Mixing and Ergodicity in Systems with Long-Range Interactions

We present a theory of collisionless relaxation in systems with long-range interactions. Contrary to Lynden-Bell's theory of violent relaxation, which assumes global ergodicity and mixing, we show that quasi-stationary states (qSS) observed in these systems exhibit broken global ergodicity. We propose that relaxation towards equilibrium occurs through a process of local mixing, where particles spread over energy shells defined by the manifold to which their trajectories are confined. To demonstrate our theory, we study the Hamiltonian Mean Field (HMF) model, a paradigmatic system with long-range interactions. Our theory accurately predicts the particle distribution functions in qSS observed in molecular dynamics simulations without any adjustable parameters. Additionally, it precisely forecasts the phase transitions observed in the HMF model.
Comment: 6 pages, 4 figures