Relaxation of quasi-stationary states in long range interacting systems and a classification of the range of pair interactions
Résumé
Systems of particles interacting with long range interactions present generically "quasi-stationary states" (QSS), which are approximately time-independent out of equilibrium states. In this proceedings, we explore the generalization of the formation of such QSS and their relaxation from the much studied case of gravity to a generic pair interaction with the asymptotic form of the potential $v(r) \sim 1/r^\gamma$ with $\gamma > 0$ in $d$ dimensions. We compute analytic estimations of the relaxation time calculating the rate of two body collisionality in a virialized system approximated as homogeneous. We show that for $\gamma < (d-1)/2$, the collision integral is dominated by the size of the system, while for $\gamma > (d-1/2)$, it is dominated by small impact parameters. In addition, the lifetime of QSS increases with the number of particles if $\gamma < d-1$ (i.e. the force is not integrable) and decreases if $\gamma > d-1$. Using numerical simulations we confirm our analytic results. A corollary of our work gives a "dynamical" classification of interactions: the dynamical properties of the system depend on whether the pair force is integrable or not.
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