Solvable Aggregation-Migration-Annihilation Processes of a Multispecies System

An aggregation-migration-annihilation model is proposed for a two-species-group system. In the system, aggregation reactions occur between any two aggregates of the same species and migration reactions between two different species in the same group and joint annihilation reactions between two species from different groups. The kinetics of the system is then investigated in the framework of the mean-field theory. It is found that the scaling solutions of the aggregate size distributions depend crucially on the ratios of the equivalent aggregation rates of species groups to the annihilation rates. Each species always scales according to a conventional or modified scaling form; moreover, the governing scaling exponents are nonuniversal and dependent on the reaction details for most cases.     

Kinetics of an n-Species Aggregation Chain Model with Complete Annihilation

The kinetic behavior of an n-species (n ≥ 3) aggregation-annihilation chain reaction model is studied. In this model, an irreversible aggregation reaction occurs between any two clusters of the same species, and an irreversible complete annihilation reaction occurs only between two species with adjacent number. Based on the rnean-field theory, we investigate the rate equations of the process with constant reaction rates to obtain the asymptotic solutions of the clustermass distributions for the system. The results show that the kinetic behavior of the system not only depends crucially on the ratio of the aggregation rate I to the annihilation rate J, but also has relation with the initial concentration of each species and the species number's odevity. We find that the cluster-mass distribution of each species obeys always a scaling law. The scaling exponents may strongly depend on the reaction rates for most cases, however, for the case in which the ratio of the aggregation rate to the annihilation rate is equal to a certain value, the scaling exponents are only dependent on the initial concentrations of the reactants.     

Monomer Migration and Annihilation Processes

We propose a two-species monomer migration-annihilation model, in which monomer migration reactions occur between any two aggregates of the same species and monomer annihilation reactions occur between two different species. Based on the mean-field rate equations, we investigate the evolution behaviors of the processes. For the case with an annihilation rate kernel proportional to the sizes of the reactants, the aggregation size distribution of either species approaches the modified scaling form in the symmetrical initial case, while for the asymmetrical initial case the heavy species with a large initial data scales according to the conventional form and the light one does not scale. Moreover, at most one species can survive finally. For the case with a constant annihilation rate kernel, both species may scale according to the conventional scaling law in the symmetrical case and survive together at the end.     

Dynamics of aggregation－annihilation process with cluster removals

We have studied the kinetic behaviours of irreversible aggregation－annihilation models with cluster removals. In the models, an irreversible aggregation reaction occurs between any two clusters of the same species and an irreversible annihilation reaction occurs simultaneously between two different species; meanwhile, the clusters of large size are gradually removed from the system. In a mean-field limit, we obtain the general solutions of the cluster-mass distributions for the cases with an arbitrary removal probability. We found that the cluster-mass distribution of either species satisfies a generalized or modified scaling form. The results also indicate that the evolution behaviours of the systems depend strongly on the details of the reaction events.     

Novel Biased Aggregation-Annihilation Model

We propose a novel two-species aggregation-annihilation model,in which irreversible aggregation reactions occur between any two aggregates of the same species and biased annihilations occur simultaneously between two different species.The kinetic scaling behavior of the model is then analytically investigated by means of the mean-field rate equation.For the system without the self-aggregation of the un-annihilated species,the aggregate size distribution of the annihilated species always approaches a modified scaling form and vanishes finally; while for the system with the self-aggregation of the un-annihilated species,its scaling behavior depends crucially on t,he details of the rate kernels.Moreover,the results also exhibit that both species are conserved together in some cases,while only the un-annihilated species survives finally in other cases.     

Competition between aggregation and migration processes of a multi-species system

We propose a solvable multi-species aggregation--migration model, in which irreversible aggregations occur between any two aggregates of the same species and reversible migrations occur between any two different species. The kinetic behaviour of an aggregation--migration system is then studied by means of the mean-field rate equation. The results show that the kinetics of the system depends crucially on the details of reaction events such as initial concentration distributions and ratios of aggregation rates to migration rate. In general, the aggregate mass distribution of each species always obeys a conventional or a generalized scaling law, and for most cases at least one species is scaled according to a conventional form with universal constants. Moreover, there is at least one species that can survive finally.     

