Quantum Solution of Classical Turbulence Decaying Energy Spectrum

This paper summarizes and elaborates on recently found {migdal2023exact} exact reduction of decaying turbulence in the Navier-Stokes equation in $3 + 1$ dimensions to a Number Theory problem of 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. We compute resulting integrals in {} and present effective index $n(t) = -t E'(t)/E(t) $ for the energy decay as a function of time Fig.{fig::NPlot} and s(k)= -k∂ₖ log E(k,t) for the energy spectrum Fig.{fig::SPIndex}. The asymptotic values are n(∞) = 5/4, s(∞) = 7/2, but the universal functions $n(t), s(t)$ are neither constant nor linear due to quantum effects (complex poles of the Mellin transform at zeros of Riemann zeta function). The theoretical value n(∞)=54 matches the grid turbulence experiments {GridTurbulence_1966} within experimental errors ~ 2%.