---
title: "Stepwise Evolution"
type: concept
related: [Vector Space, Dyson S Model Of Life, Matricies, Cell]
source: https://www.jemoka.com/posts/kbhstepwise_evolution/
confidence: high
status: active
---

To put some math behind that very, extremely simple Dyson&rsquo;s Model, we will declare a vector space \(K\) which encodes the possible set of states that our &ldquo;cell&rdquo; 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 &ldquo;next&rdquo; state is simply \(M\) applied onto the previous state).
Rolling that out, we have:
\begin{equation} P(k) = M^{k} P(0) \end{equation}
