  Violent relaxation is a process that occurs in systems with long-range
interactions. It has the peculiar feature of dramatically amplifying small
perturbations, and rather than driving the system to equilibrium it instead
leads to slowly evolving configurations known as quasi-stationary states that
fall outside the standard paradigm of statistical mechanics. Violent relaxation
was originally identified in gravity-driven stellar dynamics; here we extend
the theory into the quantum regime by developing a quantum version of the
Hamiltonian Mean Field (HMF) model which exemplifies many of the generic
properties of long-range interacting systems. The HMF model can either be
viewed as describing particles interacting via a cosine potential, or
equivalently as the kinetic XY-model with infinite range interactions, and its
quantum fluid dynamics can be obtained from a generalized Gross-Pitaevskii
equation. We show that singular caustics that form during violent relaxation
are regulated by interference effects in a universal way described by Thom's
catastrophe theory applied to waves and this leads to emergent length and time
scales not present in the classical problem. In the deep quantum regime we find
that violent relaxation is suppressed altogether by quantum zero-point motion.
Our results are relevant to laboratory studies of self-organization in cold
atomic gases with long-range interactions.

Statistical Mechanics