Critical behavior of the heterogeneous random coagulation-fragmentation processes

Abstract

In this paper, we study the asymptotic distributions of the heterogeneous random coagulation-fragmentation processes (HCFP) which model the coagulation, fragmentation and diffusion of clusters of particles on a lattice. Based on a closed form of the stationary distribution for the HCFP, we prove that the mutually dependent clusters with a finite size (finite particles) in the sub-critical stage will become independent in the critical and super-critical stages, and the asymptotic distributions of the number of clusters may converge to Gaussian or Poisson distribution according to its size, to be small or large in the critical and super-critical stages.

Coagulation Processes with Gibbsian Time Evolution

“…A particular case of a pure coagulation, when ψ(i; j) = a(i + j) + b, a, b ≥ 0, is known in the literature as the Marcus-Lushnikov stochastic process, while in [1] as well as in some other papers, the name is given to all stochastic CPs with the rates of single coagulations of the form N −1 ψ(i; j) with an arbitrary ψ(i, j ). It is important to point out that, in contrast to the latter Marcus-Lushnikov process, the basic assumption of our setting is that the rates of single coagulations do not depend on N. The equilibria of some reversible models with rates of coagulation and fragmentation depending on N were studied in [13] and [15].…”

Coagulation Processes with Gibbsian Time Evolution

“…A particular case of a pure coagulation, when ψ(i; j) = a(i+j)+b, a, b ≥ 0 is known in the literature as the Marcus-Lushnikov stochastic process, while in [1] as well as in some other papers, the name is given to all stochastic CP 's with rates of single coagulations of the form N −1 ψ(i; j), with an arbitrary ψ(i, j). It is important to point out that, in contrast to the latter Marcus-Lushnikov process, the basic assumption of our setting is that the rates of single coagulations do not depend on N. The equilibria of some reversible models with rates of coagulation and fragmentation depending on N were studied in [13], [14].…”