局部势平均场模型中非均匀态的稳定性
Stability of inhomogeneous states in mean-field models with a local potential
摘要：
众所周知，弗拉索夫方程可以很好地描述平均场系统在$N \to \infty$极限下的动力学行为。该方程具有无穷多的定态和{ \it齐次}态的情况，其中单粒子分布函数与空间变量无关，很好地解析表征。另一方面，非齐次的情况下，往往需要一些近似的分析处理：动力学，然后最好的处理在作用角变量，和潜在的产生不均匀性，通常是非常复杂的，在这些新的变量。在这里，我们对待解析的玩具模型的线性稳定性的不均匀性是由外部场。将Vlasov方程转化为作用角变量，我们推导出一个色散关系，我们完成解决两个特定的模型的不稳定性和稳定性阈值的增长率：具有额外的不对称性的哈密顿平均场模型和平均场$ϕ^4$模型。计算结果与$N$ -体动力学数值模拟结果进行了比较。当{ \it非齐次}态是定态时，我们期望在$N$体动力学中观察到准定态（QSS），其寿命与$N$代数发散。
英文摘要：
The Vlasov equation is well known to provide a good description of the dynamics of mean-field systems in the $N \to \infty$ limit. This equation has an infinity of stationary states and the case of {\it homogeneous} states, for which the single-particle distribution function is independent of the spatial variable, is well characterized analytically. On the other hand, the inhomogeneous case often requires some approximations for an analytical treatment: the dynamics is then best treated in action-angle variables, and the potential generating inhomogeneity is generally very complex in these new variables. We here treat analytically the linear stability of toy-models where the inhomogeneity is created by an external field. Transforming the Vlasov equation into action-angle variables, we derive a dispersion relation that we accomplish to solve for both the growth rate of the instability and the stability threshold for two specific models: the Hamiltonian Mean-Field model with additional asymmetry and the mean-field $ϕ^4$ model. The results are compared with numerical simulations of the $N$-body dynamics. When the {\it inhomogeneous} state is stationary, we expect to observe in the $N$-body dynamics Quasi-Stationary-States (QSS), whose lifetime diverge algebraically with $N$.
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