具有长程相互作用的空间均质系统的动力学理论：III。应用于幂律势、等离子体、恒星系统和HMF模型
Kinetic theory of spatially homogeneous systems with long-range interactions: III. Application to power-law potentials, plasmas, stellar systems, and to the HMF model
摘要：
我们将长程相互作用系统的动力学理论的一般结果应用于特定的物理系统。我们考虑在d维空间中相互作用r^{-γ}的排斥和吸引幂律势，其中γ<d.当γ>γ_c=（d-1）/2时，必须考虑强碰撞的影响，系统的演化服从Boltzmann方程或修正的Landau方程;当γ<γ_c时，强碰撞可以忽略，系统的演化服从Lenard-Balescu方程。在临界情形γ=γ_c下，我们可以用Landau方程（适当地调整截断值）作为Boltzmann方程和Lenard-Balescu方程的近似。在小尺度上的发散，出现在原来的朗道方程是正规化的强碰撞的影响。在与中和背景（例如等离子体）的排斥相互作用的情况下，在原始朗道方程中出现的大尺度发散被德拜屏蔽的集体效应正则化。在吸引相互作用（例如重力）的情况下，它被系统的空间不均匀性及其有限范围正则化。我们给出了扩散系数、摩擦系数和弛豫时间的解析表达式，它取决于指数γ的值和空间d的维数。我们在一个统一的框架内处理库仑等离子体和恒星系统的情况下，在不同的空间维度，和吸引力和排斥HMF模型的情况下。
英文摘要：
We apply the general results of the kinetic theory of systems with long-range interactions to particular systems of physical interest. We consider repulsive and attractive power-law potentials of interaction r^{-γ} with γ<d in a space of dimension d. For γ>γ_c= (d-1)/2, strong collisions must be taken into account and the evolution of the system is governed by the Boltzmann equation or by a modified Landau equation; for γ<γ_c, strong collisions are negligible and the evolution of the system is governed by the Lenard-Balescu equation. In the marginal case γ=γ_c, we can use the Landau equation (with appropriately justified cut-offs) as a relevant approximation of the Boltzmann and Lenard-Balescu equations. The divergence at small scales that appears in the original Landau equation is regularized by the effect of strong collisions. In the case of repulsive interactions with a neutralizing background (e.g. plasmas), the divergence at large scales that appears in the original Landau equation is regularized by collective effects accounting for Debye shielding. In the case of attractive interactions (e.g. gravity), it is regularized by the spatial inhomogeneity of the system and its finite extent. We provide explicit analytical expressions of the diffusion and friction coefficients, and of the relaxation time, depending on the value of the exponent γand on the dimension of space d. We treat in a unified framework the case of Coulombian plasmas and stellar systems in various dimensions of space, and the case of the attractive and repulsive HMF models.
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