Department of Mathematics
                       University of Haifa
 
                Special Talk in Applied Mathematics
 
Speaker: Dr. Shay Gueron
         (The Technion)
 
Topic:   Do fish know how to count?
         The story of Coagulation-fragmentation processes
 
Date:    Tuesday,  March 31, 1998
 
Place:   Room 1420  of the Eshkol Tower Building,  Haifa University.
 
Time:    10:30 AM              
 
                       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 given rates. The process arises in a variety of
contexts, such as polymer kinetics, astrophysics, aerosols and population
biology. It has been intensively studied for a long time. As a result,
different approximations to the model have been suggested: integral equations
and discrete systems of differential equations. These equations are motivated
by physical intuition but from a mathematical point of view they are not
derived rigorously from a stochastic model. 
 
We study the exact stochastic model of CFP. The basic idea underlying our
approach is to view the CFP as a time homogeneous interacting particles system
on the finite state space $\Omega_N$ which is the set of all partitions of $N$. 
 
We obtain the stationary distribution (invariant measure) on $\Omega_N$ for
the whole class of reversible coagulation-fragmentation processes. We also
establish a characterization of the transition rates that guarantee the
reversibility of the process. Finally, we derive explicit expressions for
important functionals of this measure, in particular, the expected group size
distribution at the steady state. 
 
By comparing our results with the Smoluchowski's differential equations and
the coagulation-fragmentation integral equation formulated by Schumann and by
Blatz and Tobolsky, we study the question of the validity of those
approximations in terms of the parameters of CFP. Particularly, we show that
in some cases the latter approximation can considerably deviate from the exact
solution.   
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