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 modeled 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 = aL/(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.