Polymer Chain Scission -- Probability Problem

From: Surendar Jeyadev (jeyadev_at_wrc.xerox.bounceback.com)
Date: 11/16/04 

Date: 16 Nov 2004 17:25:05 GMT

Any help on the following would be appreciated.
 
The problem in question is concerned with polymer chain scission and
how the distribution of the lengths of the chains evolves in time as
scission continues to occur. I am only interested in the mean values
here -- at least for the time being. The problem can be modelled as
follows.

The polymeric chains can be seen a chains made up of links, which are
broken to produce smaller chains. Links that are broken once cannot
be broken again. Let the average number (or, fraction)
of chains with 'i' links (i=0, 1, 2, ...) at time t_k be given by
n_i(t_k). The initial distribtion at t = 0, {n_i(0)} is given.
 
Assume that

  1) there is a constant flux of entities that attack the links
     and break them
  2) At any time, any of the links is equally likely ot be broken
  3) The number of links broken at at time t_k is proportional to
     the total number of links present at that time (this just takes
     into account that as the number of unbroken links reduces, the
     breaking entities will become less 'efficient' as there are
     fewer links to break due to 'dilution')

What we seek is the time evolution of the {n_i(t_k)}.
 

>From the assumptions it is clear that at evey time step a the
fraction of links are broken. If this is taken to be 'f', then
f*n_i(t_k) of the i-link chains are are broken at time t_k. The
part I am having difficulty with is determining the average
number of the broken chains that have j links ( j < i ) for
j = 0, 1, ..., i-1. In other words, I would like to determine
the "transition probability"

   n_i->j (t_k) for j = 0, 1, ..., i-1

There are i*n_i links in the population of i-link chains. Out
of these f*i*n_i are broken. The total number of ways in which
the breaks can occur is known. But is there a nice way to find
the average number of 0, 1, 2, ... link chains?

Can one get an expression like

   n_i(t_k+1) = (1-f)*n_i(t_k) + sum{0,i-1} [number of i link
                                     chains produced due to
                                     scission]

I would really like to avoid Monte Carlo, if there is a way
out.

Any pointers would be more than welcome.

thanks 

-- 
Surendar Jeyadev         jeyadev1@wrc.xerox.com
                         
                         Remove 1 for email address

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