Editing Master equation (section)

==Introduction==
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 (mathematics)|set]] of [[Classical mechanics|states]] with regard to a continuous time variable ''t''. The most familiar form of a master equation is a matrix form:
<math display="block"> \frac{d\vec{P}}{dt} = \mathbf{A}\vec{P},</math>
where <math>\vec{P}</math> is a column vector, and <math>\mathbf{A}</math> is the matrix of connections. The way connections among states are made determines the dimension of the problem; it is either
*a d-dimensional system (where d is 1,2,3,...), where any state is connected with exactly its 2d nearest neighbors, or
*a network, where every pair of states may have a connection (depending on the network's properties).

When the connections are time-independent rate constants, the master equation represents a [[kinetic scheme]], and the process is [[Markov process|Markovian]] (any jumping time probability density function for state ''i'' is an exponential, with a rate equal to the value of the connection). When the connections depend on the actual time (i.e. matrix <math>\mathbf{A}</math> depends on the time, <math>\mathbf{A}\rightarrow\mathbf{A}(t)</math> ), the process is not stationary and the master equation reads
<math display="block"> \frac{d\vec{P}}{dt} = \mathbf{A}(t)\vec{P}.</math>

When the connections represent multi exponential [[jumping time]] [[probability density function]]s, the process is [[Semi Markov process|semi-Markovian]], and the equation of motion is an [[integro-differential equation]] termed the generalized master equation:
<math display="block"> \frac{d\vec{P}}{dt}= \int^t_0 \mathbf{A}(t- \tau )\vec{P}( \tau ) \, d \tau . </math>

The [[transition rate matrix]] <math>\mathbf{A}</math> can also represent [[birth–death process|birth and death]], meaning that probability is injected (birth) or taken from (death) the system, and then the process is not in equilibrium.

When the transition rate matrix can be related to the probabilities, one obtains the [[Kolmogorov equations]].