Green-Kubo Formula for Weakly Coupled Systems with Noise

We study the Green-Kubo formula κ(ε,ς) for the heat conductivity of an infinite chain of d-dimensional finite systems (cells) coupled by a smooth nearest neigh- bor potential εV . The uncoupled systems evolve according to Hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength ς . Noting that κ (ε, ς ) exists and is finite whenever ς > 0, we are interested in what happens when the strength of the noise ς → 0. For this, we start in this work by formally expanding κ(ε,ς) in a power series in ε, κ(ε,ς) = ε2 tions satisfied by κn (ς ). We show in particular that κ2 (ς ) 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 ε−2t, 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 κ2(ς) as ς → 0 in three cases: the disordered harmonic chain, the rotor chain and a chain of strongly an harmonic oscillators. Moreover, we formally identify κ2 (ς ) with the conductivity obtained by having the chain between two reservoirs at temperature T and T + δT , in the limit δT → 0, N → ∞, ε → 0.