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We study the transport of heat along a chain of particles interacting througha harmonic potential and subject to heat reservoirs at its ends. Each particlehas two degrees of freedom and is subject to a stochastic noise that producesinfinitesimal changes in the velocity while keeping the kinetic energyunchanged. This is modelled by means of a Langevin equation with multiplicativenoise. We show that the introduction of this energy conserving stochastic noiseleads to Fourier's law. By means of an approximate solution that becomes exactin the thermodynamic limit, we also show that the heat conductivity $\kappa$behaves as $\kappa = a L/(b+\lambda L)$ for large values of the intensity$\lambda$ of the energy conserving noise and large chain sizes $L$. Hence, weconclude that in the thermodynamic limit the heat conductivity is finite andgiven by $\kappa=a/\lambda$.

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