Quasi-stationary states and the range of pair interactions

TL;DR

This paper explores the conditions under which quasi-stationary states occur in systems with long-range pair interactions, revealing dependence on the interaction's decay rate and the presence of a soft-core.

Contribution

It generalizes existing analytic models for gravity to a broader class of pair interactions and provides numerical validation of the conditions for quasi-stationary states.

Findings

Quasi-stationary states exist for interactions with decay rate a < d-1.

For a > d-1, a soft-core in the potential is necessary for quasi-stationary states.

The large-distance behavior of the interaction determines the existence of quasi-stationary states.

Abstract

"Quasi-stationary" states are approximately time-independent out of equilibrium states which have been observed in a variety of systems of particles interacting by long-range interactions. We investigate here the conditions of their occurrence for a generic pair interaction V(r \rightarrow \infty) \sim 1/r^a with a > 0, in d>1 dimensions. We generalize analytic calculations known for gravity in d=3 to determine the scaling parametric dependences of their relaxation rates due to two body collisions, and report extensive numerical simulations testing their validity. Our results lead to the conclusion that, for a < d-1, the existence of quasi-stationary states is ensured by the large distance behavior of the interaction alone, while for a > d-1 it is conditioned on the short distance properties of the interaction, requiring the presence of a sufficiently large soft-core in the interaction potential.