Physics:Nakajima–Zwanzig equation
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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
The starting point[note 1] is the quantum mechanical version of the von Neumann equation, also known as the Liouville equation:
- [math]\displaystyle{ \partial_t \rho = \frac{i}{\hbar}[\rho,H] = L \rho, }[/math]
where the Liouville operator [math]\displaystyle{ L }[/math] is defined as [math]\displaystyle{ L A = \frac{i}{\hbar}[A,H] }[/math].
The density operator (density matrix) [math]\displaystyle{ \rho }[/math] is split by means of a projection operator [math]\displaystyle{ \mathcal{P} }[/math] into two parts [math]\displaystyle{ \rho =\left( \mathcal{P}+\mathcal{Q} \right)\rho }[/math], where [math]\displaystyle{ \mathcal{Q}\equiv 1-\mathcal{P} }[/math]. The projection operator [math]\displaystyle{ \mathcal{P} }[/math] 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
- [math]\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. }[/math]
The second line is formally solved as[note 3]
- [math]\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}'). }[/math]
By plugging the solution into the first equation, we obtain the Nakajima–Zwanzig equation:
- [math]\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}')}. }[/math]
Under the assumption that the inhomogeneous term vanishes[note 4] and using
- [math]\displaystyle{ \mathcal{K}\left( t \right)\equiv\mathcal{P}L{{e}^{\mathcal{Q}Lt}}\mathcal{Q}L\mathcal{P}, }[/math]
- [math]\displaystyle{ \mathcal{P}\rho \equiv {{\rho }_\mathrm{rel}}, }[/math] as well as
- [math]\displaystyle{ \mathcal{P}^2=\mathcal{P}, }[/math]
we obtain the final form
- [math]\displaystyle{ \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
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
- ↑ [math]\displaystyle{ \mathcal{P} \rho = }[/math] (relevant part) · (constant). The relevant part is called the reduced density operator of the system, the constant part is the density matrix of the thermal bath at equilibrium.
- ↑ To verify the equation it suffices to write the function under the integral as a derivative, [math]\displaystyle{ 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]
- ↑ 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. This is true if the correlation of fluctuations on different sites caused by the thermal bath is zero.
References
- ↑ Nakajima, Sadao (1958-12-01). "On Quantum Theory of Transport Phenomena: Steady Diffusion" (in en). Progress of Theoretical Physics 20 (6): 948–959. doi:10.1143/PTP.20.948. ISSN 0033-068X. Bibcode: 1958PThPh..20..948N.
- ↑ Zwanzig, Robert (1960). "Ensemble Method in the Theory of Irreversibility". The Journal of Chemical Physics 33 (5): 1338–1341. doi:10.1063/1.1731409. Bibcode: 1960JChPh..33.1338Z.
- 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
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