Steady state in two-dimensional diffusion-controlled reactions

We investigate the conditions under which a steady state can be reached in a two-dimensional diffusion-controlled trapping reaction. If there is no interaction between trap and diffusing particles, the reaction rate decreases monotonically to zero. Here we show that a logarithmic attractive potential between trap and diffusing particles leads to a finite steady-state reaction rate. A steady state can also be reached if the diffusing particles move under the action of a uniform external field. More unexpectedly, a steady-state rate can be obtained in the absence of any “assisting field” if the trap grows due to the absorption of the diffusing particles. The reaction rates are calculated in all cases.