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:
- <math> \frac{dP_k}{dt}=\sum_\ell T_{k\ell}P_\ell, </math>
where Pk is the probability for the system to be in the state k, while the matrix <math>T_{\ell k}</math> 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
- <math>\sum_{\ell} T_{\ell k} = 0</math>
(i.e. probability is conserved), so the equation may also be written:
- <math> \frac{dP_k}{dt}=\sum_\ell(T_{k\ell}P_\ell - T_{\ell k}P_k). </math>
If the matrix <math>T_{\ell k}</math> is symmetric, ie all the microscopic transition dynamics are state-reversible so
- <math>T_{k\ell} = T_{\ell k,};</math>
this gives:
- <math> \frac{dP_k}{dt}=\sum_\ell T_{k\ell} (P_\ell - P_k). </math>
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.
