Geometric Solution of Turbulence as Diffusion in Loop Space

TL;DR

This paper introduces a loop space calculus framework that transforms nonlinear turbulence equations into linear diffusion equations, providing exact solutions and unifying various scaling laws with novel predictions supported by recent experiments.

Contribution

It presents a unified analytical approach to turbulence and related nonlinear problems using loop space calculus, connecting turbulence, string theory, and quantum chromodynamics.

Findings

Exact solution for decaying hydrodynamic turbulence (Euler ensemble)

Prediction of log-periodic oscillations in turbulence correlations

Identification of a phase transition in MHD turbulence

Abstract

Strongly nonlinear dynamics, from fluid turbulence to quantum chromodynamics, have long constituted some of the most challenging problems in theoretical physics. This review describes a unified theoretical framework, the loop space calculus, which offers an analytical approach to these problems. The central idea is a shift in perspective from pointwise fields to integrated loop observables, a transformation that recasts the governing nonlinear equations into a universal linear diffusion equation in the space of loops. This framework, supported by recent mathematical analysis, is analytically solvable and yields an exact, parameter-free solution for decaying hydrodynamic turbulence -- the Euler ensemble -- which is shown to be dual to a solvable string theory. The theory's predictions include: (i) the unification of spatial and temporal scaling laws, governed by two related, infinite spectra of intermittency and decay exponents derived from the nontrivial zeros of the Riemann zeta function; (ii) a first-order phase transition in magnetohydrodynamic (MHD) turbulence; and (iii) the formation of quantized, concentric shells in passive scalar mixing. The theory also predicts log-periodic oscillations in correlation functions -- effects not captured by standard phenomenology -- for which there is now emerging evidence from high-precision turbulence experiments. The appearance of identical mathematical structures as solutions to the turbulent regime of Yang-Mills gradient flow points to the broad applicability of this approach. The framework also yields a new type of analytic Hodge-dual matrix surface that solves the Yang-Mills fixed-point loop equation by harmonic map, opening the way for a geometric formulation of QCD string theory.