Clustering and relaxation in Hamiltonian long-range dynamics

We study the dynamics of a fully coupled network of N classical rotators, which can also be viewed as a mean-field XY Heisenberg (HMF) model, in the attractive (ferromagnetic) and repulsive (antiferromagnetic) cases. The exact free energy and the spectral properties of a Vlasov-Poisson equation give hints on the values of dynamical observables and on time relaxation properties. At high energy (high temperature T) the system relaxes to Maxwellian equilibrium with vanishing magnetization, but the relaxation time to the equilibrium momentum distribution diverges with N as NT^2 in the ferromagnetic case and as NT^3/2 in the antiferromagnetic case. The N dependence of the relaxation time is suggested by an analogy of the HMF model with gravitational and charged sheets dynamics in one dimension and is verified in numerical simulations. Below the critical temperature the ferromagnetic HMF mode shows a collective phenomenon where the rotators form a drifting cluster; we argue that the drifting speed vanishes as N^-1/2 but increases as one approaches the critical point (a manifestation of critical slowing down). For the antiferromagnetic HMF model a two-cluster drifting state with zero magnetization forms spontaneously at very small temperatures; at larger temperatures an initial density modulation produces this state, which relaxes very slowly. This suggests the possibility of exciting magnetized states in a mean-field antiferromagnetic system.