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

## TL;DR

This paper extends Irving and Zwanzig's classical limit of quantum observables to matrix-valued potentials, enabling molecular dynamics simulations to derive constitutive relations in fluid dynamics, especially at high temperatures.

## Contribution

It introduces a matrix formulation for the semi-classical limit of quantum observables in conservation laws, applicable to systems with multiple excited electron states.

## Key findings

- Derived the classical limit of quantum observables for matrix-valued potentials.
- Extended the classical limit to high-temperature regimes beyond electron eigenvalue gaps.
- Provided a framework for molecular dynamics simulations of stress and heat flux in quantum systems.

## Abstract

In the classical work by Irving and Zwanzig [Irving J.H. and Zwanzig R.W., J. Chem. Phys. 19 (1951), 1173-1180 ] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. This work derives the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the semi-classical limit of the quantum observables in the conservation laws, also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. 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 as a local grand canonical quantum ensemble determined by the initial conservation variables.