The classical limit of quantum observables in conservation laws of fluid dynamics

Mattias Sandberg, Anders Szepessy

Published 2017-02-14Version 1

Irving and Zwanzig [Irving J.H. and Zwanzig R.W., J. Chem. Phys. 19 (1951), 1173-1180 ] have shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. The classical limit of these quantum observables would be useful in order to determine constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. This work derives the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix valued potentials. The matrix formulation provides the semi-classical limit of the quantum observables in the conservation laws for a canonical ensemble, also in the case where the temperature is large compared to the electron eigenvalue gaps. The main new steps to obtain the molecular dynamics limit is to: (i) approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a non linear eigenvalue problem, (ii) define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) construct the initial particle phase-space density from the canonical quantum ensemble conditioned on macroscopic observable values for the initial density, momentum and energy.