To put some math behind that *very, extremely* simple Dyson’s Model, we will declare a vector space \(K\) which encodes the possible set of states that our “cell” can be in. Now, declare a transition matrix \(M \in \mathcal{L}(K)\) which maps from one state to another.

Finally, then, we can define a function \(P(k)\) for the \(k\) th state of our cell.

That is, then:

\begin{equation} P(k+1) = M P(k) \end{equation}

(as the “next” state is simply \(M\) applied onto the previous state).

Rolling that out, we have:

\begin{equation} P(k) = M^{k} P(0) \end{equation}