Coagulation reaction in a one-dimensional gas

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Abstract
An extension of previous work on the ballistic annihilation reaction A+A→0 to the coagulation reaction A+A→A is presented. Three possible velocities c (with probability p), -c (with probability q), and zero are considered. While the long-time behavior is controlled by moving particles when p=q, it is controlled by the stationary particles when p≠q. The comparison of the coagulation reaction with the annihilation reaction shows that the long-time results are essentially the same except for a rescaling of the time. In addition, the time dependences of the decay in the ballistic coagulation reaction when p=q and the diffusion-limited coagulation reaction are also identical, but for different physical reasons. The reason for this becomes transparent by rederiving the ballistic coagulation results using a random-walk formalism, which can perhaps be generalized to more complicated ballistic reactions.