Green-Kubo formula for weakly coupled system with dynamical noise

Description

We study the Green-Kubo (GK) formula κ(ε,ξ) for the heat conductivity of an infinite chain of d-dimensional finite systems (cells) coupled by a smooth nearest neighbour potential εV. The uncoupled systems evolve according to Hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength ξ. Noting that κ(ε,ξ) exists and is finite whenever ξ>0, we are interested in what happens when the strength of the noise ξ→0. For this, we start in this work by formally expanding κ(ε,ξ) in a power series in ε, κ(ε,ξ)=ε2∑n≥2εn−2κn(ξ) and investigating the (formal) equations satisfied by κn(ξ. We show in particular that κ2(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 ε−2t, for the cases where the latter has been established \cite{LO, DL}. For one-dimensional systems, we investigate κ2(ξ) as ξ→0 in three cases: the disordered harmonic chain, the rotor chain and a chain of strongly anharmonic oscillators. Moreover, we formally identify κ2(ξ) 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.

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