CLUSTERING AND RELAXATION IN HAMILTONIAN LONG-RANGE DYNAMICS

We study the dynamics of a fully coupled network of N classical rotato rs, 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-P oisson 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 div erges with N as NT2 in the ferromagnetic case and as NT3/2 in the anti ferromagnetic case. The N dependence of the relaxation time is suggest ed by an analogy of the HMF model with gravitational and charged sheet s dynamics in one dimension, and is verified in numerical simulations. Below the critical temperature the ferromagnetic HMF model shows a co llective phenomenon where the rotators form a drifting cluster; we arg ue that the drifting speed vanishes as N--1/2 but increases as one app roaches 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-held antiferromagnetic system.