摘要翻译：
从Liouville方程出发，导出了二维单种点涡气体的约化分布函数所满足的精确方程组。在适当的热力学极限$n~+infty$范围内，考虑解的1/n次幂展开，并忽略一些集体效应，我们导出了光滑涡度场满足的动力学方程，该方程在$O(1/n)$级有效。这个方程是以前[P.H.Chavanis,Phys.Rev.E,64,026309(2001)]从一个更抽象的投影算子形式中得到的。如果我们考虑轴对称流动，做一个马尔可夫近似，我们得到了一个更简单的动力学方程，可以进行更详细的研究。我们讨论了这些动力学方程关于$H$-定理和收敛（或不收敛）到统计平衡态的性质。我们还通过显式地计算两体相关函数在线性区域中的时间演化来研究相关的增长。在本文的第二部分，我们考虑了场涡浴中试验涡的弛豫，通过直接计算试验涡位置增量的第二（扩散）矩和第一（漂移）矩，得到了Fokker-Planck方程。我们的方法的一个特殊之处是得到具有明确物理意义的一般方程，这些方程适用于不一定是轴对称的流动，并考虑到非马尔可夫效应。然而，我们的方法的一个局限性是它忽略了集体效应。
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英文标题：
《Kinetic theory of two dimensional point vortices from a BBGKY-like
  hierarchy》
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作者：
P.H. Chavanis
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最新提交年份：
2008
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分类信息：

一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
  Starting from the Liouville equation, we derive the exact hierarchy of equations satisfied by the reduced distribution functions of the single species point vortex gas in two dimensions. Considering an expansion of the solution in powers of 1/N in a proper thermodynamic limit $N\to +\infty$, and neglecting some collective effects, we derive a kinetic equation satisfied by the smooth vorticity field which is valid at order $O(1/N)$. This equation was obtained previously [P.H. Chavanis, Phys. Rev. E, 64, 026309 (2001)] from a more abstract projection operator formalism. If we consider axisymmetric flows and make a markovian approximation, we obtain a simpler kinetic equation which can be studied in great detail. We discuss the properties of these kinetic equations in regard to the $H$-theorem and the convergence (or not) towards the statistical equilibrium state. We also study the growth of correlations by explicitly calculating the time evolution of the two-body correlation function in the linear regime. In a second part of the paper, we consider the relaxation of a test vortex in a bath of field vortices and obtain the Fokker-Planck equation by directly calculating the second (diffusion) and first (drift) moments of the increment of position of the test vortex. A specificity of our approach is to obtain general equations, with a clear physical meaning, that are valid for flows that are not necessarily axisymmetric and that take into account non-Markovian effects. A limitations of our approach, however, is that it ignores collective effects.
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PDF链接：
https://arxiv.org/pdf/704.3953