טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentErlihson Michael
SubjectCoagulation-Fragmentation Process - Equilibrium and
Transient Behavior
DepartmentDepartment of Applied Mathematics
Supervisor Mr. Boris Granovsky


Abstract

The coagulation-fragmentation process (CFP) models the stochastic evolution of a population of N particles distributed into groups of different sizes that coagulate and fragment at a given rates. The model arises at different contexts of applications. Some examples are polymer kinetics, astrophysics, aerosols and biology. This process was intensively studied for a long time. As a result, different approximations to the model were suggested.  This work deals with the exact model this is viewed as a time homogeneous Markov process on the state space ΩN   the set of all partitions of a given integer N.
We will start from recalling the definition of a reversible CFP(k)  admitting interactions of up to k groups, as am generalization of a classical CFP(2). The equilibrium of the process considered is fully defined by a parameter function a(j),  j=1,…, on the set of integers. We observe that for all 2≤k≤N the CFP(k)'s appear to have the same invariant measure on the set of partitions of a given integer N (the number of particles).
Our main result is the central limit theorem for the number of groups νN at the steady state of a class of reversible CFP-s with the parameter function a(j)=jp-1, p>0, j=1,2,… for the number f particles N→∞, so we need to investigate the asymptotical behavior of the probability function of  νN. For this purpose we use Khintchine method for the derivation of the asymptotic formulae. In the spirit of the method, we construct a representation of the probability function of the number of groups via the probability function of the sum of independent identically distributed random variables. As a result, we prove the local and central limit theorems for the number of groups at equilibrium, as   N→∞. To achieve this, we employ new (for this field) tool: Poisson summation formula.
Central limit theorem makes possible to provide a verbal description of the picture of the equilibrium distribution of CFP-a considered, as the number of particles tends to infinity. Metropolis algorithm simulation for the invariant measure of  νsupports theoretical results, obtained in this work (central limit theorem for the number of groups).
Formally, particular cases of the invariant probability measure of reversible CGP-s confirm to a variety of quite different contexts. Following this we compare the results of our study with the classical ones for random permutation, the Ewens sampling formula and random combinatorial structures.  The second part of the work addresses to the transient behavior of CFP's. We give the formal definition of number of groups process (GNP) based on CFP and obtain the expressions for transition rates which provide Markovity of GNP. We also discover the transient behavior of the expectation of the number of groups for one of them.