关键词: kinetic behaviour, cluster growth, scaling law, rate equation

Abstract: We propose two solvable cluster growth models, in which an irreversible aggregation spontaneously occurs between any two clusters of the same species; meanwhile, monomer birth or death of species A occurs with the help of species B. The system with the size-dependent monomer birth/death rate kernel $K(i,j) = Jij^v$ is then investigated by means of the mean-field rate equation. The results show that the kinetic scaling behaviour of species A depends crucially on the value of the index $v$. For the model with catalysis-driven monomer birth, the cluster-mass distribution of species $A$ obeys the conventional scaling law in the $v \leq 0$ case, while it satisfies a generalized scaling form in the $v > 0$ case; moreover, the total mass of species A is a nonzero value in the $v < 0$ case while it grows continuously with time in the $v \geq 0$ case. For the model with catalysis-driven monomer death, the cluster-mass distribution also approaches the conventional scaling form in the $v < 0$ case, while the conventional scaling description of the system breaks down in the $v \geq 0$ case. Additionally, the total mass of species A retains a nonzero quantity in the $v < 0$ case, but it decreases to zero with time in the $v \geq 0$ case.

Key words: kinetic behaviour, cluster growth, scaling law, rate equation