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Fourier's law from a chain of coupled planar harmonic oscillators under energy conserving noise

We study the transport of heat along a chain of particles interacting through a harmonic potential and subject to heat reservoirs at its ends. Each particle has two degrees of freedom and is subject to a stochastic noise that produces infinitesimal changes in the velocity while keeping the kinetic energy unchanged. This is modelled by means of a Langevin equation with multiplicative noise. We show that the introduction of this energy conserving stochastic noise leads to Fourier's law. By means of an approximate solution that becomes exact in the thermodynamic limit, we also show that the heat conductivity $κ$ behaves as $κ= a L/(b+λL)$ for large values of the intensity $λ$ of the energy conserving noise and large chain sizes $L$. Hence, we conclude that in the thermodynamic limit the heat conductivity is finite and given by $κ=a/λ$.