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}