M.Sc Student | Erlihson Michael |
---|---|

Subject | Coagulation-Fragmentation Process - Equilibrium and Transient Behavior |

Department | Department of Applied Mathematics |

Supervisor | Mr. Boris Granovsky |

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 Ω_{N}
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 ν_{N} at the steady state of a class of
reversible CFP-s with the parameter function a(j)=j^{p-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 ν_{N}.
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 ν_{N }supports 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.