论文标题
二维点涡流的动力学理论和波动散文定理
Kinetic theory of two-dimensional point vortices and fluctuation-dissipation theorem
论文作者
论文摘要
我们完成了先前工作中启动的二维（2D）点涡度的动力学理论。我们使用更简单，更多的身体形式主义。我们考虑提交给小的外部随机扰动的2D点涡流系统，并确定系统对扰动的响应。我们通过测试涡流的极化得出扩散系数和漂移。我们介绍了一个涉及扩散项和漂移项的一般福克 - 普兰克方程。当可以忽略极化的漂移时，我们获得了由外部噪声产生的世俗装扮扩散（SDD）方程。当外部扰动是由$ n $ point涡流的离散集合而产生的，当忽略集体效应时，我们会获得lenard-balescu样动力学方程，从而减少到类似Landau的动力学方程。我们考虑一个多物种涡流系统。我们讨论了单个和多物种情况下动力学阻断的过程。当场涡流处于统计平衡（热浴）时，我们为2D点涡流的波动散动作定理的正确表达与系统响应函数有关。在这种情况下，漂移系数和扩散系数满足类似爱因斯坦的关系，而Fokker Planck方程将减少到类似Smoluchowski的方程。我们提到了2D点涡流和恒星系统之间的类比。特别是，在2D流体动力学中，点涡流的漂移是Chandrasekhar动力学摩擦在天体物理学中的对应物。我们还考虑了由$ N $耦合随机Langevin方程描述的2D Brownian Point涡旋的气体，并确定其平均值和介质进化。在本文中，我们处理单向流的情况，但我们的结果可以直接概括为轴对称流。
We complete the kinetic theory of two-dimensional (2D) point vortices initiated in previous works. We use a simpler and more physical formalism. We consider a system of 2D point vortices submitted to a small external stochastic perturbation and determine the response of the system to the perturbation. We derive the diffusion coefficient and the drift by polarization of a test vortex. We introduce a general Fokker-Planck equation involving a diffusion term and a drift term. When the drift by polarization can be neglected, we obtain a secular dressed diffusion (SDD) equation sourced by the external noise. When the external perturbation is created by a discrete collection of $N$ point vortices, we obtain a Lenard-Balescu-like kinetic equation reducing to a Landau-like kinetic equation when collective effects are neglected. We consider a multi-species system of point vortices. We discuss the process of kinetic blocking in the single and multi-species cases. When the field vortices are at statistical equilibrium (thermal bath), we establish the proper expression of the fluctuation-dissipation theorem for 2D point vortices relating the power spectrum of the fluctuations to the response function of the system. In that case, the drift coefficient and the diffusion coefficient satisfy an Einstein-like relation and the Fokker Planck equation reduces to a Smoluchowski-like equation. We mention the analogy between 2D point vortices and stellar systems. In particular, the drift of a point vortex in 2D hydrodynamics is the counterpart of the Chandrasekhar dynamical friction in astrophysics. We also consider a gas of 2D Brownian point vortices described by $N$ coupled stochastic Langevin equations and determine its mean and mesoscopic evolution. In the present paper, we treat the case of unidirectional flows but our results can be straightforwardly generalized to axisymmetric flows.
