从泛函重整化群看Kuramoto-Sivashinsky方程的标度律
Scaling regimes of the Kuramoto-Sivashinsky equation from the functional renormalization group
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中文总结 AI 辅助
研究一维随机Kuramoto-Sivashinsky方程的重整化群方法，指出基于尖锐截断的微扰Wilsonian RG方法无效，采用泛函重整化群框架建立流方程，计算关联函数得出三种普适标度律，研究确定性方程确定KPZ标度律出现尺度。
中文摘要 AI 辅助
我们重新审视了一维随机Kuramoto-Sivashinsky（KS）方程的重整化群（RG）方法，表明先前基于具有尖锐截断的微扰Wilsonian RG方法无效，虽定性正确但因动量导数问题导致内在发散。采用泛函重整化群（FRG）框架可解决，我们建立了KS模型的流方程，证明其在大尺度流向Kardar-Parisi-Zhang（KPZ）不动点。计算了广泛动量和频率范围内的全两点关联函数，展示了三种普适标度律，还研究了确定性KS方程在噪声消失极限下的大尺度行为及KPZ标度律出现的尺度。
英文摘要
We revisit the renormalization group (RG) approach to the one-dimensional stochastic Kuramoto-Sivashinsky (KS) equation and show that previous approaches based on perturbative Wilsonian RG with a sharp cutoff are not valid, even though they yield a qualitatively correct picture. The reason is that taking momentum derivatives while using the sharp cutoff is not well-defined in some cases and leads to intrinsic divergencies. This is a well-known problem of Wilsonian RG, which can be simply cured by using a smooth cutoff, and employing the functional renormalization group (FRG) framework. We establish the flow equations for the KS model within the FRG, and demonstrate that it flows to the Kardar-Parisi-Zhang (KPZ) fixed point at large scales. We then calculate the full two-point correlation function over a wide range of momenta and frequencies. We show that it exhibits three universal scaling regimes that we characterize: the KPZ regime (with dynamical exponent $z=3/2$), the Edwards-Wilkinson regime (with $z=2$) and the recently discovered inviscid regime (with $z=1$). The latter develops over an extended range of large wavenumbers and originates from the vanishing of the effective viscosity. Lastly, we investigate the large-scale behavior of the deterministic KS equation by studying the limit of vanishing noise and we determine the scales where the KPZ regime can emerge in the deterministic case.
