In Reichl, 2016; pg405 the author gives a derivation of the master equation - which I will outline below in my own words:
- We start with:$$P(n,t+\Delta t)=\sum_m^M P(m,t)P(n,t+\Delta t\mid m,t)\tag{$\star$}$$
- We then Taylor expand $P(n,t+\Delta t\mid m,t)$ as a function of $\Delta t$ to get: $$ P(n,t+\Delta t\mid m,t)=\delta_{m,n} \left[ 1-\Delta t \sum^M_{l=1} w_{m,l}(t)\right]+w_{m,n}(t)\Delta t+\ldots$$ where we have enforced conservation of probability
- Sub this back into our expression for $P(n,t+\Delta t)$ , rearrange and take the limit as $\Delta t\rightarrow 0$ gives: $$\frac{\partial P(n,t)}{dt}=\sum_m^M\left[ P(m,t) W_{m,n}(t)-P(n,t) w_{n,m}(t)\right] $$
However many sources (including Reichl) hint at a restriction to Markovian process. Given that ($\star$) is simply true and everything else seems is standard I cannot see where the Markovian property comes into this - if it does? Please can someone explain this to me.