The Mori-Zwanzig formalism for the derivation of a fluctuating heat conduction model from molecular dynamics

Energy transport equations are derived directly from full molecular dynamics models as coarse-grained description. With the local energy chosen as the coarse-grained variables, we apply the Mori-Zwanzig formalism to derive a reduced model, in the form of a generalized Langevin equation. A Markovian embedding technique is then introduced to eliminate the history dependence. In sharp contrast to conventional energy transport models, this derivation yields {\it stochastic} dynamics models for the spatially averaged energy. We discuss the approximation of the random force using both additive and multiplicative noises, to ensure the correct statistics of the solution.