  We study the steady state of diffusion-limited coalescence, A+A<-->A, in the
presence of a trap and with a background drift. In one dimension this model can
be analyzed exactly through the method of inter-particle distribution functions
(IPDF). Because of the irreversible trap the steady state of the system is a
non-equilibrium state. An interesting phase transition, controlled by the drift
away from the trap, takes place: from a non-trivial steady state, when the
drift is weak, to a trivial steady state (the vacuum), as the drift increases
beyond some critical point. Surprisingly, regardless of the drift strength, the
computed IPDF resembles that of an homogeneous equilibrium system, without the
trap. We suggest that this is due to "shielding": the particle nearest to the
trap shields the remaining particles from the effects of the trap. Finally, we
compare the exact solution to that of a reaction-diffusion equation, and we
determine the optimal values of the appropriate rate coefficients.
