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Stoichiometric number and mass changes for finite coagulation and fragmentation events

Statement

In the finite discrete model of Finite discrete coagulation and fragmentation with a mass reservoir, an in-range coagulation event changes the retained number by $- 1$ and retained mass by $0$ . An overflow event changes retained number by $- 2$ , retained mass by $- (m_{i} + m_{j})$ , and reservoir mass by $m_{i} + m_{j}$ . A fragmentation event of species $i$ changes number by $\sum_{k < i} b_{i} (k) - 1$ and mass by $0$ .

Facts & Assumptions

Given: The event definitions, stoichiometric increments, and daughter normalization in Finite discrete coagulation and fragmentation with a mass reservoir.

[F1]

Finite discrete coagulation and fragmentation with a mass reservoir: an in-range aggregation event has increment $e_{i + j} - e_{i} - e_{j}$ ; overflow removes both parents and adds their mass to the reservoir; fragmentation has increment $- e_{i} + \sum_{k < i} b_{i} (k) e_{k}$ with $\sum_{k < i} k b_{i} (k) = i$ .

Proof

1.1givenF1algebra

For $i + j \leq N$ , summing the coordinates of $e_{i + j} - e_{i} - e_{j}$ gives $1 - 1 - 1 = - 1$ , and pairing it with the mass vector $(m_{k})$ gives $(i + j) a - i a - j a = 0$ .

1.2givenF1algebra

For $i + j > N$ , the two parent removals give number change $- 2$ and retained-mass change $- (m_{i} + m_{j})$ ; the reservoir increment is exactly $m_{i} + m_{j}$ , so augmented mass change is zero.

2.1givenF1algebra∎

Fragmentation changes number by $- 1 + \sum_{k < i} b_{i} (k)$ . Its mass change is $- i a + \sum_{k < i} k a b_{i} (k) = a (- i + i) = 0$ by the daughter normalization.

Depends on

Finite discrete coagulation and fragmentation with a mass reservoir

Used by

A binary split preserves first moment and increases number by one Example

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Sources

Philippe Laurençot and Stéphane Mischler, From the discrete to the continuous coagulation-fragmentation equations, §1 (standard reference, not scraped)
Jacek Banasiak, Analytic fragmentation semigroups and classical solutions to coagulation-fragmentation equations, §1 (standard reference, not scraped)