The Quantum Pendulum of Turbulent Flow: Decaying turbulence as a thermal state of an integrable system

Abstract

We present a theory of decaying turbulence in which the statistics of a classical flow becomes quantum mechanics in loop space. The loop functional, the average of $e^{\mathrm{i}\Gamma /\nu }$ for the circulation $\Gamma$ around a loop, is the wave function, and the Navier-Stokes equation becomes the linear loop equation. An exact operator identity represents the functional as the loop-ordered exponential of the covariant derivative $\partial +\mathrm{i}v/\nu$, with the loop parameter as time and viscosity as Planck's constant. The turbulent state is the equal-weight density matrix on a decaying trajectory, a quantum form of the ergodic hypothesis. As in quantum field theory, turbulence at infinite Reynolds number is defined by regularization and renormalization; the loop equation regularized on $N$-sided polygons has an exact manifold of decaying solutions, the Euler ensemble, labeled by rational angles and spins. As a fermionic trace, its partition function and conditional vorticity correlations become, in the local limit, Gibbs averages of two Mathieu quantum pendulums, an integrable system. The energy spectrum, given by explicit quadratures, falls as $k^{-1}\log (1/kL)$ at small $k$ and as $k^{-7/2}$ at large $k$, with $L=\sqrt{\tilde{\nu }(t+t_0)}$ and turbulent viscosity $\tilde{\nu }$. At $kL=\sqrt{2}$ its slope diverges as a double logarithm: the decay at fixed $k$ is singular at a finite time, and the vorticity correlation oscillates with period $\sqrt{2}\pi L$. The reduction to the pendulums, with the continuum and grand canonical limits at every nonzero separation, is formally verified in Lean 4 in forthcoming work by Y. Jiang, A. Migdal et al. The theory rests on stated physical hypotheses, as statistical mechanics rests on the Gibbs distribution; its comparison with simulations and experiment is so far qualitative.

arXiv comment48 pages, 4 figures, radically revised paper, after exact solution in continuum limit was discovered, verified and investigated, with redefined vorticity correlation as conditional probability. The reeduction of the Euler ensemble to the quantum pendulum is rigorously proven in forthcoming paper with Google Reesearch, using Lean 4 language