Formation and relaxation of quasi-stationary states in particle systems with power law interactions

Abstract

We explore the formation and relaxation of so-called quasi-stationary states (QSS) for particle distributions in three dimensions interacting via an attractive radial pair potential $V(r \rightarrow \infty) \sim 1/r^\gamma$ with $\gamma > 0$, and either a soft-core or hard-core regularization at small $r$. In the first part of the paper we generalize, for any spatial dimension $d \geq 2$, Chandrasekhar's approach for the case of gravity to obtain analytic estimates of the rate of collisional relaxation due to two body collisions. The resultant relaxation rates indicate an essential qualitative difference depending on the integrability of the pair force at large distances: for $\gamma >d-1$ the rate diverges in the large particle number $N$ (mean field) limit, unless a sufficiently large soft core is present; for $\gamma < d-1$, on the other hand, the rate vanishes in the same limit even in the absence of any regularization. In the second part of the paper we compare our analytical predictions with the results of extensive parallel numerical simulations in $d=3$, for a range of different exponents $\gamma$ and soft cores leading to the formation of QSS. We find, just as for the previously well studied case of gravity (which we also revisit), excellent agreement between the parametric dependence of the observed relaxation times and our analytic predictions. Further, as in the case of gravity, we find that the results indicate that, when large impact factors dominate, the appropriate cut-off is the size of the system (rather than, for example, the mean inter-particle distance). Our results provide strong evidence that the existence of QSS is robust only for long-range interactions with a large distance behavior $\gamma < d-1$; for $\gamma \geq d-1$ the existence of such states will be conditioned strongly on the short range properties of the interaction.