Master equation

A master equation is a phenomenologically based first order differential equation that describes the time evolution of the probabilities of a system.

For states of a discrete set of states is the master equation:

Where the probability is that the system is in state k, and k is assumed to be constant transition probability from state l to state. Analogously, the master equation for continuous states ( and corresponding probability densities ) formulate, only with an integration rather than a summation as discrete states.

In probability theory, this is regarded as a continuous Markov process, in which corresponds to the integrated master equation of the Chapman - Kolmogorov equation.

If the matrix is ​​symmetric ( ie all microscopic transitions are reversible and the transition probabilities equal in both directions), then:

And thus:

The master equation in the above form has been derived in the first quantum statistics of Wolfgang Pauli and therefore is also called the Pauli master equation. It is a differential equation for the state probabilities, that is, the diagonal elements of the density matrix. There are also generalizations that the off-diagonal elements include ( master equation in Lindblad form). A further generalization is the Nakajima - Twenty - equation in the Mori- Twenty formalism.

General called in statistical mechanics master equations basic equations (often in the above form of a balance equation ) for the probability distributions from which then can be derived easily be solved by approximations and border crossings equations such as differential equations of the type of the Fokker -Planck equation ( which also includes the diffusion equation ) in the continuum limit. Behind these approximations but is still valid, the microscopic master equation, hence the name.