Nakajima-Zwanzig equation

The Nakajima Twenty equation ( named after the two physicists Sadao Nakajima and Robert Twenty ) is an integro-differential equation which the time evolution of the " relevant " content of a quantum system describes. It is formulated in the density operator formalism and can be considered as a generalization of the master equation.

The equation is part of the Mori- Twenty - theory in statistical mechanics of irreversible processes (named after addition Hazime Mori ). The momentum is with the aid of a projection operator in a slow, collective share split (relevant portion) and in a rapidly fluctuating irrelevant content. The aim is to develop dynamic equations for the collective share.

Derivation

Starting with the quantum mechanical Liouville equation ( von Neumann equation)

With the Liouvilleoperator defined by.

The density operator ( matrix density ) is divided by the projection operator in two portions, with. The projection operator is projected on the above-mentioned relevant portion, for an equation of motion is to be derived.

The Liouville - von Neumann equation can therefore by

Are shown.

The second line is formally

Solved. Inserted into the first equation yields the Nakajima - Twenty - equation:

Assuming that the inhomogeneous term disappears and the shortcut

And of the use of, one obtains the final form