Master equation

In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determined by a transition rate matrix. The equations are a set of differential equations over time of the probabilities that the system occupies each of the different states.

A master equation is a phenomenological set of first-order differential equations describing the time evolution of (usually) the probability of a system to occupy each one of a discrete set of states with regard to a continuous time variable t. The most familiar form of a master equation is a matrix form:

where ${\displaystyle {\vec {P}}}$ is a column vector (where element i represents state i), and ${\displaystyle \mathbf {A} }$ is the matrix of connections. The way connections among states are made determines the dimension of the problem; it is either

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