Mass distribution in n-polymer stochastic aggregation and large-mass behaviour

The exact solutions of the rate equations of the n-polymer stochastic aggregation involving two types of clusters, active and passive for the kernel \prod_k=1^n s_i_k\left(s_i_k=i_k\right) and \sum_k=1^n s_i_k\left(s_i_k=i_k\right), are obtained. The large-mass behaviours of the final mass distribution of the active and passive clusters have scaling-like forms, although the models exhibit different properties. Respectively, they have different decay exponents \gamma=\frac2 n+12(n-1) and \gamma=q+\frac2 n+12(n-1) for \prod_k=1^n s_i_k\left(s_i_k=i_k\right) and \gamma=\frac32(n-1) and \gamma=q+\frac32(n-1) for\sum_k=1^n s_i_k\left(s_i_k=i_k\right), which include exponents of two-polymer stochastic aggregation. We also find that gelation is suppressed for kernel \prod_k=1^n s_i_k\left(s_i_k=i_k\right) which is different from the deterministic aggregation.