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Green-Kubo formula for weakly coupled system with dynamical noise

We study the Green-Kubo (GK) formula $κ(\varepsilon, ξ)$ 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 $ξ$. Noting that $κ(\varepsilon, ξ)$ exists and is finite whenever $ξ> 0$, we are interested in what happens when the strength of the noise $ξ\to 0$. For this, we start in this work by formally expanding $κ(\varepsilon, ξ)$ in a power series in $\varepsilon$, $κ(\varepsilon, ξ) = \varepsilon^2 \sum_{n\ge 2} \varepsilon^{n-2} κ_n (ξ)$ and investigating the (formal) equations satisfied by $κ_n (ξ$. We show in particular that $κ_2 (ξ)$ 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 $κ_2 (ξ)$ as $ξ\to 0$ in three cases: the disordered harmonic chain, the rotor chain and a chain of strongly anharmonic oscillators. Moreover, we formally identify $κ_2 (ξ)$ with the conductivity obtained by having the chain between two reservoirs at temperature $T$ and $T+δT$, in the limit $δT\to 0$, $N \to \infty$, $\varepsilon \to 0$.