Nucleation model for multiparticle reactions with finite reaction rates in one dimension

We study one-dimensional reactions $\ensuremath{\mu}A\ensuremath{\rightarrow}\ensuremath{\nu}A$ ($\ensuremath{\mu}g\ensuremath{\nu}$)with nucleation and finite reaction rate ($k\ensuremath{\ll}$) in one dimension for the particle density decay by means of a Monte Carlo simulation and analytic modeling. The anomalous case $\ensuremath{\mu}=2$ was studied in our previous work. The marginal case $\ensuremath{\mu}=3$ is described without logaritmic corrections in the mean field approach. The case $\ensuremath{\mu}g3$ is well described by classical rate equations. The rate equation for the particle density is derived for all $\ensuremath{\mu}$. We present a mean field approach for the early time regime (reaction-controlled limit) for any initial density. Also, the mean field aproximation is derived in a simple way from the rate equation for any time at low densities.