Problem with the derivation of the Navier-Stokes equation by means of Zwanzig-Mori technique: Correction and solution

TL;DR

This paper identifies and corrects a flaw in the derivation of the Navier-Stokes equation using Zwanzig-Mori techniques, ensuring the derivation aligns with physical expectations by properly handling correlation functions.

Contribution

The paper corrects the derivation of the Navier-Stokes equation via projection operator methods by properly treating correlation functions, resolving previous inaccuracies.

Findings

Incorrect second-order friction term was due to improper correlation function treatment

Corrected derivation yields a zero second-order term, aligning with physical principles

Ensures the Navier-Stokes equation is accurately derived from microscopic dynamics

Abstract

The derivation of the Navier-Stokes equation starting from the Liouville equation using projector techniques yields a friction term which is nonlinear in the velocity. As has been explained in the 1. version of this paper, when the second-order part of the term is non-zero, this leads to an incorrect formula for the equation. In this 2. version, it is shown that the problem is due to an inadequate treatment of one of the correlation functions appearing . Repeating the calculation leads to zero second-order part. The Navier-Stokes equation is correctly derived by projection operator technique.