# Dual Theory of Decaying Turbulence

- Alexander Migdal

We investigate the recently found cite{migdal2023exact} 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 quantum statistical theory of free fermions on a ring, with an external field related to the random fractions of $pi$.

We find the solution of this system in the statistical limit $Nto infty$ in terms of a complex trajectory (instanton) providing a saddle point to the path integral over the charge 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 assumptions, approximations, or fitted parameters.

We compute resulting integrals in Mathematica{} and present effective indexes for the energy decay as a function of time Fig.ref{fig::NPlot} and the energy spectrum as a function of the wavevector at fixed time Fig.ref{fig::SPIndex}.

In particular, the asymptotic value of the effective index in energy decay $n(infty) = frac{7}{4}$, but the universal function $n(t)$ is neither constant nor linear.

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