Low dimensional features of the HMF model

The order parameter of the Hamiltonian Mean Field (HMF) model, which describes the motion of N globally coupled rotors, is a two-component vector [Mx = ∑i N cos(θi)/N, My = ∑i N sin{θi)/N). Its dynamics is found to be "cyclic": the vector approaches a fixed norm and rotates around the origin. We here show that, using the equations of motion of the HMF model and making a crucial Ansatz, one can derive an infinite set of second order differential equations involving the "moments" of the particle distribution (Mx (k) = ∑i N cos(kθi)/N, My (k) = ∑i N sin(kθi)/N). The first of such equations (k = 1) describes the motion of the order parameter It is also shown that, at equilibrium, the amplitude of the moments rapidly decreases with k and, hence, only a few moments (expecially at high enough energy) should describe the dynamics of the system. This could explain the prominent low-dimensional features found in the dynamics of the HMF model. Numerical experiments partly confirm this picture, but show the presence of strong instabilities in the set of differential equations. Finally, we find a specific solution of the infinite set of equations, which has some correspondence with numerical observations