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

Energy transport equations are derived directly from a many-particle system as a coarse-grained (CG) description. This effort is motivated by the observation that the conventional heat equation is unable to describe the heat conduction process at the nano-mechanical scale. With the local energy density chosen as the CG 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 employed to eliminate the history dependence. Meanwhile, auxiliary variables are introduced to establish auxiliary equations that govern the dynamics of the energy flux. In sharp contrast to conventional energy transport models, this derivation yields stochastic dynamical models for the spatially averaged energy. The random force in the generalized Langevin equation is typically modeled by additive white Gaussian noise. As an initial attempt, we consider multiplicative white Gaussian noise, to ensure the correct statistics of the non-Gaussian solution.