Green-Kubo formula for weakly coupled system with dynamical noise.

Description

We consider an infinite system of cells coupled into a chain by a smooth nearest neighbor potential $\ve V$. The uncoupled system (cells) evolve according to Hamiltonian dynamics perturbed stochastically with an energy conserving noise of strenght $\noise$. We study the Green-Kubo (GK) formula $\kappa(\ve,\noise)$ for the heat conductivity of this system which exists and is finite for $\noise >0$, by formally expanding $\kappa(\ve,\noise)$ in a power series in $\ve$, $\kappa(\ve,\noise) = \sum_{n\ge 2} \ve^n \kappa_n(\noise)$. We show that $\kappa_2(\noise)$ is the same as the conductivity obtained in the weak coupling (van Hove) limit where time is rescaled as $\ve^{-2}t$. $\kappa_2(\noise)$ is conjectured to approach as $\noise \to 0$ a value proportional to that obtained for the weak coupling limit of the purely Hamiltonian chain. We also show that the $\kappa_2(\noise)$ from the GK formula, is the same as the one obtained from the flux of an open system in contact with Langevin reservoirs. Finally we show that the limit $\noise\to 0$ of $\kappa_2(\noise)$ is finite for the pinned anharmonic oscillators due to phase mixing caused by the non-resonating frequencies of the neighboring cells. This limit is bounded for coupled rotors and vanishes for harmonic chain with random pinning.

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