Master equation

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In physics, a master equation is a phenomenological first-order differential equation 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 P_k is the probability for the system to be in the state k, while the matrix $T_{\ell k}$ 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:

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

In this form, it closely resembles Liouville's equation in classical mechanics, and Lindblad's equation in quantum mechanics.

If the matrix $T_{\ell k}$ is symmetric, ie all the microscopic transition dynamics are state-reversible so

$T_{k\ell} = T_{\ell k,};$

this gives:

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

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).

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