Classical Goldstone modes in long-range interacting systems

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.