Quantum Solution of Classical Turbulence. Decaying Energy Spectrum

Alexander Migdal

doi:10.20944/preprints202312.1012.v9

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preprints.org > physical sciences > fluids and plasmas physics > doi: 10.20944/preprints202312.1012.v9

Preprint Article Version 9 Preserved in Portico This version is not peer-reviewed

Quantum Solution of Classical Turbulence. Decaying Energy Spectrum

Alexander Migdal

Version 1 : Received: 13 December 2023 / Approved: 13 December 2023 / Online: 14 December 2023 (08:24:06 CET)
Version 2 : Received: 24 December 2023 / Approved: 26 December 2023 / Online: 26 December 2023 (09:54:09 CET)
Version 3 : Received: 7 March 2024 / Approved: 7 March 2024 / Online: 7 March 2024 (14:17:24 CET)
Version 4 : Received: 24 March 2024 / Approved: 25 March 2024 / Online: 26 March 2024 (08:24:14 CET)
Version 5 : Received: 26 March 2024 / Approved: 27 March 2024 / Online: 27 March 2024 (14:15:31 CET)
Version 6 : Received: 27 March 2024 / Approved: 28 March 2024 / Online: 28 March 2024 (13:31:01 CET)
Version 7 : Received: 2 April 2024 / Approved: 3 April 2024 / Online: 3 April 2024 (14:11:11 CEST)
Version 8 : Received: 7 April 2024 / Approved: 8 April 2024 / Online: 8 April 2024 (11:36:43 CEST)
Version 9 : Received: 11 April 2024 / Approved: 12 April 2024 / Online: 12 April 2024 (14:39:31 CEST)
Version 10 : Received: 13 April 2024 / Approved: 15 April 2024 / Online: 17 April 2024 (07:19:09 CEST)

How to cite: Migdal, A. Quantum Solution of Classical Turbulence. Decaying Energy Spectrum. Preprints 2023, 2023121012. https://doi.org/10.20944/preprints202312.1012.v9 Migdal, A. Quantum Solution of Classical Turbulence. Decaying Energy Spectrum. Preprints 2023, 2023121012. https://doi.org/10.20944/preprints202312.1012.v9

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MDPI and ACS Style

Migdal, A. Quantum Solution of Classical Turbulence. Decaying Energy Spectrum. Preprints 2023, 2023121012. https://doi.org/10.20944/preprints202312.1012.v9

APA Style

Migdal, A. (2024). Quantum Solution of Classical Turbulence. Decaying Energy Spectrum. Preprints. https://doi.org/10.20944/preprints202312.1012.v9

Chicago/Turabian Style

Migdal, A. 2024 "Quantum Solution of Classical Turbulence. Decaying Energy Spectrum" Preprints. https://doi.org/10.20944/preprints202312.1012.v9

Abstract

This paper brings to fruition recently found \cite{migdal2023exact} exact reduction of decaying turbulence in the Navier-Stokes equation in $3 + 1$ dimensions to a Number Theory problem of finding the statistical limit of the Euler ensemble. We reformulate the Euler ensemble as a Markov chain and show the equivalence of this formulation to the \textbf{quantum} statistical theory of $N$ fermions on a ring, with an external field related to the random fractions of $\pi$. We find the solution of this system in the turbulent limit $N\to \infty, \nu \to 0$ in terms of a complex trajectory (instanton) providing a saddle point to the path integral over the density of these fermions. This results in an analytic formula for the observable correlation function of vorticity in wavevector space. This is a full solution of decaying turbulence from the first principle without approximations or fitted parameters. The energy spectrum decays as $k^{-\frac{7}{2}}$. We compute resulting integrals in \Mathematica{} and present effective index $n(t) = -t\partial_t \log E $ for the energy decay as a function of time Fig.\ref{fig::NPlot}. The asymptotic value of the effective index in energy decay $n(\infty) = \frac{5}{4}$, but the universal function $n(t)$ is neither constant nor linear.

Keywords

Turbulence; fractal; anomalous dissipation; fixed point; velocity circulation; loop equations; Euler Phi; prime numbers; Fermi particles; instanton

Subject

Physical Sciences, Fluids and Plasmas Physics

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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