Nakajima–Zwanzig equation


The Nakajima–Zwanzig equation is an integral equation describing the time evolution of the "relevant" part of a quantum-mechanical system. It is formulated in the density matrix formalism and can be regarded a generalization of the master equation.
The equation belongs to the Mori–Zwanzig theory within the statistical mechanics of irreversible processes. By means of a projection operator the dynamics is split into a slow, collective part and a rapidly fluctuating irrelevant part. The goal is to develop dynamical equations for the collective part.

Derivation

The starting point is the quantum mechanical Liouville equation
where the Liouville operator is defined as.
The density operator is split by means of a projection operator
into two parts
,
where. The projection operator projects onto the aforementioned relevant part, for which an equation of motion is to be derived.
The Liouville – von Neumann equation can thus be represented as
The second line is formally solved as
By plugging the solution into the first equation, we obtain the Nakajima–Zwanzig equation:
Under the assumption that the inhomogeneous term vanishes and using
we obtain the final form