Diffusion-controlled annihilation in the presence of particle sources: Exact results in one dimension

The steady-state particle density n¯ and the relaxation time τ of homogeneous density fluctuations are calculated for one-dimensional systems in which particles move diffusively and annihilate irreversibly, and steady sources of either single particles (model I) or pairs of neighboring particles (model II) are also present. For small particle-production rates h, we find n¯∼${\mathrm{h}}^{1/\mathrm{\delta }}$ and τ∼${\mathrm{h}}^{\mathrm{-}\mathrm{\Delta }}$ with δ=3, Δ=(2/3) for model I and δ=2, Δ=1 for model II. If we interpret particles as solitons, model II is used to account for some aspects of the experimental data on the photoinduced absorption of trans-(CH${)}_{\mathrm{x}}$.