arXiv 2406.145752024-06-16cond-mat.stat-mech

$d \rightarrow \infty$ 下的湍流理论：重整化群的四环近似

Theory of Turbulence for $d \rightarrow \infty$: Four-Loop Approximation of the Renormalization Group

Loran Ts. Adzhemyan, Yury Kirienko

备注：: 13 pages

AI总结：

本研究在充分发展湍流的重整化群框架下，针对高维 $d\rightarrow\infty$ 情形计算四阶微扰 $\beta$ 函数，确定重整化群不动点位置与红外稳定指数 $\omega$，通过费曼图抵消简化计算，得到 $\omega$ 的四阶 $\varepsilon$ 展开式。

摘要：

在充分发展湍流随机模型的重整化群方法框架内，针对高维空间 $d \rightarrow \infty$ 的情形，计算了微扰论四阶下的 $\beta$ 函数。确定了 $\varepsilon$ 展开四阶下重整化群不动点的位置，并计算了表征该点红外稳定性的指数 $\omega$。我们证明了通过相互抵消可大幅减少费曼图数量的可能性。所得结果使我们能够得到指数 $\omega$ 的 $\varepsilon$ 展开的四项：$$ \omega= 2\\,\varepsilon + \frac{2}{3}\\,\varepsilon^2 + \frac{10}{9}\\,\varepsilon^3 + \frac{56}{27}\\,\varepsilon^4. $$

英文摘要：

Within the framework of the renormalization group approach in the stochastic model of fully developed turbulence, the $β$-function has been calculated in the fourth order of perturbation theory for high-dimensional spaces $d \rightarrow \infty$. The position of the fixed point of the renormalization group in the fourth order of the $\varepsilon$-expansion has been determined, and the index $ω$, which defines the infrared stability of this point, has been calculated. We demonstrate the possibility of significantly reducing the number of Feynman diagrams through mutual cancellation. The results obtained allow us to find the four terms of the $\varepsilon$-expansion of the index $ω$: $$ ω= 2\,\varepsilon + \frac{2}{3}\,\varepsilon^2 + \frac{10}{9}\,\varepsilon^3 + \frac{56}{27}\,\varepsilon^4. $$

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摘要：

英文摘要：

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