= Cylindrical pump-loss balance
{title2=$\int_S n^2dS=(\gamma/\Gamma)\int_S n\,dS$}
For a stationary pumped condensate in a closed cross-section, integrate the <gain-loss Madelung equations>. Vanishing lateral <condensate number current> and constant integrated axial current imply the displayed balance. It fixes a density-weighted mean, not the pointwise density. If the integrated axial current $J(z)$ varies, the exact relation instead is $\int_S n^2dS=(\gamma/\Gamma)N(z)-[\hbar/(2\Gamma)]J'(z)$. Thus lateral no-flux conditions alone are insufficient for a slice-by-slice balance in an axially inhomogeneous state.
Back to article page
