Finite discrete coagulation and fragmentation with a mass reservoir

Fix a basic mass $a>0$ and an integer $N\ge 2$. Species $i\in \{1,\dots ,N\}$ has mass $m_i=ia$ and number concentration $n_i\ge 0$. A symmetric nonnegative array $K_{ij}=K_{ji}$ specifies binary aggregation. In the ordered-pair convention, the event rate for $(i,j)$ is $K_{ij}{n}_i{n}_j\mathrm{/}2$; summing over ordered pairs counts an unequal pair twice and an equal pair once with its required factor $1\mathrm{/}2$.

If $i+j\le N$, an aggregation event has stoichiometric increment $e_{i+j}-e_i-e_j$, where $e_i$ is the $i$th coordinate vector. If $i+j>N$, it removes the two parents from the retained bins and adds mass $m_i+m_j$ to a scalar overflow reservoir $L$; the reservoir has no return dynamics. Thus this branch is a truncation model and is not a physical state law for particles that later fragment.

For fragmentation, each species $i\ge 2$ has event rate ${\beta }_i{n}_i$, where ${\beta }_i\ge 0$. The monomer does not fragment: set ${\beta }_1=0$ and impose no daughter normalization for $i=1$. Its expected daughter counts are $b_i(k)\ge 0$ for $1\le k<i$, with ${\sum }_{k<i}k{b}_i(k)=i$. One event has increment $-e_i+{\sum }_{k<i}{b}_i(k)e_k$. The mean equations are the finite sums of these event rates times their increments, together with the stated overflow-reservoir equation. When $b_i$ is specified as an actual random daughter law, these equations describe its expected event increment.