Abstract
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 dimensiond>2) the thermal conductivity and the bulk and shear viscosities, leading to Navier-Stokes behavior on a large scale. Ford<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.
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Visscher, P.B. Renormalization-group derivation of Navier-Stokes equation. J Stat Phys 38, 989–1013 (1985). https://doi.org/10.1007/BF01010426
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DOI: https://doi.org/10.1007/BF01010426