# Nakajima–Zwanzig equation

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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.

## Derivation

The starting point[1] is the quantum mechanical Liouville equation (von Neumann equation)

$\partial_t \rho = \frac{i}{\hbar}[\rho,H] = L \rho,$

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

The density operator (density matrix) $\rho$ is split by means of a projection operator $\mathcal{P}$ into two parts $\rho =\left( \mathcal{P}+\mathcal{Q} \right)\rho$, where $\mathcal{Q}\equiv 1-\mathcal{P}$. The projection operator $\mathcal{P}$ 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

${\partial_t}\left( \begin{matrix} \mathcal{P} \\ \mathcal{Q} \\ \end{matrix} \right)\rho =\left( \begin{matrix} \mathcal{P} \\ \mathcal{Q} \\ \end{matrix} \right)L\left( \begin{matrix} \mathcal{P} \\ \mathcal{Q} \\ \end{matrix} \right)\rho +\left( \begin{matrix} \mathcal{P} \\ \mathcal{Q} \\ \end{matrix} \right)L\left( \begin{matrix} \mathcal{Q} \\ \mathcal{P} \\ \end{matrix} \right)\rho.$

The second line is formally solved as

$\mathcal{Q}\rho ={{e}^{\mathcal{Q}Lt}}Q\rho (t=0)+\int_{0}^{t}dt'{e}^{\mathcal{Q}Lt'}\mathcal{Q}L\mathcal{P}\rho (t-{t}').$

By plugging the solution into the first equation, we obtain the Nakajima–Zwanzig equation:

$\partial_t \mathcal{P}\rho =\mathcal{P}L\mathcal{P}\rho +\underbrace{\mathcal{P}L{{e}^{\mathcal{Q}Lt}}Q\rho (t=0)}_{=0}+\mathcal{P}L\int_{0}^{t}{dt'{{e}^{\mathcal{Q}Lt'}}\mathcal{Q}L\mathcal{P}\rho (t-{t}')}.$

Under the assumption that the inhomogeneous term vanishes[2] and using

$\mathcal{K}\left( t \right)\equiv\mathcal{P}L{{e}^{\mathcal{Q}Lt}}\mathcal{Q}L\mathcal{P},$
$\mathcal{P}\rho \equiv {{\rho }_\mathrm{rel}},$ as well as
$\mathcal{P}^2=\mathcal{P},$

we obtain the final form

$\partial_t{\rho }_\mathrm{rel}=\mathcal{P}L{{\rho}_\mathrm{rel}}+\int_{0}^{t}{dt'\mathcal{K}({t}'){{\rho }_\mathrm{rel}}(t-{t}')}.$

## References

• E. Fick, G. Sauermann: The Quantum Statistics of Dynamic Processes Springer-Verlag, 1983, ISBN ISSN 3-540-50824-4.
• Heinz-Peter Breuer, Francesco Petruccione: Theory of Open Quantum Systems. Oxford, 2002 ISBN ISSN 970-0-19-852063-4
• Hermann Grabert Projection operator techniques in nonequilibrium statistical mechanics, Springer Tracts in Modern Physics, Band 95, 1982
• R. Kühne, P. Reineker: Nakajima-Zwanzig's generalized master equation: Evaluation of the kernel of the integro-differential equation, Zeitschrift für Physik B (Condensed Matter), Band 31, 1978, S. 105–110, Abstract

### Original works

• Sadao Nakajima (1958), "On Quantum Theory of Transport Phenomena" (in German), Progress of Theoretical Physics 20 (6): pp. 948–959
• Robert Zwanzig (1960), "Ensemble Method in the Theory of Irreversibility" (in German), Journal of Chemical Physics 33 (5): pp. 1338–1341
• original paper

## Notes

1. ^ A derivation analogous to that presented here is found, for instance, in Breuer, Petruccione The theory of open quantum systems, Oxford University Press 2002, S.443ff
2. ^ Such an assumption can be made if we assume that the irrelevant part of the density matrix is 0 at the initial time, so that the projector for t=0 is the identity.