Master Equation
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:Date: 29 July 2019



Derivation
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**Step 1** Master Equation is **simply** Conservation of Probability

.. math::
	\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*}