We study the Green-Kubo formula $${\kappa (\varepsilon, \varsigma)}$$ for the heat conductivity of an infinite chain of d-dimensional finite systems (cells) coupled by a smooth nearest neighbor potential $${\varepsilon V}$$ . The uncoupled systems evolve according to Hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength $${\varsigma}$$ . Noting that $${\kappa (\varepsilon, \varsigma)}$$ exists and is finite whenever $${\varsigma > 0}$$ , we are interested in what happens when the strength of the noise $${\varsigma \to 0}$$ . For this, we start in this work by formally expanding $${\kappa (\varepsilon, \varsigma)}$$ in a power series in $${\varepsilon}$$ , $${\kappa (\varepsilon, \varsigma) = \varepsilon^2 \sum_{n\geq 2} \varepsilon^{n-2} \kappa_n (\varsigma)}$$ and investigating the (formal) equations satisfied by $${\kappa_n (\varsigma)}$$ . We show in particular that $${\kappa_2 (\varsigma)}$$ 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 (Liverani and Olla, in JAMS 25:555–583, 2012; Dolgopyat and Liverani, in Commun Math Phys 308:201–225, 2011). For one-dimensional systems, we investigate $${\kappa_2 (\varsigma)}$$ as $${\varsigma \to 0}$$ in three cases: the disordered harmonic chain, the rotor chain and a chain of strongly an harmonic oscillators. Moreover, we formally identify $${\kappa_2 (\varsigma)}$$ with the conductivity obtained by having the chain between two reservoirs at temperature T and $${T+\delta T}$$ , in the limit $${\delta T \to 0}$$ , $${N \to \infty}$$ , $${\varepsilon \to 0}$$ .