Nakajima–Zwanzig equation

The Nakajima–Zwanzig equation (named after the physicists Sadao Nakajima and Robert Zwanzig) 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 (named after Hazime Mori). By means of a projection operator the dynamics is split into a slow, collective part (relevant part) and a rapidly fluctuating irrelevant part. The goal is to develop dynamical equations for the collective part.

The starting point is the quantum mechanical Liouville equation (von Neumann equation)

where the Liouville operator ${\displaystyle L}$ is defined as ${\displaystyle LA={\frac {i}{\hbar }}[A,H]}$.

The density operator (density matrix) ${\displaystyle \rho }$ is split by means of a projection operator ${\displaystyle {\mathcal {P}}}$ into two parts ${\displaystyle \rho =\left({\mathcal {P}}+{\mathcal {Q}}\right)\rho }$, where ${\displaystyle {\mathcal {Q}}\equiv 1-{\mathcal {P}}}$. The projection operator ${\displaystyle {\mathcal {P}}}$ projects onto the aforementioned relevant part, for which an equation of motion is to be derived.

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