Violent relaxation in quantum fluids with long-range interactions

Abstract

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 quasistationary 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 scales and timescales 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.

Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model

“…In the optical soliton literature this solution is called the fundamental soliton and the coordinate z is replaced by z − vt representing a shape-preserving waveform propagating at the group velocity v (there is also a multiplicative phase factor we shall not specify here) [11]. By contrast, the LRI in the HMF model lead to a nonlocal nonlinearity [116,117]. The HMF model lives on a ring so that the wavefunction obeys periodic boundary conditions Ψ(θ, t) = Ψ(θ + 2π, t) and it also does not include an explicit potential V ext (x).…”

“…Motivation for studying the quantum problem comes both from cold atom experiments and the Bose star picture of dark matter mentioned above. The equilibrium states of the quantum HMF model have been examined in [115,116] and the dynamics, including violent relaxation, were recently studied in [117] where it was found that the automatic coarse-graining of phase space at the level of Planck's constant h can strongly modify the relaxation in the deep quantum regime. Furthermore, in its quantum form the HMF model bears a resemblance to the CS model mentioned above, which, when defined on a finite domain with periodic boundary conditions, has a pairwise interaction V (θ i −θ j ) = 1/ sin 2 (θ i −θ j ).…”

Caustics in quantum many-body dynamics

“…tion (nonlinear Schrödinger equation) which can equally describe a self-interacting condensate [125] or nonlinear optics in a fibre [101]. Nevertheless, all these examples are morally similar to the classical wave scenario found in optics and hydrodynamics, whether linear or nonlinear.…”

Determination of Intervening Pillar Thickness Based on the Cusp Catastrophe Model

“…e catastrophe theory, as a branch of the nonlinear theory, can study the characteristics of a system changing with the change of its control parameters, especially when the parameters change the performance of the system to cause catastrophe under certain conditions [10][11][12]. At present, the catastrophe theory has been widely used in economics and physics [13,14]. Due to failure of rock mass with the characteristics of energy accumulation and sudden release, in recent years, many achievements have been made in the research of rock mass instability based on the catastrophe theory.…”