Physics:Nakajima–Zwanzig equation

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Short description: 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

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

  1. 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
  2. [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.
  3. 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]
  4. 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

  1. 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. Bibcode1958PThPh..20..948N. 
  2. Zwanzig, Robert (1960). "Ensemble Method in the Theory of Irreversibility". The Journal of Chemical Physics 33 (5): 1338–1341. doi:10.1063/1.1731409. Bibcode1960JChPh..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

External links