Perturbative anomalous exponents from Kolmogorov multipliers
Abstract
Intermittency, manifested through anomalous scaling, remains one of the central unresolved problems in turbulence theory, with few analytical approaches extending beyond idealized linear transport models. We introduce a perturbative framework for anomalous scaling in turbulent transport based on multiplier statistics, rather than zero-mode calculations. We demonstrate the approach using a shell model combining deterministic and Kraichnan-like stochastic components. We reduce the problem to the analysis of a stationary Fokker-Planck equation for Kolmogorov multipliers, defined as ratios of successive scalar amplitudes. Its solution yields the invariant measure through a perturbative expansion around a Gaussian distribution. Using the resulting multiplier statistics, we compute explicit anomalous scaling exponents for structure functions of arbitrary order, including odd, even, and non-integer moments. Although demonstrated here for a shell model, the framework suggests a systematic perturbative route toward analytical theories of intermittency in turbulence.
Versions
Related
- Optimal square size for separating the operator and completely bounded norms of Schur multipliers on S^4
- Jacobi last multipliers and metrisability of Liénard differential equations
- Granger causality detection with Kolmogorov-Arnold networks
- Cryogenic performance of Silicon Photo-Multipliers at LN2 temperature and at 3 K in the RES-NOVA cryostat
- Comparative Robustness of Mach-Zehender and Microring Photonic Matrix Multipliers Under Laser RIN and Thermal Drift
BibTeX
@misc{mailybaev2026perturbative,
title = {{Perturbative anomalous exponents from Kolmogorov multipliers}},
author = {Alexei A. Mailybaev and Simon Thalabard},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2605.26359},
eprint = {2605.26359},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.26359},
}
RIS
TY - UNPB
TI - Perturbative anomalous exponents from Kolmogorov multipliers
AU - Alexei A. Mailybaev
AU - Simon Thalabard
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2605.26359
UR - https://arxiv.org/abs/2605.26359
AB - Intermittency, manifested through anomalous scaling, remains one of the central unresolved problems in turbulence theory, with few analytical approaches extending beyond idealized linear transport models. We introduce a perturbative framework for anomalous scaling in turbulent transport based on multiplier statistics, rather than zero-mode calculations. We demonstrate the approach using a shell model combining deterministic and Kraichnan-like stochastic components. We reduce the problem to the analysis of a stationary Fokker-Planck equation for Kolmogorov multipliers, defined as ratios of successive scalar amplitudes. Its solution yields the invariant measure through a perturbative expansion around a Gaussian distribution. Using the resulting multiplier statistics, we compute explicit anomalous scaling exponents for structure functions of arbitrary order, including odd, even, and non-integer moments. Although demonstrated here for a shell model, the framework suggests a systematic perturbative route toward analytical theories of intermittency in turbulence.
ER -