Phase-space origin of superfluid stability in ring Bose-Einstein condensates

We present a kinetic description of superfluid currents in ring-shaped Bose-Einstein condensates based on the Wigner phase-space formalism. Starting from the Gross-Pitaevskii equation in a toroidal geometry, we derive a Vlasov-type equation for the angular Wigner function, in which the mean-field interaction generates an effective force proportional to the density gradient. Within this framework, we obtain the dispersion relation of collective modes and recover the Bogoliubov spectrum in the long-wavelength limit. We show that the Landau criterion for superfluidity can be interpreted as the absence of resonant phase-space trajectories satisfying the condition $\omega = q v_\ell$. In a ring geometry, the quantization of angular momentum leads to a discrete set of velocities, which suppresses the availability of resonant states and strongly inhibits Landau damping. In contrast, in the continuous limit $R \to \infty$, the spectrum becomes quasi-continuous and the standard Landau damping mechanism is recovered, establishing a direct connection between kinetic resonances and the energetic criterion for superfluidity. We further analyze the role of Bogoliubov depletion by considering a finite-width angular momentum distribution. Although resonant states formally exist in this case, we show that, for flow velocities below the sound velocity, the phase-space distribution does not provide the gradients required for energy transfer, and the superfluid current remains dynamically stable. Our results provide a unified phase-space interpretation of superfluidity, highlighting the role of angular momentum quantization and the structure of the distribution function in determining the stability of persistent currents.