Nakajima–Zwanzig equation

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 $LA={\frac {i}{\hbar }}[A,H]$ .

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^[2]

{\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^[3] 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 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

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

↑ 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
↑ To verify the equation it suffices to write the function under the integral as a derivative, de^QLt'Qe^L(t-t') = -e^QLt'QLPe^L(t-t')dt' .
↑ 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.

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