$10\mathrm{.}7\mathrm{.}1$ Linear rate processes.

We consider a large bath of monomer, whose concentration remains

constant, and a smaller number of particles exhibiting random growth

by the reactions

$P_n+M-{?}^{k_g}{P}_{n+1},r_n=k_g{c}_M{f}_n,n=1,2,$dots,$n_T$

We consider the monomer concentration, $c_M$ to be constant, and let

$f_n$ denote the number of particles in size class $n=1,2,$dots,$n_T.$ We

consider the reaction $($growth$)$ rate constant $k_g$ to be independent of

particle size, but this assumption is merely for convenience and is not

important to the development. We denote the probability density

$p(f_1,f_2,$dots,$f_{n_T},t)$

as the probability that at time $t,$ the system has $f_1$ particles in size class

$1,f_2$ particles in size class $2,$ and so on up to the largest size class $n_T.$

The deterministic model for this growth process follows immediately

from the production rates of Reactions $10\mathrm{.}21.$

Deterministic.

$\frac{d}{dt}{f}_1=-f_1$

$\frac{d}{dt}{f}_n=f_{n-1}-f_n,n=2,dots{n}_T-1$

$\frac{d}{dt}{f}_{n_T}=f_{n_T-1}$

in which we use $$ to denote the constant $=k_g{c}_M.$ For the random

model, the equation governing the probability density is

$\frac{d}{dt}p(f_1,f_2,$dots,$f_{n_T},t)=$

$-(f_1+f_2+cdots{f}_{n_{T-1}})p(f_1,f_2,$dots,$f_{n_T},t)$

$+(f_1+1)p(f_1+1,f_2-1,$dots,$f_{n_T},t)$

$+(f_2+1)p(f_1,f_2+1,f_3-1,$dots,$f_{n_T},t)$

$+$cdots$+(f_{n_T-1}+1)p(f_1,f_2,$dots,$f_{n_T-1}+1,f_{n_T}-1,t)$