Nakajima–Zwanzig equation
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[note 1] is the quantum mechanical Liouville equation (von Neumann equation)
where the Liouville operator is defined as .
The density operator (density matrix) is split by means of a projection operator into two parts , where . The projection operator 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
The second line is formally solved as[note 2]
By plugging the solution into the first equation, we obtain the Nakajima–Zwanzig equation:
Under the assumption that the inhomogeneous term vanishes[note 3] and using
- as well as
we obtain the final form
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
- To verify the equation it suffices to write the function under the integral as a derivative,
- 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
- Nakajima, Sadao (1 December 1958). "On Quantum Theory of Transport Phenomena: Steady Diffusion". Progress of Theoretical Physics. 20 (6): 948–959. doi:10.1143/PTP.20.948. ISSN 0033-068X.
- Zwanzig, Robert (1960). "Ensemble Method in the Theory of Irreversibility". The Journal of Chemical Physics. 33 (5): 1338–1341. doi:10.1063/1.1731409.
- 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
- "Nakajima-Zwanzig-Gleichung". PhysikWiki (in German). Retrieved 20 December 2018.