Nakajima–Zwanzig equation - Alchetron, the free social encyclopedia

The Nakajima–Zwanzig equation (named after the physicists Sadao Nakajima and Robert Zwanzig) 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 is the quantum mechanical Liouville equation (von Neumann equation)

∂ t ρ = i ℏ [ ρ , H ] = L ρ ,

where the Liouville operator L is defined as L A = i ℏ [ A , H ] .

The density operator (density matrix) ρ is split by means of a projection operator P into two parts ρ = ( P + Q ) ρ , where Q ≡ 1 − P . The projection operator P 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

∂ t ( P Q ) ρ = ( P Q ) L ( P Q ) ρ + ( P Q ) L ( Q P ) ρ .

The second line is formally solved as

Q ρ = e Q L t Q ρ ( t = 0 ) + ∫ 0 t d t ′ e Q L t ′ Q L P ρ ( t − t ′ ) .

By plugging the solution into the first equation, we obtain the Nakajima–Zwanzig equation:

∂ t P ρ = P L P ρ + P L e Q L t Q ρ ( t = 0 ) ⏟ = 0 + P L ∫ 0 t d t ′ e Q L t ′ Q L P ρ ( t − t ′ ) .

Under the assumption that the inhomogeneous term vanishes and using

K ( t ) ≡ P L e Q L t Q L P , P ρ ≡ ρ r e l , as well as P 2 = P ,

we obtain the final form

∂ t ρ r e l = P L ρ r e l + ∫ 0 t d t ′ K ( t ′ ) ρ r e l ( t − t ′ ) .

References

Nakajima–Zwanzig equation Wikipedia

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