Current bottleneck in an inhomogeneous condensate (${\rho }_L{v}_L={\rho }_R{v}_R$)

Stationary one-dimensional flow has constant particle current $j=\rho v$. A lower-density segment must flow faster at the same current and also has smaller local speed of sound $\sqrt{g\rho /m}$. It can therefore lose local phonon energetic stability before a denser segment.

Combining zero-current bulk densities with quantized circulation gives an approximate winding bound. If equal halves have densities ${\rho }_0(1-a)$ and ${\rho }_0(1+a)$, $0<a<1$, the more restrictive local condition is $n^2<(m{R}^2{\mu }_0/{\hslash }^2)(1-a)/(1+a{)}^2$. A self-consistent finite-current calculation must also allow kinetic pressure to change those densities.