  Presently, the main methods for describing a non-equilibrium
charge-transporting steady state are based on time-evolving it from the initial
zero-current situation. An alternative class of theories would give the
statistical non-equilibrium density operator from principles of statistical
mechanics, in a spirit close to Gibbs ensembles for equilibrium systems,
leading to a variational principle for the non-equilibrium steady state. We
discuss the existing attempts to achieve this using the maximum entropy
principle based on constraining the average current. We show that the
current-constrained theories result in a zero induced drop in electrostatic
potential, so that such ensembles cannot correspond to the time-evolved density
matrix, unless left- and right-going scattering states are mutually incoherent.
