“Nakajima–Zwanzig equation”的意思、由来-开放百科全书

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 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^[3] is the quantum mechanical Liouville equation (von Neumann equation)

The density operator (density matrix)

is split by means of a projection operator

where

. The projection operator

projects onto the aforementioned relevant part, for which an equation of motion is to be derived.

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

See also

Notes

1. ^{{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=}}
2. ^{{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}}
3. ^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
4. ^To verify the equation it suffices to write the function under the integral as a derivative,

5. ^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.

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|last=|first=|date=|website=PhysikWiki|language=German|archive-url=|archive-date=|dead-url=|access-date=20 December 2018}}

2 : Quantum mechanics|Statistical mechanics