Renormalization-group derivation of Navier-Stokes equation
type: Item
instance of: scholarly article
title: Renormalization-group derivation of Navier-Stokes equation
zbMATH Open document ID: 0625.76036
DOI: 10.1007/BF01010426
author: P. B. Visscher
published in: Journal of Statistical Physics
publication date: 1985-01-01T00:00:00Z
review text: The Navier-Stokes equation is proved from first principles (rotational symmetry and conservation of momentum, mass, and energy) using renormalization-group ideas. That is, we consider a system described by one (classical) conserved vector field and two conserved scalar fields, and demonstrate that on a large scale it obeys the Navier-Stokes equation. No assumptions about the physical meanings of the fields are required; in particular, no results from thermodynamics are used. The result comes about because the Euler equation is an exact fixed point of an appropriate scale-coarsening transformation, and the coefficients of the eigenvectors of the transformation with the largest (``most relevant'') eigenvalues include (in dimension \(d>2)\) the thermal conductivity and the bulk and shear viscosities, leading to Navier-Stokes behavior on a large scale. For \(d<2\), the largest eigenvalue corresponds to a convection term, and the Navier-Stokes equation is incorrect. Our method differs from previous renormalization approaches in using time-coarsening as well as space-coarsening transformations. This allows renormalization trajectories to be determined exactly, and allows the determination of the macroscopic behavior of specific microscopic models. The Navier- Stokes equation we obtain is almost, but not exactly, the same as the conventional one; distinguishing between them experimentally would require measurement of the very small asymmetry of the Brillouin line in a simple fluid.
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76A02
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 4016444
zbMATH Keywords: Navier-Stokes equation
zbMATH Keywords: rotational symmetry
zbMATH Keywords: conservation of momentum
zbMATH Keywords: renormalization-group
zbMATH Keywords: Euler equation
zbMATH Keywords: scale-coarsening transformation
zbMATH Keywords: eigenvalues
zbMATH Keywords: space-coarsening transformations
zbMATH Keywords: Brillouin line
MaRDI profile type: Publication
cites work: Foundations of Statistical Mechanics
full work available at URL: https://doi.org/10.1007/bf01010426
OpenAlex ID: W2050274322
description: scientific article; zbMATH DE number 4016444