  One-dimensional non-equilibrium models of particles subjected to a
coagulation-diffusion process are important in understanding non-equilibrium
dynamics, and fluctuation-dissipation relation. We consider in this paper
transport properties in finite and semi-infinite one-dimensional chains. A set
of particles freely hop between nearest-neighbor sites, with the additional
condition that, when two particles meet, they merge instantaneously into one
particle. A localized source of particle-current is imposed at the origin as
well as a non-symmetric hopping rate between the left and right directions
(particle drift). This model was previously studied with exact results for the
particle density by Hinrichsen et al. [1] in the long-time limit. We are
interested here in the crossover process between a scaling regime and long-time
behavior, starting with a chain filled of particles. As in the previous
reference [1], we employ the empty-interval-particle method, where the
probability of finding an empty interval between two given sites is considered.
However a different method is developed here to treat the boundary conditions
by imposing the continuity and differentiability of the interval probability,
which allows for a closed and unique solution, especially for any given initial
particle configuration. In the finite size case, we find a crossover between
the scaling regime and two different exponential decays for the particle
density as function of the input current. Precise asymptotic expressions for
the particle-density, and coagulation rate are given.
