For a classical system with long-range interactions, a soft mode exists whenever a stationary state spontaneously breaks a continuous symmetry of the Hamiltonian. Besides that, if the corresponding coordinate associated to the symmetry breaking is periodic, then the same energy of the different stationary states and finite $N$ thermal fluctuations result in a superdiffusive motion of the center of mass for total zero momentum, that tends to a normal diffusion for very long times. As examples of this, we provide a two-dimensional self-gravitating system, a free electron laser, and the Hamiltonian mean-field (HMF) model. For the latter, a detailed theory for the motion of the center of mass is given. We also discuss how the coupling of the soft mode to the mean-field motion of individual particles may lead to strong chaotic behavior for a finite particle number, as illustrated by the HMF model.

中文翻译：

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远程交互系统中的经典Goldstone模式

对于具有长距离相互作用的经典系统，只要稳态自发地破坏哈密顿量的连续对称性，就会存在软模式。除此之外，如果与对称破坏相关的相应坐标是周期性的，那么不同稳态和有限能量的相同能量$ñ$热涨落导致质心针对总零动量的超扩散运动，这往往会在很长一段时间内正常扩散。作为示例，我们提供了二维自重系统，自由电子激光器和哈密顿平均场（HMF）模型。对于后者，给出了质心运动的详细理论。我们还讨论了如何将软模式耦合到单个粒子的平均场运动，从而如何在有限的粒子数下导致强烈的混沌行为，如HMF模型所示。