Diffusion-limited coalescence,A+A⇌A,with a trap

We study diffusion-limited coalescence, $A+A\rightleftharpoons A,$ in one dimension, and derive an exact solution for the steady state in the presence of a trap. Without the trap, the system arrives at an equilibrium state that satisfies detailed balance, and can therefore be analyzed by classical equilibrium methods. The trap introduces an irreversible element, and the stationary state is no longer an equilibrium state. The exact solution is compared to that of a reaction-diffusion equation—the habitual approximation method of choice. The reaction-diffusion equation can also be obtained exactly in this case, making the present model particularly useful.