长程相互作用系统的动态稳定性：Nyquist方法在HMF模型中的应用
Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model
摘要：
本文将Nyquist方法应用于Hamilton平均场（HMF）模型，以解决空间齐次分布函数关于Vlasov方程的线性动力学稳定性问题。我们考虑了Maxwell（等温）和Tsallis（多方）分布的情形，证明了系统在临界动力学温度T_c以上是稳定的，在此温度以下是不稳定的，然后考虑了由两个偏心Maxwell分布叠加而成的对称双峰分布，并在稳定性图中证明了重入相的存在。当我们考虑不对称双峰分布时，在不对称因子Delta>1.09的临界值以上，重入相消失。我们还考虑了具有排斥相互作用的HMF模型。在这种情况下，单峰分布总是稳定的。对于不对称双峰分布，1<Delta<25.6时有一个重入相，25.6<Delta 43.9时有一个二重入相<43.9 and no re-entrant phase for Delta>。最后，我们将结果推广到任意相互作用势，并提到了HMF模型、库仑等离子体和引力系统之间的联系。讨论了线性动力学稳定性与形式非线性动力学稳定性之间的关系，并证明了它们在空间均匀分布下的等价性。我们还提供了一个非齐次系统的动力学稳定性判据。
英文摘要：
We apply the Nyquist method to the Hamiltonian Mean Field (HMF) model in order to settle the linear dynamical stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature T_c and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered Maxwellians, and show the existence of a re-entrant phase in the stability diagram. When we consider an asymmetric double-humped distribution, the re-entrant phase disappears above a critical value of the asymmetry factor Delta>1.09. We also consider the HMF model with a repulsive interaction. In that case, single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for 1<Delta<25.6, a double re-entrant phase for 25.6<Delta<43.9 and no re-entrant phase for Delta>43.9. Finally, we extend our results to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational systems. We discuss the relation between linear dynamical stability and formal nonlinear dynamical stability and show their equivalence for spatially homogeneous distributions. We also provide a criterion of dynamical stability for inhomogeneous systems.
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