Kinetic theory of vortex gases in low-dimensional superfluids

To study strong turbulence in two-dimensional (2D) quantum gases, it must be taken into account the excitation of topological defects, the quantum vortices, and their dynamical interaction. The quantum vortices appear naturally in perturbed superfluids, particularly in atomic Bose-Einstein condensates (BECs).

These structures are stationary solutions of the Gross-Pitaeskii equation (GPE), which is the equation that describes the evolution of quantum fluids, like BECs. Given that quantum vortices are a manifestation of quantum turbulence, a better understanding of turbulence could come from having a description of an ensemble of these excitations dynamically interacting.

We start by going back to the GPE and study the dynamics of a single vortex, where we propose a correction to the vortex wavefunction that takes into consideration its motion in the BEC. With this correction, we are able to derive a vortex mass. Subsequently, we study the interaction of two vortices and obtain an interaction pairwise potential.

Finally, we are in a position to focus on the Vlasov equation, from which we derive a kinetic model for a one species and two species vortex gas, encapsulating its statistical and dynamical features. We observe that the dispersion relation does not depend on the sign of the vortex gas, but only on its initial velocity and density, being constant when initialized with zero initial velocity or with an initial velocity perpendicular to the wave vector. This allows us to have a completely analytical kinetic description of strong quantum turbulence in low-dimensional quantum fluids.