Editing Master equation (section)

==Detailed description of the matrix and properties of the system==
Let <math>\mathbf{A}</math> be the matrix describing the transition rates (also known as kinetic rates or [[reaction rate]]s). As always, the first subscript represents the row, the second subscript the column. That is, the source is given by the second subscript, and the destination by the first subscript. This is the opposite of what one might expect, but is appropriate for conventional [[matrix multiplication]].

For each state ''k'', the increase in occupation probability depends on the contribution from all other states to ''k'', and is given by:
<math display="block"> \sum_\ell A_{k\ell}P_\ell, </math>
where <math> P_\ell </math> is the probability for the system to be in the state <math> \ell </math>, while the [[matrix (mathematics)|matrix]] <math>\mathbf{A}</math> is filled with a grid of transition-rate [[Constant (mathematics)|constant]]s.  Similarly, <math>P_k</math> contributes to the occupation of all other states <math> P_\ell, </math>
<math display="block"> \sum_\ell A_{\ell k}P_k, </math>

In probability theory, this identifies the evolution as a [[continuous-time Markov process]], with the integrated master equation obeying a [[Chapman–Kolmogorov equation]].

The master equation can be simplified so that the terms with ''ℓ'' = ''k'' do not appear in the summation.  This allows calculations even if the main diagonal of <math>\mathbf{A}</math> is not defined or has been assigned an arbitrary value.
<math display="block">
 \frac{dP_k}{dt}
        = \sum_\ell(A_{k\ell}P_\ell)
        = \sum_{\ell\neq k}(A_{k\ell}P_\ell) + A_{kk}P_k
        = \sum_{\ell\neq k}(A_{k\ell}P_\ell - A_{\ell k}P_k). </math>

The final equality arises from the fact that 
<math display="block">  \sum_{\ell, k}(A_{\ell k}P_k) = \frac{d}{dt} \sum_\ell(P_{\ell}) = 0 </math> 
because the summation over the probabilities <math> P_{\ell} </math> yields one, a constant function. Since this has to hold for any probability <math>\vec{P}</math> (and in particular for any probability of the form <math> P_{\ell} = \delta_{\ell k}</math> for some k) we get
<math display="block">  \sum_{\ell}(A_{\ell k}) =  0 \qquad \forall k.</math> 
Using this we can write the diagonal elements as
<math display="block"> A_{kk} = -\sum_{\ell\neq k}(A_{\ell k}) \Rightarrow A_{kk} P_k = -\sum_{\ell\neq k}(A_{\ell k} P_k) .</math>

The master equation exhibits [[detailed balance]] if each of the terms of the summation disappears separately at equilibrium—i.e. if, for all states ''k'' and ''ℓ'' having equilibrium probabilities <math>\pi_k</math> and <math>\pi_\ell</math>,
<math display="block">A_{k \ell} \pi_\ell = A_{\ell k} \pi_k .</math>

These symmetry relations were proved on the basis of the [[time reversibility]] of  microscopic dynamics ([[microscopic reversibility]]) as [[Onsager reciprocal relations]].