Random Heterogeneous Coagulation-Fragmentation Process

In this paper, We study the heterogeneous coagulation-fragmentation model by stochastic process theory mainly.First, We set up the heterogeneous coagulation-fragmentation stochastic model strictly, and then prove that there existence and uniqueness of the Feller process which corresponding to the infinite dimensional coagulation-fragmentation by the tool of solution of martingale problem. Under some condition on coagulation kernels and fragmentation kernels, We not only present the closed form of the invariant measure of the heterogeneous coagulation-fragmentation process, but also give out some properties of the invariant measure under the condition of equilibrium, which is a very important result for Physicist. At the same time, it is the basic of the study of the hydrodynamic limit of the heterogeneous coagulation-fragmentation. When the total number of the particles N large enough, we prove that the correlation between different size clusters is very weak, and the distribution of the number of finite size cluster convergence to the Poisson distribution .Second, We prove that the heterogeneous coagulation-fragmentation process may have some kind of the hydrodynamic limit under weak limit by tool of correlation function, and also present the integral equation of the hydrodynamic limit. This result present the mainly difference between heterogeneous coagulation-fragmentation and homogeneous coagulation-fragmentation. For the homogeneous coagulation fragmentation, it is no means to speak the hydrodynamic limit under the scale changing of time or space.Finally, We study the polymer model corresponding to the heterogeneous coagulation fragmentation. If a fragmentation strength is presented on the fragmentation kernel, We prove that there is a critical curve for the occurrence of gelation. As the same of invariant measure, critical behavior is also a interesting problem of Physicist of particle system, So our result is very important to the study of the microscopic models. When the total number of the particles N go to infinite , We prove that the distribution of the number of small, medium and largest clusters converge to Gaussian, Poisson and 1—0 distribution in the supercritical (post-gelation), respectively.This paper contains six sections. In section 1, we give some introduction on all kinds of coagulation-fragmentation model. In section 2, we show that there is a unique Feller process corresponding to the heterogeneous coagulation-fragmentation model in infinite dimensional. In section 3, we give out the closed form of the invariant measure and some properties of the invariant measure of the random heterogeneous coagulation-fragmentation process. In section 4, we prove that the hydrodynamic limit of the the heterogeneous coagulation-fragmentation process. In section 5, we consider the asymptotic probability distribution of size of the reversible random heterogeneous coagulation-fragmentation process. In the last section, We list some problems on the heterogeneous coagulation-fragmentation model which We will continue to study.