Communications in Mathematical Sciences

Volume 17 (2019)

Number 2

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

Pages: 539 – 563



Weiqi Chu (Department of Mathematics, Pennsylvania State University, University Park, Pa., U.S.A.)

Xiantao Li (Department of Mathematics, Pennsylvania State University, University Park, Pa., U.S.A.)


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.


molecular dynamics, nanoscale energy transport, Mori–Zwanzig formalism

2010 Mathematics Subject Classification

60H10, 65C20

The authors’ research was supported by the National Science Foundation DMS-1522617, DMS-1619661 and DMS-1819011.

Received 27 March 2018

Received revised 27 December 2018

Accepted 27 December 2018

Published 8 July 2019