1}). Now, let us consider:
\begin{equation}
Z\_{\lambda}(\psi) \mathrel{:=} \int d\mu\_{C}(\varphi) e^{-\lambda V(\varphi)+\langle C^{-1}\psi, \varphi\rangle} \tag{3}\label{3}
\end{equation}
where $\langle \cdot, \cdot \rangle$ is a given inner product. This is called the partition function of the system with source term $C^{-1}\psi$. The partition function (and also its logarithm) is a powerful tool to calculate correlation functions. With a little algebra, we can prove that:
\begin{equation}
\mathcal{G}\_{\lambda,C}(\psi) = -\frac{1}{2}\langle \psi, C^{-1}\psi\rangle + \log Z\_{\lambda}(\psi) \tag{4}\label{4}
\end{equation}
and thus we can also use the effective action to study correlation functions. It is often the case that the covariance $C$ can be written as a sum of covariances $C= C\_{1}+\dotsb + C\_{n}$. Taking $n=2$ for simplicity, plugging it into $\mathcal{G}\_{\lambda V,C}$ and using the property that a sum of Gaussian random variables is again a Gaussian random we get:
\begin{equation}
\mathcal{G}\_{\lambda V,C\_{1}+C\_{2}}(\psi) = \mathcal{G}\_{\mathcal{G}\_{\lambda V, C\_{1}}, C\_{2}}(\psi) \tag{5}\label{5}
\end{equation}
which is the semi-group property of the effective action. Recursively, this also hold for $n>2$. Thus, it is sufficient to study the map:
\begin{equation}
-\log \int d\mu\_{C\_{i}}(\varphi)e^{-\lambda V\_{i}(\varphi+\psi)} \tag{6}\label{6}
\end{equation}
where $V\_{i}$ is properly chosen. 


The point that is not clear to me is the following. In statistical mechanics, one is primarily interested in evaluating the partition function, even though other quantities (such as the ones defined above) are also relevant. I'd like to better understand the relation between the above objects and the actual partition function of the system (not the one with source terms) and what is the difference between approaching a problem using the effective action or the partition function itself, is there is any. 



Effective action, partition function and the renormalization group
These correspondences might be helpful: we have the *partition function* $Z(\beta)$, the *effective action* $W(\beta)$, the *classical action* $I\_\beta[\phi]$ for physical variables $\phi(x,\tau)$ in Euclidean time $\tau$ with period $\beta$ equal to inverse temperature. These are related by
$$ e^{-W(\beta)}=Z(\beta)=\int D\phi\, e^{-I\_\beta[\phi]}$$
$$\qquad\qquad = \int D\phi \exp\left[-\int\_0^\beta d\tau\,\int d^3 x\,\bigl(\tfrac{1}{2}(\nabla\phi)^2+V(\phi)\bigr)\right].$$
The thermodynamic *free energy* is $F(\beta)=\beta^{-1}W(\beta)$, the *entropy*
\begin{equation}
S(\beta)=\beta^2 \frac{dF(\beta)}{d\beta}=\left(\beta\frac{d}{d\beta}-1\right)W(\beta).
\end{equation}


My answer to the question in the OP, "what is the difference between approaching a problem using the effective action or the partition function itself" would be --- there is no difference, one is the logarithm of the other.


