Why Does Classical Turbulence Obey an Area Law?

TL;DR

This paper derives a stochastic Navier-Stokes equation from quantum mechanics principles, explaining the origin of viscosity and circulation statistics in turbulence through a quantum-to-classical transition framework.

Contribution

It introduces a quantum mechanical derivation of the stochastic Navier-Stokes equation, linking quantum Lindblad operators to classical turbulence viscosity and circulation laws.

Findings

Derived a stochastic Navier-Stokes equation from quantum Lindblad dynamics.

Verified the Migdal area law for circulation statistics numerically.

Established the quantum origin of viscosity and circulation quantization in turbulence.

Abstract

In incompressible flow the viscous force is solenoidal, whereas the Madelung transform of a spinless Schr\"odinger equation produces only gradient forces. The two are orthogonal, so viscosity cannot arise from Hamiltonian quantum mechanics alone; an open quantum treatment is required. Reducing the $N$-body density matrix to its one-body component and closing the dynamics via Born-Markov yields Lindblad jump operators with $k^2$ scattering rates, which we unravel via quantum state diffusion (QSD) into a norm-preserving stochastic nonlinear Schr\"odinger equation. Dissipation and stochastic forcing are not separate ingredients: both come from the same Lindblad operators, and their amplitudes are locked by the QSD structure. The Madelung transform of this equation, under incompressibility, gives a stochastic Navier-Stokes equation whose viscosity is set by the mean free path and whose noise correlator satisfies the fluctuation-dissipation relation by construction, in agreement with the Landau-Lifshitz framework. The recovery is conditional: the viscous identification holds at the ensemble level via the vortex decomposition of the velocity field; the single-trajectory identification remains open. The zeros of the wavefunction carry quantised circulation; their codimension-2 topology yields the Migdal area law for circulation statistics under a Poisson assumption, here through a different mechanism than the loop-functional saddle point and verified numerically even in the quantum regime where the de~Broglie length exceeds the Kolmogorov scale.