n physics, a master equation is a phenomenological set of first-order differential equations describing the time evolution of the probability of a system to occupy each one of a discrete set of states:
\frac{dP_k}{dt}=\sum_\ell T_{k\ell}P_\ell,
where Pk is the probability for the system to be in the state k, while the matrix T is filled with a grid of transition-rate constants.
The notation \scriptstyle T_{\ell k} indicates an element from this matrix. It is the rate constant that corresponds to the transition from state k to state ℓ. Because T is square, the indices ℓ and k may be arbitrarily defined as rows or columns. Here, the first subscript is row, the second is column. The order of the subscripts, which refer to source and destination states, are opposite of the normal convention for elements of a matrix. That is, in other contexts, T12 could be interpreted as the 1 \rightarrow 2 transition. However, it is convenient to write the subscripts in the opposite order when using Einstein notation, so the subscripts in T12 should be interpreted as 1 \leftarrow 2.
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 the T is not defined or has been assigned an arbitrary value.
\frac{dP_k}{dt}=\sum_\ell(T_{k\ell}P_\ell - T_{\ell k}P_k)=\sum_{\ell\neq k}(T_{k\ell}P_\ell - T_{\ell k}P_k).
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 \scriptstyle\pi_k and \scriptstyle\pi_\ell, \scriptstyle T_{k \ell} \pi_\ell = T_{\ell k} \pi_k.
Many physical problems in classical, quantum mechanics and problems in other sciences, can be reduced to the form of a master equation, thereby performing a great simplification of the problem (see mathematical model).
The Lindblad equation in quantum mechanics is a generalization of the master equation describing the time evolution of a density matrix. Though the Lindblad equation is often referred to as a master equation, it is not one in the usual sense, as it governs not only the time evolution of probabilities (diagonal elements of the density matrix), but also of variables containing information about quantum coherence between the states of the system (non-diagonal elements of the density matrix).
