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arXiv 0903.0956cond-mat.stat-mechastro-ph.COphysics.flu-dyn
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具有白噪声初速度的Burgers方程的一些统计性质

Some statistical properties of the Burgers equation with white-noise initial velocity

P. Valageas

备注：
29 pages, published in J. Stat. Phys
发表信息：
J. Stat. Phys., (2009), 137, 729
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摘要：

我们重新审视一维Burgers方程的白噪声初速度的无粘极限。我们推导出速度和拉格朗日增量的概率分布，测量任何长度的间隔$x$。这也给出了速度结构函数。接下来，对于初始密度均匀的情况，我们得到密度在任何尺度$x$上的分布，并且我们导出密度两点相关和功率谱。最后，我们考虑拉格朗日位移场，我们推导出的拉格朗日映射的增量分布。我们检查，这给出了众所周知的质量函数的冲击。对于所有的分布，我们描述了在大尺度和小尺度极限中获得的极限尺度函数。我们还讨论了这些结果如何推广到其他初始条件，或更高的维度，并作出了启发式的多重分形形式主义的连接。我们注意到，点状质量的形成通常导致密度分布的普遍小尺度标度，这在宇宙学背景下被称为“稳定聚类ansatz”（其中Burgers动力学也被称为“粘附模型”）。

英文摘要：

We revisit the one-dimensional Burgers equation in the inviscid limit for white-noise initial velocity. We derive the probability distributions of velocity and Lagrangian increments, measured on intervals of any length $x$. This also gives the velocity structure functions. Next, for the case where the initial density is uniform, we obtain the distribution of the density, over any scale $x$, and we derive the density two-point correlation and power spectrum. Finally, we consider the Lagrangian displacement field and we derive the distribution of increments of the Lagrangian map. We check that this gives back the well-known mass function of shocks. For all distributions we describe the limiting scaling functions that are obtained in the large-scale and small-scale limits. We also discuss how these results generalize to other initial conditions, or to higher dimensions, and make the connection with a heuristic multifractal formalism. We note that the formation of point-like masses generically leads to a universal small-scale scaling for the density distribution, which is known as the ``stable-clustering ansatz'' in the cosmological context (where the Burgers dynamics is also known as the ``adhesion model'').

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