Violent relaxation in two-dimensional flows with varying interaction range.

Understanding the relaxation of a system towards equilibrium is a longstanding 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 $\sim r^{\alpha-2}$ for $\alpha>0$. We find that changing $\alpha$ with a prescribed initial state leads to different flow patterns: for small $\alpha$, a coarsening process leads to the formation of a sharp interface between two regions of homogenized $\alpha$-vorticity; for large $\alpha$, 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 $\left(\frac{2\pi}{L}\right)^\alpha \frac{E}{Z}$ 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.