Nakajima twenty equation

The Nakajima twenty equation (named after the two physicists Sadao Nakajima and Robert Zwanzig ) is an integrodifferential equation that describes the time evolution of the “relevant” part of a quantum mechanical system. It is formulated in density operator formalism and can be viewed as a generalization of the master equation .

The equation is part of the Mori twenty theory in the statistical mechanics of irreversible processes (also named after Hazime Mori ). With the help of a projection operator, the dynamics are broken down into a slow, collective part ( relevant part ) and a rapidly fluctuating irrelevant part. The aim is to develop dynamic equations for the collective part.

Derivation

Starting with the quantum mechanical Liouville equation (von Neumann equation)

{\ displaystyle {d} _ {t} \ rho = {\ frac {i} {\ hbar}} [\ rho, H] = L \ rho}

with the Liouville operator defined by . ${\ displaystyle L}$ ${\ displaystyle LA = {\ frac {i} {\ hbar}} [A, H]}$

The density operator (density matrix) is split into two parts by the projection operator , with . The projection operator projects onto the relevant part mentioned above , for which an equation of motion is to be derived. ${\ displaystyle \ rho}$ ${\ displaystyle {\ mathcal {P}}}$ ${\ displaystyle \ rho = \ left ({\ mathcal {P}} + {\ mathcal {Q}} \ right) \ rho}$ ${\ displaystyle {\ mathcal {Q}} \ equiv 1 - {\ mathcal {P}}}$ ${\ displaystyle {\ mathcal {P}}}$

The Liouville - von Neumann equation can therefore go through

{\ displaystyle {{d} _ {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}

being represented.

The second line is formalized by

{\ 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} ')}}

solved. Inserted into the first equation, you get the Nakajima twenty equation:

{\ displaystyle {\ text {d}} _ {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} ')}}

Assuming that the inhomogeneous term disappears and the abbreviation

{\ displaystyle {\ mathcal {K}} \ left (t \ right) = {\ mathcal {P}} L {{e} ^ {{\ mathcal {Q}} Lt}} {\ mathcal {Q}} L {\ mathcal {P}}}

,

${\ displaystyle {\ mathcal {P}} \ rho \ equiv {{\ rho} _ {rel}}}$ as well as the utilization of one obtains the final shape ${\ displaystyle {\ mathcal {P}} ^ {2} = {\ mathcal {P}}}$

{\ displaystyle {\ text {d}} _ {t} {\ rho} _ {rel} = {\ mathcal {P}} L {{\ rho} _ {rel}} + \ int _ {0} ^ { t} {dt '{\ mathcal {K}} ({t}') {{\ rho} _ {rel}} (t- {t} ')}}

literature

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 0-19-852063-8 .
Hermann Grabert: Projection operator techniques in nonequilibrium statistical mechanics. (= Springer Tracts in Modern Physics. Volume 95). 1982.
R. Kühne, P. Reineker: Nakajima-Zwanzig's generalized master equation: Evaluation of the kernel of the integro-differential equation. In: Zeitschrift für Physik B (Condensed Matter). Volume 31, 1978, pp. 105-110. doi : 10.1007 / BF01320131

Original work

Sadao Nakajima: On Quantum Theory of Transport Phenomena . In: Progress of Theoretical Physics . tape 20 , no. 6 , 1958, pp. 948-959 .
Robert Zwanzig: Ensemble Method in the Theory of Irreversibility . In: Journal of Chemical Physics . tape 33 , no. 5 , 1960, pp. 1338-1341 .

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Original article

Notes and individual references

↑ The derivation is similar to that shown here, for example. B. in H.-P. Breuer, F. Petruccione: The theory of open quantum systems. Oxford University Press, 2002, pp. 443ff.
↑ This can be done if one assumes that the irrelevant part of the density matrix is 0 at the start time, i.e. the projector for t = 0 is the identity.

[1] The derivation is similar to that shown here, for example. B. in H.-P. Breuer, F. Petruccione: The theory of open quantum systems. Oxford University Press, 2002, pp. 443ff.

[2] This can be done if one assumes that the irrelevant part of the density matrix is 0 at the start time, i.e. the projector for t = 0 is the identity.