Nakajima–Zwanzig equation – Wikipedia
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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[edit]
The starting point[note 1] is the quantum mechanical version of the von Neumann equation, also known as the Liouville 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
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
The second line is formally solved as[note 3]
By plugging the solution into the first equation, we obtain the Nakajima–Zwanzig equation:
Under the assumption that the inhomogeneous term vanishes[note 4] and using
- as well as
we obtain the final form
See also[edit]
- ^ 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
- ^ (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,
- ^ 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[edit]
- 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[edit]
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