Monte Carlo Approximation of Coagulation-Fragmentation Equations

Flavius Guias

Institut fuer Angewandte Mathematik

Im Neuenheimer Feld 294

D-69120 Heidelberg

Many processes in  nature can be described in terms of particle
systems governed by binary interactions of coagulation-fragmentation
type. More precisely,  we consider the particles characterized by an
additive size parameter (volume or mass)  k=1,2,...  , calling for  k=1  the
simple particles as   monomers  and, in general, the particles
composed of  k  monomers as  k -  mers . The possible interactions
are:

-the formation of  i+j -mers by  collisions of  i -mers with  j -mers

-the splitting of  i+j -mers into  i -mers and  j -mers.

In order
to describe the dynamics in time of such processes, it
is convenient to consider as variables the concentrations  u^k(t)  of  k -mers
and to
write down  a system of differential equations (the so-called
  coagulation-fragmentation  equations )
for the variables  u^k ,   k=1,2,...  , which  takes in account the
previously described binary reactions.

In our approach we consider a Monte Carlo method in order to approximate
the solutions of the coagulation-fragmentation  equations.

For the case of bounded coagulation rates (independently of  i  and  j ),
we construct a
 N -particle system whose dynamics is  given by a Markov
jump process describing the coagulation and fragmentation of particles.

  Dynkin's formula  for the infinitesimal operator enables us to
obtain the dynamics for the concentration of  k -mers for each
 k=1,..., N , which are nothing else than the coagulation-fragmentation  equations
perturbed by some martingale terms. The estimations of the second
moments of the martingales imply a convergence result
of the concentrations derived from the particle system to the
solution of the coagulation-fragmentation equations in the mean square
norm and on some sequence spaces.

If the growth order of the coagulation rates
(with respect to the size of the particles)
is in a certain range, we give an
existence and uniqueness proof by showing convergence in sequence spaces
of solutions of approximating systems with bounded coefficients.
This enables us to state convergence results of the Monte Carlo
method to the respective solutions by adjusting properly the speed
of convergence to infinity of the total number of particles in terms
of the upper bound
of the coagulation coefficients.