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We study the diffusion-limited process $A+A\to A$ in one dimension, withfinite reaction rates. We develop an approximation scheme based on the methodof Inter-Particle Distribution Functions (IPDF), which was formerly used forthe exact solution of the same process with infinite reaction rate. Theapproximation becomes exact in the very early time regime (or thereaction-controlled limit) and in the long time (diffusion-controlled)asymptotic limit. For the intermediate time regime, we obtain a simpleinterpolative behavior between these two limits. We also study the coalescenceprocess (with finite reaction rates) with the back reaction $A\to A+A$, and inthe presence of particle input. In each of these cases the system reaches anon-trivial steady state with a finite concentration of particles. Theoreticalpredictions for the concentration time dependence and for the IPDF are comparedto computer simulations. P. A. C. S. Numbers: 82.20.Mj 02.50.+s 05.40.+j05.70.LnComment: 13 pages (and 4 figures), plain TeX, SISSA-94-0

Diffusion-limited coalescence with finite reaction rates in one dimension