Holes in the Model (void)

While, in an optimal world, we could simulate everything down to our best understanding of physics, for computational reasons, we cannot do this, and we must make simplifying assumptions for the model, which allows us a model which is useful as opposed to a model which doesn’t exist. The same ideology is chosen by DiffEqRxn.

Chemicals From “Nowhere”

Let’s say that we know that some chemical from elsewhere in the system is being produced at a given rate \(p_1\). Using the syntax from the previous chapter would make this ambiguous / impossible, we’re always relative to some other chemical in those equations.

To remedy this issue, we introduce the void keyword.

This keyword has some special behaviors, and, in exchange, some special rules.

Rules for Using void

1. Void can only occupy it’s side of the equation, so void + A would not be valid in this model, and the model should fail to parse.

  1. The side opposite the void can be neither void nor anything other than produced chemical species

void behavior

Getting chemicals from the void

Chemicals which are “created from the void” are defined as so:

void ==> A {p_1}

This declares that the chemical species A is made at a rate of \(p_1\), where \(p_1\) is in \(\rm{units} \cdot \rm{sec}^{-1}\).

Expressed in differential equations this statement is equivalent to:

\[\frac{\mathrm{d} A}{\mathrm{d} t} = p_1\]

Moving chemicals into the void

If we need a chemical to be lost at a certain rate, but not be converted to another chemical, we can send it to the void:

A ==> void {l_1}

As opposed to the previous case, this case doesn’t decrease at a constant rate but rather proportional to how much A there is in the system. Expressed in differential equations, this looks like:

\[\frac{\mathrm{d} A}{\mathrm{d} t} = l_1 \cdot A\]

Note

Were it to be a constant rate, if \(l_1 > p_1\), then the solution would be non-sensical