On the derivation of Fourier’s law in stochastic energy exchange systems

Abstract. We present a detailed derivation of Fourier’s law in a class of stochastic energy exchange systems with periodic lattice structure. We provide two separate derivations of the thermal conductivity proving it is identical to the frequency of energy exchanges. The first derivation relies on the diffusion of the Helfand moment, which is determined solely by static correlations. The second approach relies on a gradient expansion of the probability measure around a non-equilibrium stationary state. The linear part of the heat current is determined by local thermal equilibrium distributions which solve a Boltzmann-like equation. A numerical scheme is presented with computations of the conductivity along our two methods. The results are in outstanding agreement with our theory.