Master Equation

Date:

29 July 2019

Derivation

Step 1 Master Equation is simply Conservation of Probability

\[\begin{split}\begin{align*} & p_i(t)= \text{ probability that the system is in states }i\text{ at time } t\\ & \dot{p_i}(t) = -\sum\limits_{j\ne i} p_i(t) \; k_{i\to j} + \sum\limits_{i\ne j} p_j(t)\; k_{j\to i}\\ &where \;\; k_{i\to j}=\text{ transition probability}\\ & \text{Matrix Form: } \boxed{ \dot{p} =- Mp} \end{align*}\end{split}\]