View source for Nakajima–Zwanzig equation

{{Use dmy dates|date=October 2014}}
{{Short description|Integral equation in quantum simulations}}
{{Inline citations|date=December 2018}}

The '''Nakajima–Zwanzig equation''' (named after the physicists who developed it, Sadao Nakajima<ref>{{Cite journal|last=Nakajima|first=Sadao|date=1958-12-01|title=On Quantum Theory of Transport Phenomena: Steady Diffusion|url=https://academic.oup.com/ptp/article/20/6/948/1930693|journal=Progress of Theoretical Physics|language=en|volume=20|issue=6|pages=948–959|doi=10.1143/PTP.20.948|issn=0033-068X|via=}}</ref> and [[Robert Zwanzig]]<ref>{{Cite journal|last=Zwanzig|first=Robert|date=1960|title=Ensemble Method in the Theory of Irreversibility|journal=The Journal of Chemical Physics|volume=33|issue=5|pages=1338–1341|doi=10.1063/1.1731409}}</ref>) 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<ref group="note">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</ref> is the quantum mechanical [[Liouville equation]] ([[von Neumann equation]])
:<math>\partial_t \rho = \frac{i}{\hbar}[\rho,H] = L \rho,</math> 
where the Liouville operator <math>L</math> is defined as <math>L A = \frac{i}{\hbar}[A,H]</math>.

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

The second line is formally solved as<ref group="note">To verify the equation it suffices to write the function under the integral as a derivative, <math>de^{\mathcal{Q}Lt'}\mathcal{Q}e^{L(t-t')} = -e^{\mathcal{Q}Lt'}\mathcal{Q}L\mathcal{P}e^{L(t-t')}dt' .</math> </ref>
:<math>\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}').</math>

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

:<math>\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}')}.</math>

Under the assumption that the inhomogeneous term vanishes<ref group="note">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.</ref> and using
:<math>\mathcal{K}\left( t \right)\equiv\mathcal{P}L{{e}^{\mathcal{Q}Lt}}\mathcal{Q}L\mathcal{P},</math>
:<math>\mathcal{P}\rho \equiv {{\rho }_\mathrm{rel}},</math> as well as
:<math>\mathcal{P}^2=\mathcal{P},</math> 
we obtain the final form
:<math>\partial_t{\rho }_\mathrm{rel}=\mathcal{P}L{{\rho}_\mathrm{rel}}+\int_{0}^{t}{dt'\mathcal{K}({t}'){{\rho }_\mathrm{rel}}(t-{t}')}.</math>
:

== See also ==

* [[Redfield equation]]

== Notes ==
<references group="note" />

== References ==
<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 ==
*{{Cite web|url=http://wiki.physikerwelt.de/wiki/Nakajima-Zwanzig-Gleichung|title=Nakajima-Zwanzig-Gleichung|website=PhysikWiki|language=German|access-date=20 December 2018}}

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[[Category:Quantum mechanics]]
[[Category:Statistical mechanics]]