Reduction of Decaying Navier--Stokes Turbulence to the Quantum Pendulum
Abstract
The statistics of a classical flow is a problem of quantum mechanics in loop space, and not by analogy: the loop functional of any flow is the loop-ordered exponential of the covariant derivative ∂+iv/ν. We follow this representation from the Navier--Stokes equation to the Euler ensemble, the fixed point of the loop equation regularized by polygons with N sides. Near the turning angle β=π an exact operator identity, followed by an asymptotic continuum step and without a saddle-point approximation, reduces the ensemble to two quantum pendulums, the Mathieu equation; this supersedes our earlier WKB evaluation. A Monte Carlo simulation up to N=105 confirms the reduction. Extreme turbulence has singularities at small scales, and, as in quantum field theory, we define it by this regularization and a renormalization: the leading power of N is removed, and the turbulent viscosity ν̃ remains as the single physical scale. This is a proposed definition of turbulence at infinite Reynolds number; its continuum spectrum is computed under a stated convergence condition, and its identification with physical turbulence is a hypothesis. The energy spectrum is a function of κ=k√(ν̃(t+t0)) that falls as 1/k and then as k−2. At κ=√2 it has a weak nonanalyticity: its second derivative diverges logarithmically. At a fixed wave number the decay is therefore singular at the finite time t*=2/(ν̃k2)−t0, when t*>0, and the vorticity correlations oscillate in space with the period √2 π√(ν̃(t+t0)). Of the limits of the ensemble with structure, only the pendulum limit is taken at a critical point and has a bulk dissipation that vanishes with the viscosity. A companion paper compares it with simulations and experiments.
Related
- Dynamic Interpolation and Higher-Order Turbulence Scales in 3D Navier-Stokes Flows
- Spatial symmetry invariance of three-dimensional Navier-Stokes turbulence
- Work in progress: A Lean-verified conditional reduction of periodic Navier-Stokes regularity on T^3 to a single integrability hypothesis
- Global Strong Solution to Multi-Dimensional Compressible Quantum Navier-Stokes Equation in Bounded Domains with Large Initial Data
- Large data global strong solutions to the quantum Navier-Stokes equations with degenerate viscosity
BibTeX
@misc{migdal2026reduction,
title = {{Reduction of Decaying Navier--Stokes Turbulence to the Quantum Pendulum}},
author = {Alexander Migdal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.05745},
eprint = {2607.05745},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.05745},
note = {CC0},
}
RIS
TY - UNPB
TI - Reduction of Decaying Navier--Stokes Turbulence to the Quantum Pendulum
AU - Alexander Migdal
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2607.05745
UR - https://arxiv.org/abs/2607.05745
AB - The statistics of a classical flow is a problem of quantum mechanics in loop space, and not by analogy: the loop functional of any flow is the loop-ordered exponential of the covariant derivative ∂+iv/ν. We follow this representation from the Navier--Stokes equation to the Euler ensemble, the fixed point of the loop equation regularized by polygons with N sides. Near the turning angle β=π an exact operator identity, followed by an asymptotic continuum step and without a saddle-point approximation, reduces the ensemble to two quantum pendulums, the Mathieu equation; this supersedes our earlier WKB evaluation. A Monte Carlo simulation up to N=10^5 confirms the reduction. Extreme turbulence has singularities at small scales, and, as in quantum field theory, we define it by this regularization and a renormalization: the leading power of N is removed, and the turbulent viscosity ν̃ remains as the single physical scale. This is a proposed definition of turbulence at infinite Reynolds number; its continuum spectrum is computed under a stated convergence condition, and its identification with physical turbulence is a hypothesis. The energy spectrum is a function of κ=k√(ν̃(t+t_0)) that falls as 1/k and then as k^−2. At κ=√2 it has a weak nonanalyticity: its second derivative diverges logarithmically. At a fixed wave number the decay is therefore singular at the finite time t_*=2/(ν̃k^2)−t_0, when t_*>0, and the vorticity correlations oscillate in space with the period √2 π√(ν̃(t+t_0)). Of the limits of the ensemble with structure, only the pendulum limit is taken at a critical point and has a bulk dissipation that vanishes with the viscosity. A companion paper compares it with simulations and experiments.
ER -