3.1.9 Conclusions

3.1.7 Properties of the Pauli equation

Non-Equilibrium Particle Dynamics

dSrelð Þt

We used <sup>d</sup>

equilibrium.

(Ek <sup>¼</sup> <sup>ℏ</sup><sup>2</sup>

22

k2

dt ¼ �k<sup>B</sup> <sup>∑</sup>

¼ 1 2 k<sup>B</sup> ∑ n ∑ m

3.1.8 Example: transition rates

n ∑ m

dt <sup>∑</sup>np1ð Þ¼ <sup>n</sup>; <sup>t</sup> <sup>d</sup>

typical case is the collisions expressed by a†

=2m,ω<sup>q</sup> ¼ c∣q∣)

En ¼ Ek, and the final state ∣mi ¼ ∣k<sup>0</sup>

particles ∣ki ¼ ∣k, σi, the matrix element v k; σ; k<sup>0</sup>

The transition rate wnm obeys the condition of detailed balance, wmn ¼ wnm, the

An important property is that it describes irreversible evolution with time. For the

wnm <sup>p</sup>1ð Þ� <sup>m</sup>; <sup>t</sup> <sup>p</sup>1ð Þ <sup>n</sup>; <sup>t</sup> ln <sup>p</sup>1ð Þ <sup>n</sup>; <sup>t</sup> � <sup>k</sup><sup>B</sup> <sup>∑</sup>

<sup>n</sup>jH<sup>0</sup> h i <sup>j</sup><sup>m</sup> <sup>¼</sup> <sup>m</sup>jH0þ h i <sup>j</sup><sup>n</sup> <sup>∗</sup> <sup>¼</sup> <sup>m</sup>jH<sup>0</sup> h i <sup>j</sup><sup>n</sup> <sup>∗</sup> : (88)

wnm <sup>p</sup>1ð Þ� <sup>n</sup>; <sup>t</sup> <sup>p</sup>1ð Þ <sup>m</sup>; <sup>t</sup> ln <sup>p</sup>1ð Þ <sup>n</sup>; <sup>t</sup> � ln <sup>p</sup>1ð Þ <sup>m</sup>; <sup>t</sup> <sup>≥</sup>0:

dt 1 ¼ 0 and interchanged n with m in the half

n

<sup>1</sup> between the (momentum)

<sup>q</sup>b<sup>q</sup> (90)

<sup>q</sup> þ h:c: (91)

<sup>þ</sup>q,k must fulfill

, qi, energy Em ¼ Ek<sup>0</sup> þ ℏω<sup>q</sup> for emission in the

p1ð Þ n; t p1ð Þ n; t <sup>∂</sup>p1ð Þ <sup>n</sup>; <sup>t</sup> ∂t

(89)

inverse transition has the same rate. This follows because H<sup>0</sup> is hermitean,

of the expression. Since ln x is a monotonic function of x, the relation

ð Þ x<sup>1</sup> � x<sup>2</sup> ð Þ ln x<sup>1</sup> � ln x<sup>2</sup> ≥0 holds. Considering states n, m where transitions are possible, equilibrium (dSrelð Þt =dt ¼ 0) occurs if p1ð Þ¼ m; t p1ð Þ n; t ; else Srelð Þt increases with time. Equipartition corresponds to the microcanonical ensemble in

We consider transitions between eigenstates of H<sup>0</sup> owing to interaction. A

eigenstates ∣ki of H0. This is discussed in the following section on kinetic theory. Another example is minimal coupling known from QFT between a Dirac fermionic

field (electron) and the Maxwell bosonic field (photons), with

H<sup>0</sup> ¼ ∑ k Eka†

Hint ¼ ∑ k, <sup>k</sup><sup>0</sup> , q v kk<sup>0</sup> ; q a†

k1 a† k2 ak0 2 ak0

<sup>k</sup>a<sup>k</sup> þ ∑ q

(spin and polarization variables are not indicated separately), and the interaction

The transition rates (87) are calculated between the initial state ∣ni ¼ ∣ki, energy

vacuum state. For absorption, the corresponding process can be given. For free

momentum conservation. Together with the conservation of energy in Eq. (87), the second-order transition rate vanishes. Only in fourth order, different contributions (Compton scattering, pair creation) are possible. If considering an radiating atom, the electrons are moving in orbits around the nucleus, ∣ki ¼ ∣nlm, σi. Momentum conservation is not required, and the standard expressions (Fermi's Golden rule) for absorption and emission of light by an atom are obtained. The corresponding

ℏωqb†

k0akb†

; σ<sup>0</sup> ; <sup>q</sup> <sup>∝</sup>δk<sup>0</sup>

relevant entropy SrelðÞ¼� t kB∑np1ð Þ n; t ln p1ð Þ n; t we find

Quantum master equations and the Pauli equation are fundamental expressions to describe nonequilibrium phenomena, such as one-step processes of excitation and deexcitation, two-level systems, nuclear decay, chemical reactions, and also conductivity where electrons are scattered by ions, etc. A basic assumption is the subdivision into a system and a bath. In Born-Markov approximation, correlations between system and bath (back-reactions) are neglected. Projection to diagonal elements of the reduced density matrix or the Rotating wave approximation lead to irreversible equations of evolution (dissipator) as derived by Zwanzig, Lindblad, Kossakowski, and others. Further developments of the theory are, e.g., the Nakajima-Zwanzig equation or the Quantum Fokker-Planck equation [4]. A fundamental problem is the subdivision in relevant (system) and irrelevant (bath) degrees of freedom. If correlations between the system and bath become relevant, the corresponding degrees of freedom of the bath must be included in the set of system variables.
