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Master equation

In 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:

$\frac{dP_k}{dt}=\sum_\ell T_{k\ell}P_\ell,$

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where P_k is the probability for the system to be in the state k, while the matrix $\scriptstyle T_{\ell k}$ is filled with a grid of transition-rate constants.

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

Note that

$\sum_{\ell} T_{\ell k} = 0$

(i.e. probability is conserved), so the equation may also be written as

$\frac{dP_k}{dt}=\sum_\ell(T_{k\ell}P_\ell - T_{\ell k}P_k).$

allowing us to omit the term ℓ = k from the summation. Thus, in the latter form of the master equation there is no need to define the diagonal elements of T.

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).

Another generalization of the master equation is the Fokker-Planck equation which describes the time evolution of a continuous probability distribution.