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:


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

where Pk 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:


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

In this form, it closely resembles Liouville's equation in classical mechanics, and Lindblad's equation in quantum mechanics. The master equation exhibits detailed balance if each of the terms of the summation disappears separately at equilibrium — ie if, for all states k and ℓ having equilibrium probabilities $$\scriptstyle\pi_k$$ and $$\scriptstyle\pi_\ell$$, $$\scriptstyle T_{k \ell} \pi_\ell = T_{\ell k} \pi_k$$.

If the matrix $$\scriptstyle T_{\ell k}$$ 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.