  We study the Green-Kubo (GK) formula $\kappa (\varepsilon, \xi)$ for the heat
conductivity of an infinite chain of $d$-dimensional finite systems (cells)
coupled by a smooth nearest neighbour potential $\varepsilon V$. The uncoupled
systems evolve according to Hamiltonian dynamics perturbed stochastically by an
energy conserving noise of strength $\xi$. Noting that $\kappa (\varepsilon,
\xi)$ exists and is finite whenever $\xi> 0$, we are interested in what happens
when the strength of the noise $\xi \to 0$. For this, we start in this work by
formally expanding $\kappa (\varepsilon, \xi)$ in a power series in
$\varepsilon$, $\kappa (\varepsilon, \xi) = \varepsilon^2 \sum_{n\ge 2}
\varepsilon^{n-2} \kappa_n (\xi)$ and investigating the (formal) equations
satisfied by $\kappa_n (\xi$. We show in particular that $\kappa_2 (\xi)$ is
well defined when no pinning potential is present, and coincides formally with
the heat conductivity obtained in the weak coupling (van Hove) limit, where
time is rescaled as $\varepsilon^{-2}t$, for the cases where the latter has
been established \cite{LO, DL}. For one-dimensional systems, we investigate
$\kappa_2 (\xi)$ as $\xi\to 0$ in three cases: the disordered harmonic chain,
the rotor chain and a chain of strongly anharmonic oscillators. Moreover, we
formally identify $\kappa_2 (\xi)$ with the conductivity obtained by having the
chain between two reservoirs at temperature $T$ and $T+\delta T$, in the limit
$\delta T\to 0$, $N \to \infty$, $\varepsilon \to 0$.
