Quantum Solution of Classical Turbulence. Decaying Energy Spectrum

This paper brings to fruition recently found {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 quantum statistical theory of N fermions on a ring, with an external field related to the random fractions of π. We find the solution of this system in the turbulent limit N→ ∞, ν → 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-7/2. We compute resulting integrals in {} and present effective index n(t) = -t∂ₜ log E for the energy decay as a function of time Fig.{fig::NPlot}. The asymptotic value of the effective index in energy decay n(∞) = 7/4, but the universal function $n(t)$ is neither constant nor linear.