Geometric Solution of Turbulence as Diffusion in Loop Space
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914621093314560 |
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| author | Migdal, Alexander |
| author_facet | Migdal, Alexander |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02165 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric Solution of Turbulence as Diffusion in Loop Space Migdal, Alexander Fluid Dynamics High Energy Physics - Theory Mathematical Physics 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 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. |
| title | Geometric Solution of Turbulence as Diffusion in Loop Space |
| topic | Fluid Dynamics High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2511.02165 |