湍流的规范理论
A Gauge Theory of Turbulence:
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中文总结 AI 辅助
该论文将不可压缩Navier-Stokes方程形式化为规范理论，通过BRST对称性和Higgs机制解释湍流间歇性，并利用Gribov-inspired log-Poisson层级精确满足Kolmogorov定律，为量子场论方法研究湍流奠定基础。
中文摘要 AI 辅助
我们将不可压缩流体的Navier-Stokes方程的规范结构形式化，将$\nabla\cdot\mathbf{v}=0$解释为类似于电磁学中Coulomb规范的规范固定。我们构造了Martin-Siggia-Rose作用量及其完整的BRST对称性，为Gribov参数$\gamma$和单值相位$\theta$引入BRST双重态。通过Higgs机制，$\gamma$获得真空期望值$\gamma_{0}$，为涡量和鬼场产生质量尺度；我们计算了单圈有效势并分析了真空稳定性。间歇性——由指数$\zeta_{n}$度量——通过Higgs场$\sigma$在凝聚体周围的涨落来描述，通过一个受Gribov启发的log-Poisson层级，精确满足$\zeta_{3}=1$（Kolmogorov的4/5定律）。对DNS数据的拟合倾向于丝状涡量结构，其中$D_{f}\approx1$（单参数）或$D_{f}\approx2.1$，$\Delta\approx0.44$（双参数）。在$d=3$中的单圈反常维度$\Delta_{\sigma}\approx0.50$支持了$D_{f}=2\Delta_{\sigma}$的识别。该形式主义将经典流体力学与规范理论和自发对称破缺统一起来，为湍流和间歇性的研究开辟了量子场论方法的途径。
英文摘要
We formalize the gauge structure of the Navier--Stokes equation for incompressible fluids, interpreting $\nabla\cdot\mathbf{v}=0$ as a gauge fixing analogous to the Coulomb gauge in electromagnetism. We construct the Martin--Siggia--Rose action and its full BRST symmetry, introducing BRST doublets for the Gribov parameter $γ$ and the monodromy phase $θ$. Through a Higgs mechanism, $γ$ acquires a vacuum expectation value $γ_{0}$, generating a mass scale for the vorticity and the ghosts; we compute the one-loop effective potential and analyze vacuum stability. Intermittency --- measured by the exponents $ζ_{n}$ --- is described by fluctuations of the Higgs field $σ$ around the condensate, via a Gribov-inspired log-Poisson hierarchy that satisfies exactly $ζ_{3}=1$ (Kolmogorov's four-fifths law). Fits to DNS data favor filamentary vorticity structures, with $D_{f}\approx1$ (one parameter) or $D_{f}\approx2.1$, $Δ\approx0.44$ (two parameters). The one-loop anomalous dimension in $d=3$, $Δ_σ\approx0.50$, supports the identification $D_{f}=2Δ_σ$. The formalism unifies classical hydrodynamics with gauge theory and spontaneous symmetry breaking, opening the study of turbulence and intermittency to quantum field theory methods.
发表机构
- Instituto de Física, Universidade do Estado do Rio de Janeiro（里约热内卢州立大学物理研究所）
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