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.
Laboratoire de Physique, École Normale Supérieure de Lyon, Université de Lyon, CNRS, 46 Allée d'Italie, F-69364 Lyon, cedex 07, France.
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Laboratoire de Physique, École Normale Supérieure de Lyon, Université de Lyon, CNRS, 46 Allée d'Italie, F-69364 Lyon, cedex 07, France.
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Dipartimento di Fisica e Astronomia and CSDC, Università di Firenze, INFN and CNISM, via G. Sansone, 1 50019 Sesto Fiorentino, Italy.
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