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: rac{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_{\ell k} is filled with a grid of transition-rate Constant s.

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:

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

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:

: rac{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.


SEE ALSO