Above I wrote:

> Specifically, the firing rate of the process is equal to its rate constant times the product of its input population counts.

And:

> That the firing rates are proportional to the input population sizes is called the _law of mass action_.

I see this needs to be corrected. This product rule holds only when none of the input species to a transition occur with multiplicity more than one. The general case is given by a modified product rule.

The operative principle here is explained in:

* John C. Baez and Jacob Biamonte, [Quantum techniques for stochastic mechanics](https://arxiv.org/abs/1209.3632), arXiv:1209.3632 [quant-ph].

The modified product rule actual follows from a more abstract principle.

Picture the state space of the Petri net as a graph with nodes \\(\mathbb{N}^k\\).

Each transition induces a (possibly infinite) collection of edges into this graph. These edges are generated by the "delta vector" that is produced by one firing of the transition.

For example, in the SI model, the delta vector is (-1,1).

Note: the technical name is not delta vector but stoichiometric vector. So I took some poetic license.

Furthermore, each edge \\(e\\) from state \\(x\\) to state \\(y\\) gets weighted by the _number of ways_ of moving directly from \\(x\\) to \\(y\\).

And this counting procedure is predicated on the idea that the individuals in the population are _distinguishable_.

For example, suppose that we had a transition:

\\[t: A + B \rightarrow Z\\]

The delta vector for \\(t\\) is (-1, -1, +1).

For example, take the SI model, and suppose that edge \\(e\\) goes from \\(x = (2,4,6)\\) to \\(y = (1,3,7)\\).

The underlying procedure for a single step of the transition involved choosing one of 2 members from A and one of 3 members from B. So there are 6 ways that this transition can be performed, and the weight for the edge is 6. There you have the product rule.

But now consider the case of:

\\[t: A + A \rightarrow X\\]

Now the delta vector is (-2, +1).

Now the underlying procedure involves choosing a member from species A, and then choosing a second member from A after it has been reduced by one from the first step.

So if \\(e\\) is an edge from \\(x\\) to \\(y\\), and state \\(x\\) contains \\(k\\) members of A, then the weight given to \\(e\\) is \\(k \cdot (k-1)\\).

This is called the falling power.

The modified product rule then is clear: take the product of falling powers, where there is one falling power for each distinct species that occurs as input to a transition.

The law of mass action -- which simply uses the product rule -- follows as a limiting approximation for large population sizes.

> Specifically, the firing rate of the process is equal to its rate constant times the product of its input population counts.

And:

> That the firing rates are proportional to the input population sizes is called the _law of mass action_.

I see this needs to be corrected. This product rule holds only when none of the input species to a transition occur with multiplicity more than one. The general case is given by a modified product rule.

The operative principle here is explained in:

* John C. Baez and Jacob Biamonte, [Quantum techniques for stochastic mechanics](https://arxiv.org/abs/1209.3632), arXiv:1209.3632 [quant-ph].

The modified product rule actual follows from a more abstract principle.

Picture the state space of the Petri net as a graph with nodes \\(\mathbb{N}^k\\).

Each transition induces a (possibly infinite) collection of edges into this graph. These edges are generated by the "delta vector" that is produced by one firing of the transition.

For example, in the SI model, the delta vector is (-1,1).

Note: the technical name is not delta vector but stoichiometric vector. So I took some poetic license.

Furthermore, each edge \\(e\\) from state \\(x\\) to state \\(y\\) gets weighted by the _number of ways_ of moving directly from \\(x\\) to \\(y\\).

And this counting procedure is predicated on the idea that the individuals in the population are _distinguishable_.

For example, suppose that we had a transition:

\\[t: A + B \rightarrow Z\\]

The delta vector for \\(t\\) is (-1, -1, +1).

For example, take the SI model, and suppose that edge \\(e\\) goes from \\(x = (2,4,6)\\) to \\(y = (1,3,7)\\).

The underlying procedure for a single step of the transition involved choosing one of 2 members from A and one of 3 members from B. So there are 6 ways that this transition can be performed, and the weight for the edge is 6. There you have the product rule.

But now consider the case of:

\\[t: A + A \rightarrow X\\]

Now the delta vector is (-2, +1).

Now the underlying procedure involves choosing a member from species A, and then choosing a second member from A after it has been reduced by one from the first step.

So if \\(e\\) is an edge from \\(x\\) to \\(y\\), and state \\(x\\) contains \\(k\\) members of A, then the weight given to \\(e\\) is \\(k \cdot (k-1)\\).

This is called the falling power.

The modified product rule then is clear: take the product of falling powers, where there is one falling power for each distinct species that occurs as input to a transition.

The law of mass action -- which simply uses the product rule -- follows as a limiting approximation for large population sizes.