Nakajima–Zwanzig equation

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (December 2018) (Learn how and when to remove this template message)

The Nakajima–Zwanzig equation (named after the physicists who developed it, Sadao Nakajima^[1] and Robert Zwanzig^[2]) 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 formalism within the statistical mechanics of irreversible processes (named after ). 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^{[note 1]} is the quantum mechanical version of the Liouville equation, known as the von Neumann equation:

${\displaystyle \partial _{t}\rho ={\frac {i}{\hbar }}[\rho ,H]=L\rho ,}$

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 ${\displaystyle {\mathcal {P}}}$ selects the aforementioned relevant part from the density operator,^{[note 2]} for which an equation of motion is to be derived.

The Liouville – von Neumann equation can thus be represented as

${\displaystyle {\partial _{t}}\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)\rho =\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)L\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)\rho +\left({\begin{matrix}{\mathcal {P}}\\{\mathcal {Q}}\\\end{matrix}}\right)L\left({\begin{matrix}{\mathcal {Q}}\\{\mathcal {P}}\\\end{matrix}}\right)\rho .}$

The second line is formally solved as^{[note 3]}

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

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

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

Under the assumption that the inhomogeneous term vanishes^{[note 4]} and using

${\displaystyle {\mathcal {K}}\left(t\right)\equiv {\mathcal {P}}L{{e}^{{\mathcal {Q}}Lt}}{\mathcal {Q}}L{\mathcal {P}},}$

${\displaystyle {\mathcal {P}}\rho \equiv {{\rho }_{\mathrm {rel} }},}$ as well as

${\displaystyle {\mathcal {P}}^{2}={\mathcal {P}},}$

we obtain the final form

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

See also[]

• Redfield equation

Notes[]

References[]

• E. Fick, G. Sauermann: The Quantum Statistics of Dynamic Processes Springer-Verlag, 1983, ISBN 3-540-50824-4.
• Heinz-Peter Breuer, Francesco Petruccione: Theory of Open Quantum Systems. Oxford, 2002 ISBN 9780198520634
• 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, doi:10.1007/BF01320131

External links[]

• "Nakajima-Zwanzig-Gleichung". PhysikWiki (in German). Retrieved 20 December 2018.