Violent relaxation in two-dimensional flows with varying interaction range

Understanding the relaxation of a system towards equilibrium is a long-standing problem in statistical mechanics. Here we address the role of long-range interactions in this process by considering a class of two-dimensional flows where the interaction between fluid particles varies with the distance as ∼r(α-2) for α>0. We find that changing α with a prescribed initial state leads to different flow patterns: for small α, a coarsening process leads to the formation of a sharp interface between two regions of homogenized α-vorticity; for large α, the flow is attracted to a stable dipolar structure through a filamentation process. Assuming that the energy E and the enstrophy Z are injected at a typical scale smaller than the domain scale L, we argue that convergence towards the equilibrium state is expected when the parameter (2π/L)(α)E/Z tends to one, while convergence towards a dipolar state occurs systematically when this parameter tends to zero. This suggests that weak long-range interacting systems are more prone to relax towards an equilibrium state than strong long-range interacting systems.