Two catalyzed-birth models of n-species （n ≥ 2） aggregates with exchange-driven growth processes are proposed and compared. In the first one, the exchange reaction occurs between any two aggregates Ak^m and Af^m of...Two catalyzed-birth models of n-species （n ≥ 2） aggregates with exchange-driven growth processes are proposed and compared. In the first one, the exchange reaction occurs between any two aggregates Ak^m and Af^m of the same species with the rate kernels Km（k,j）= Kmkj （m = 1, 2,... ,n, n ≥ 2）, and aggregates of A^n species catalyze a monomer-birth of A^l species （l = 1, 2 , n - 1） with the catalysis rate kernel Jl（k,j） -Jlkj^v. The kinetic behaviors are investigated by means of the mean-field theory. We find that the evolution behavior of aggregate-size distribution ak^l（t） of A^l species depends crucially on the value of the catalysis rate parameter v： （i） ak^l（t） obeys the conventional scaling law in the case of v ≤ 0, （ii） ak^l（t） satisfies a modified scaling form in the case of v 〉 0. In the second model, the mechanism of monomer-birth of An-species catalyzed by A^l species is added on the basis of the first model, that is, the aggregates of A^l and A^n species catalyze each other to cause monomer-birth. The kinetic behaviors of A^l and A^n species are found to fall into two categories for the different v： （i） growth obeying conventional scaling form with v ≤ 0, （ii） gelling at finite time with v 〉 0.展开更多
We propose an aggregation model of a two-species system to mimic the growth of cities＇ population and assets, in which irreversible coagulation reactions and exchange reactions occur between any two aggregates of th...We propose an aggregation model of a two-species system to mimic the growth of cities＇ population and assets, in which irreversible coagulation reactions and exchange reactions occur between any two aggregates of the same species, and the monomer-birth reactions of one species occur by the catalysis Of the other species. In the case with population-catalyzed birth of assets, the rate kernel of an asset aggregate Bκ of size k grows to become an aggregate Bκ＋1 through a monomer-birth catalyzed by a population aggregate Aj of size j is J（κ,j） = Jkjλ. And in mutually catalyzed birth model, the birth rate kernels of population and assets are H（k,j）=Hkjη and J（k,j） = Jkjλ, respectively. The kinetics of the system is investigated based on the mean-field theory. In the model of population-catalyzed birth of aseets, the long-time asymptotic behavior of the assets aggregate size distribution obeys the conventional or modified scaling form. In mutually catalyzed birth system, the asymptotic behaviors of population and assets obey the conventional scaling form in the case of η=λ =0, and they obey the modified scaling form in the case of η=0, λ=1. In the case of η = λ= 1, the total mass of population aggregates and that of asset aggregates both grow much faster than those in population-catalyzed birth of assets model, and they approaches to infinite values in finite time.展开更多
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10275048 and 10305009 and the Natural Science Foundation of Zhejiang Province of China under Grant No. 102067
文摘Two catalyzed-birth models of n-species （n ≥ 2） aggregates with exchange-driven growth processes are proposed and compared. In the first one, the exchange reaction occurs between any two aggregates Ak^m and Af^m of the same species with the rate kernels Km（k,j）= Kmkj （m = 1, 2,... ,n, n ≥ 2）, and aggregates of A^n species catalyze a monomer-birth of A^l species （l = 1, 2 , n - 1） with the catalysis rate kernel Jl（k,j） -Jlkj^v. The kinetic behaviors are investigated by means of the mean-field theory. We find that the evolution behavior of aggregate-size distribution ak^l（t） of A^l species depends crucially on the value of the catalysis rate parameter v： （i） ak^l（t） obeys the conventional scaling law in the case of v ≤ 0, （ii） ak^l（t） satisfies a modified scaling form in the case of v 〉 0. In the second model, the mechanism of monomer-birth of An-species catalyzed by A^l species is added on the basis of the first model, that is, the aggregates of A^l and A^n species catalyze each other to cause monomer-birth. The kinetic behaviors of A^l and A^n species are found to fall into two categories for the different v： （i） growth obeying conventional scaling form with v ≤ 0, （ii） gelling at finite time with v 〉 0.
基金The project supported by National Natural Science Foundation of China under Grant Nos.10275048 and 10175008, and the Natural Science Foundation of Zhejiang Province of China under Grant No. 102067
文摘We propose an aggregation model of a two-species system to mimic the growth of cities＇ population and assets, in which irreversible coagulation reactions and exchange reactions occur between any two aggregates of the same species, and the monomer-birth reactions of one species occur by the catalysis Of the other species. In the case with population-catalyzed birth of assets, the rate kernel of an asset aggregate Bκ of size k grows to become an aggregate Bκ＋1 through a monomer-birth catalyzed by a population aggregate Aj of size j is J（κ,j） = Jkjλ. And in mutually catalyzed birth model, the birth rate kernels of population and assets are H（k,j）=Hkjη and J（k,j） = Jkjλ, respectively. The kinetics of the system is investigated based on the mean-field theory. In the model of population-catalyzed birth of aseets, the long-time asymptotic behavior of the assets aggregate size distribution obeys the conventional or modified scaling form. In mutually catalyzed birth system, the asymptotic behaviors of population and assets obey the conventional scaling form in the case of η=λ =0, and they obey the modified scaling form in the case of η=0, λ=1. In the case of η = λ= 1, the total mass of population aggregates and that of asset aggregates both grow much faster than those in population-catalyzed birth of assets model, and they approaches to infinite values in finite time.
