= Solution
Here equilibrium means a time-independent <number density> and current, with a common temporal <quantum phase>. Write $\psi(\mathbf x,t)=e^{-i\mu t/\hbar}\Psi(\mathbf x)$ with real $\mu$, since an imaginary temporal frequency would change the density. The stationary pumped <Gross–Pitaevskii equation> is
$$
\boxed{\mu\Psi=-\frac{\hbar^2}{2m}\nabla^2\Psi
+V_0|\Psi|^2\Psi+i(\gamma-\Gamma|\Psi|^2)\Psi}.
$$
The <chemical potential> is the common phase-rotation frequency times $\hbar$; this open system has no conserved particle-number constraint fixing it as in the conservative problem. Local gain and loss need not cancel separately in an inhomogeneous equilibrium, because a stationary <condensate number current> can transport particles between gain and loss regions.
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