  In stochastic systems with weak noise, the logarithm of the stationary
distribution becomes proportional to a large deviation rate function called the
quasi-potential. The quasi-potential, and its characterization through a
variational problem, lies at the core of the Freidlin-Wentzell large deviations
theory%.~\cite{freidlin1984}.In many interacting particle systems, the particle
density is described by fluctuating hydrodynamics governed by Macroscopic
Fluctuation Theory%, ~\cite{bertini2014},which formally fits within
Freidlin-Wentzell's framework with a weak noise proportional to $1/\sqrt{N}$,
where $N$ is the number of particles. The quasi-potential then appears as a
natural generalization of the equilibrium free energy to non-equilibrium
particle systems. A key physical and practical issue is to actually compute
quasi-potentials from their variational characterization for non-equilibrium
systems for which detailed balance does not hold. We discuss how to perform
such a computation perturbatively in an external parameter $\lambda$, starting
from a known quasi-potential for $\lambda=0$. In a general setup, explicit
iterative formulae for all terms of the power-series expansion of the
quasi-potential are given for the first time. The key point is a proof of
solvability conditions that assure the existence of the perturbation expansion
to all orders. We apply the perturbative approach to diffusive particles
interacting through a mean-field potential. For such systems, the variational
characterization of the quasi-potential was proven by Dawson and Gartner%.
~\cite{dawson1987,dawson1987b}. Our perturbative analysis provides new explicit
results about the quasi-potential and about fluctuations of one-particle
observables in a simple example of mean field diffusions: the
Shinomoto-Kuramoto model of coupled rotators%. ~\cite{shinomoto1986}. This is
one of few systems for which non-equilibrium free energies can be computed and
analyzed in an effective way, at least perturbatively.
