Paper detail

Energy cascade with small-scales thermalization, counterflow metastability and anomalous velocity of vortex rings in Fourier-truncated Gross-Pitaevskii equation

The statistical equilibria of the dynamics of the Gross-Pitaevskii Equation (GPE) with a finite range of spatial Fourier modes are characterized using a new algorithm, based on a stochastically forced Ginzburg-Landau equation (SGLE), that directly generates grand canonical distributions.. The SGLE-generated distributions are validated against finite-temperature GPE-thermalized states and exact low-temperature results. A standard second-order $λ$-transition is exhibited. A new mechanism of GPE thermalization through a direct cascade of energy is found using initial conditions with mass and energy distributed at large scales. A long transient with partial thermalization at small-scales is observed. Vortices are shown to disappear as a prelude to final thermalization and their annihilation is related to the contraction of vortex rings due to mutual friction. Increasing the amount of dispersion at truncation wavenumber is shown to slowdown thermalization and vortex annihilation. A bottleneck that produces spontaneous effective self truncation with partial thermalization is characterized in the limit of large dispersive effects. Metastable counter-flow states are generated using the SGLE algorithm. Spontaneous nucleation of vortex ring is observed and the corresponding Arrhenius law is characterized. Dynamical counter-flow effects on vortex evolution are investigated. Longitudinal effects are produced and measured. A dilatation of vortex rings is obtained for larger counterflows. The vortex ring longitudinal velocity has a strong dependence on temperature, an effect that is related to the presence of finite-amplitude Kelvin waves. This anomalous vortex ring velocity is quantitatively reproduced by assuming equipartition of energy of the Kelvin waves. Orders of magnitude are given for the predicted effects in weakly interacting Bose-Einstein condensates and superfluid $^4{\rm He}$.