Scaling Behavior of an Aggregation-Migration Model

We study the kinetic behavior of a two-species aggregation-migration model in which an irreversible aggregation occurs between any two clusters of the same species and a reversible migration occurs simultaneously between two different species. For a simple model with constant aggregation rates and with the migration rates K_A(i;j)=K'_A(i;j) ∝ij^v₁ and K_B(i;j)=K'_B(i;j) ∝ij^v₂, we find that the evolution behavior of the system depends crucially on the values of the indexes v₁ and v₂. The aggregate size distribution of either species obeys a conventional scaling law for most cases. Moreover, we also generalize the two-species system to the multi-species case and analyze its kinetic behavior under the symmetrical conditions.     

Random deposition of two annihilating species in the (1+1) dimension

We present simulation results for the one-dimensional random deposition of two annihilating species A and B, falling with probabilities p and q (p+q=1), which then react to produce an inert product, i.e., A+B-->0. Two different annihilation rules are defined: top annihilation and nearest-neighbor annihilation (NNA), leading to distinct scaling behaviors. In particular, the values of the scaling exponents for NNA are found to be dependent on probability p, suggesting different universality classes.     

Kinetic Behavior of Aggregation-Exchange Growth Process with Catalyzed-Birth

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 1through a monomer-birth catalyzed by a population aggregate Aj of size j is J(k,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 assets,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 ofη =λ= 0,and they obey the modified scalingform 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.     

Cluster-cluster aggregation in binary mixtures

The structure and aggregation kinetics of three-dimensional clusters composed of two different monomeric species at three concentrations are thoroughly investigated by means of extensive, large-scale computer simulations. The aggregating monomers have all the same size and occupy the cells of a cubic lattice. Two bonding schemes are considered: (a) the binary diffusion-limited cluster-cluster aggregation (BDLCA) in which only the monomers of different species stick together, and (b) the invading binary diffusion-limited cluster-cluster aggregation (IBDLCA) in which additionally monomers of one of the two species are allowed to bond. In the two schemes, the mixed aggregates display self-similarity with a fractal dimension d(f) that depends on the relative molar fraction of the two species and on concentration. At a given concentration, when this molar fraction is small, d(f) approaches a value close to the reaction-limited cluster-cluster aggregation of one-component systems, and when the molar fraction is 0.5, d(f) becomes close to the value of the diffusion-limited cluster-cluster aggregation model. The crossover between these two regimes is due to a time-decreasing reaction probability between colliding particles, particularly at small molar fractions. Several dynamical quantities are studied as a function of time. The number of clusters and the weight-average cluster size display a power-law behavior only at small concentrations. The dynamical exponents are obtained for molar fractions above 0.3 but not at or below 0.2, indicating the presence of a critical transition between a gelling to a nongelling system. The cluster-size distribution function presents scaling for molar fractions larger than 0.2.     

Kinetics of a Migration－Driven Aggregation－Fragmentation Process

We propose a reversible model of the migration-driven aggregation-fragmentation process with the symmetric migration rate kernels K(k;j)=K'(k;j)=λkj^υ and the constant aggregation rates I₁, I₂ and fragmentation rates J₁, J₂. Based on the mean-field theory, we investigate the evolution behavior of the aggregate size distributions in several cases with different values of index υ. We find that the fragmentation reaction plays a more important role in the kinetic behaviors of the system than the aggregation and migration. When J₁=0 and J₂ =0, the aggregate size distributions a_k(t) and b_k(t) obey the conventional scaling law, while when J₁>0 and J₂>0, they obey the modified scaling law with an exponential scaling function. The total mass of either species remains conserved.     

Phase transition in annihilation-limited processes

A system of particles is studied in which the stochastic processes are one-particle type-change (or one-particle diffusion) and multi-particle annihilation. It is shown that, if the annihilation rate tends to zero but the initial values of the average number of the particles tend to infinity, so that the annihilation rate times a certain power of the initial values of the average number of the particles remain constant (the double scaling) then if the initial state of the system is a multi-Poisson distribution, the system always remains in a state of multi-Poisson distribution, but with evolving parameters. The large time behavior of the system is also investigated. The system exhibits a dynamical phase transition. It is seen that for a k-particle annihilation, if k is larger than a critical value k_c, which is determined by the type-change rates, then annihilation does not enter the relaxation exponent of the system; while for k < k_c, it is the annihilation (in fact k itself) which determines the relaxation exponent.