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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 8

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

Pages: 2191 – 2221

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n8.a5

#### Authors

#### Abstract

In the classical work by Irving and Zwanzig [J.H. Irving and R.W. Zwanzig, *J. Chem. Phys.*, 19, 1173–1180, 1951] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. In this work we derive the corresponding classical molecular dynamics limit by extending Irving and Zwanzig’s result to matrixvalued potentials for a general quantum particle system. The matrix formulation provides the 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 are: (i) to approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a nonlinear eigenvalue problem, (ii) to define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) to determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) to construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables.

#### Keywords

conservation laws, stress tensor, heat flux, molecular dynamics, Weyl quantization

#### 2010 Mathematics Subject Classification

35L65, 35Q70, 81Q20, 82C10

The research of A.S. and M.S. was supported by Swedish Research Council 621-2014-4776. The research of P.P. was supported by ARO MURI Award No. W911NF-14-024.

Received 13 September 2018

Accepted 8 July 2019

Published 3 February 2020