Scaling laws for stationary Navier-Stokes-Fourier flows and the unreasonable effectiveness of hydrodynamics at the molecular level

TL;DR

This paper derives universal scaling laws for stationary compressible Navier-Stokes-Fourier flows far from equilibrium, confirmed by molecular dynamics simulations, highlighting NSF equations' surprising accuracy at the molecular level.

Contribution

The authors develop and validate general scaling laws for stationary NSF flows, demonstrating their robustness and applicability to molecular-scale systems.

Findings

Scaling laws accurately collapse data from molecular dynamics simulations.

Stationary density and temperature depend on pressure and a kinetic excess velocity.

NSF equations effectively describe molecular flows despite finite-size effects.

Abstract

Hydrodynamics provides a universal description of the emergent collective dynamics of vastly different many-body systems, based solely on their symmetries and conservation laws. Here we harness this universality, encoded in the Navier-Stokes-Fourier (NSF) equations, to find general scaling laws for the stationary uniaxial solutions of the compressible NSF problem far from equilibrium. We show for general transport coefficients that the steady density and temperature fields are functions of the pressure and a kinetic field that quantifies the quadratic excess velocity relative to the ratio of heat flux and shear stress. This kinetic field obeys in turn a spatial scaling law controlled by pressure and stress, which is inherited by the stationary density and temperature fields. We develop a scaling approach to measure the associated master curves, and confirm our predictions through compelling data collapses in large-scale molecular dynamics simulations of paradigmatic model fluids. Interestingly, the robustness of the scaling laws in the face of significant finite-size effects reveals the surprising accuracy of NSF equations in describing molecular-scale stationary flows. Overall, these scaling laws provide a novel characterization of stationary states in driven fluids.