Superfluidity

Remarks

The argument: let the fluid flow with velocity $\mathbf{v}$ relative to a wall. An excitation of momentum $\mathbf{p}$, measured in the fluid frame, has energy $\varepsilon(p) + \mathbf{p} \cdot \mathbf{v}$ in the wall frame. It can appear spontaneously only if this energy is negative, which requires $\abs{\mathbf{v}} > \varepsilon(p)/p$. Example: for the weakly interacting Bose gas of Gross–Pitaevskii theory, the Bogoliubov spectrum is $\varepsilon(p) = \sqrt{c^2 p^2 + (p^2 / 2m)^2}$ with sound speed $c > 0$; then $\varepsilon(p)/p = \sqrt{c^2 + p^2/(4m^2)} \ge c$, so $v_c = c > 0$. Non-example: for the ideal Bose gas, $\varepsilon(p) = p^2 / 2m$, so $\varepsilon(p)/p = p/(2m) \to 0$ and $v_c = 0$; it has a condensate but is not superfluid. In the hydrodynamic description, the superfluid is described by a wave function $\psi = \sqrt{n} \, e^{i\theta}$ and has velocity $\mathbf{v}_s = (\hbar / m) \nabla \theta$, where $\hbar$ is the reduced Planck constant. At positive temperature the two-fluid model splits the fluid into a superfluid part and a normal viscous part. In helium-4 the observed critical speeds are usually much lower than Landau's value, because vortices are created first.