Classical Turbulence as Quantum Geometry

The loop equation for circulation PDF as functional of the loop shape is derived from the Hopf equation. This equation is exactly equivalent to the Schrödinger equation in loop space, with viscosity playing the role of Planck's constant. This equivalence, in principle, allows direct numerical simulation of classical turbulence on quantum computers. Topologically stable time-dependent Euler flows, which we call Kelvinons, are introduced. They have a conserved velocity circulation around stationary loop. Kelvinons are fixed points of the loop equation in the WKB limit of vanishing viscosity at fixed energy dissipation, which corresponds to strong turbulence. This makes them responsible for asymptotic PDF tails of velocity circulation, perfectly matching numerical simulations. Area law and the asymptotic scaling law for mean circulation at a large area are derived from the loop equation. The representation of the solution of the loop equation in terms of a singular stochastic equation for momentum loop trajectory is presented.



Alexander Migdal


New York University, Abu Dhabi