Nakajima–Zwanzig equation – Wikipedia

Integral equation in quantum simulations

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 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[edit]

The starting point^{[note 1]} is the quantum mechanical version of the von Neumann equation, also known as the Liouville equation:

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

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

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

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

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

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

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

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

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

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

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[edit]

References[edit]

• 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[edit]