3.1.6 The Pauli equation

We consider a system whose state is described by the observable A, and which takes the value a. This can be a set of numbers in the classical case that describe the degrees of freedom we use as relevant variables. In the quantum case, this is a set of relevant observables that describe the state of the system. The eigenvalue a corresponds to a state vector ∣ai in the Hilbert space.

At time t, we expect a probability distribution p1ð Þ a; t to find the system in state a, if the property A is measured. The change of the probability p1ð Þ a; t with time is described by a master equation or balance equation

$$\frac{d}{dt}p\_1(a,t) = \sum\_{a' \neq a} \left[ w\_{aa'} p\_1(a',t) - w\_{a'a} p\_1(a,t) \right]. \tag{72}$$

In the context of the time evolution of a physical system, this master equation is also denoted as Pauli equation. We derive it from a microscopical approach using perturbation theory. The statistical operator ρð Þt follows the von Neumann equation of motion (8) with the Hamiltonian

Non-Equilibrium Particle Dynamics

$$H = H^0 + \lambda H'\tag{73}$$

On the right-hand side, we can drop the projector Dn. Its action disappears

second contribution is of second order in λ and will be dropped here because we consider only the lowest order in λ (ρirrelð Þt is also of the order λ). This is denoted as

<sup>i</sup><sup>ℏ</sup> <sup>H</sup>0; <sup>ρ</sup>irrelð Þ<sup>t</sup> � � <sup>¼</sup> <sup>1</sup>

; ρrel ½ � ð Þ t<sup>1</sup> e

; ρrel ½ � ð Þ t<sup>1</sup> e

e �i

nm we have

<sup>ℏ</sup>ð Þ Em�En ð Þ <sup>t</sup>1�<sup>t</sup> <sup>þ</sup> <sup>e</sup>

<sup>¼</sup> <sup>2</sup>πλ<sup>2</sup> ℏ

wnn0p<sup>1</sup> n<sup>0</sup> ð Þ� ; t wn<sup>0</sup>

mn <sup>p</sup>1ð Þ� <sup>n</sup>; <sup>t</sup> <sup>p</sup>1ð Þ <sup>m</sup>; <sup>t</sup> � �

�i <sup>ℏ</sup>ð Þ Em�En ð Þ <sup>t</sup>1�<sup>t</sup> h idt1: (85)

np1ð Þ <sup>n</sup>; <sup>t</sup> � �: (86)

H0 nm � � � � 2

δð Þ En � Em : (87)

<sup>ℏ</sup>H0ð Þ <sup>t</sup>1�<sup>t</sup> λH<sup>0</sup>

<sup>ℏ</sup>H0ð Þ <sup>t</sup>1�<sup>t</sup> H<sup>0</sup>

This result describes a memory effect. The change of ρrelð Þt is determined by the values ρrelð Þ t<sup>1</sup> at all previous times t<sup>1</sup> ≤t. In the Markov approximation, we replace ρrelð Þ t<sup>1</sup> by ρrelð Þt so that memory effects are neglected. This is justified in the limit

<sup>i</sup><sup>ℏ</sup> <sup>λ</sup>H<sup>0</sup>

�i

�i <sup>ℏ</sup>H0ð Þ <sup>t</sup>1�<sup>t</sup> h idt1: (83)

<sup>ℏ</sup>H0ð Þ <sup>t</sup>1�<sup>t</sup> ; <sup>ρ</sup>relð Þ<sup>t</sup> h i h i dt1: (84)

1

With this expression for ρirrelð Þt , we find a closed equation for ρrelð Þt ,

;e i

; e i <sup>ℏ</sup>H0ð Þ <sup>t</sup>1�<sup>t</sup> H<sup>0</sup>

This expression has similar structure as the QME (39) an can be treated in the same way. The right-hand side Dρrelð Þt is related to the dissipator after subtracting

Explicit expressions for the time evolution of the density matrix are obtained by projection on the basis ∣ni. With the matrix elements njρrel h i ð Þj t m ¼ δn,mp1ð Þ n; t as

; ρirrel ½ � ð Þt , the

; ρrel ½ � ð Þt : (81)

<sup>ℏ</sup>H0ð Þ <sup>t</sup>1�<sup>t</sup> dt1: (82)

because ρrel is diagonal. It is seen that ρirrelð Þt is of the order λ.

ρirrelðÞþt ερirrelðÞ�t

1 iℏ ðt

ðt

ðt

well as <sup>n</sup>jH<sup>0</sup>j<sup>m</sup> � � <sup>¼</sup> <sup>δ</sup>n,mEn and <sup>n</sup>jH<sup>0</sup> h i <sup>j</sup><sup>m</sup> <sup>¼</sup> <sup>H</sup><sup>0</sup>

dt <sup>p</sup>1ð Þ¼� <sup>n</sup>; <sup>t</sup>

λ2 <sup>ℏ</sup><sup>2</sup> <sup>∑</sup> m H0 nmH<sup>0</sup>

Performing the integral over t1, we find [with the Dirac identity

� ðt �∞ e <sup>ε</sup>ð Þ <sup>t</sup>1�<sup>t</sup> e i

<sup>x</sup> � iπδð Þ x ] the Pauli equation

n0 6¼n

þ

� �

1 �iωnm þ ϵ

The transition rates are given by Fermi's Golden rule,

dt <sup>p</sup>1ð Þ¼ <sup>n</sup>; <sup>t</sup> <sup>∑</sup>

d

d

�∞ e <sup>ε</sup>ð Þ <sup>t</sup>1�<sup>t</sup> H<sup>0</sup>

�∞ e <sup>ε</sup>ð Þ <sup>t</sup>1�<sup>t</sup> H<sup>0</sup>

λ ! 0 because then ρrelð Þt changes only slowly with time. Then

�∞ e <sup>ε</sup>ð Þ <sup>t</sup>1�<sup>t</sup> e i

The solution is simple by integration,

ρirrelðÞ¼ t

The proof is given by insertion.

λ2 <sup>ℏ</sup><sup>2</sup> <sup>D</sup><sup>n</sup>

λ2 <sup>ℏ</sup><sup>2</sup> <sup>D</sup><sup>n</sup>

∂ ∂t

∂ ∂t

lim<sup>ϵ</sup>!þ<sup>0</sup>

1 <sup>x</sup>þi<sup>ϵ</sup> � <sup>P</sup> <sup>1</sup>

wnm <sup>¼</sup> lim<sup>ϵ</sup>!<sup>0</sup>

21

λ2 <sup>ℏ</sup><sup>2</sup> <sup>H</sup><sup>0</sup> nm � � � � <sup>2</sup> 1 iωnm þ ϵ

ρrelðÞ¼� t

ρrelðÞ¼� t

the Lamb shift contribution.

Born approximation. We have

Nonequilibrium Statistical Operator

DOI: http://dx.doi.org/10.5772/intechopen.84707

∂ ∂t

In the remaining projection 1ð Þ � <sup>D</sup><sup>n</sup> <sup>H</sup>0; <sup>ρ</sup>irrelð Þ<sup>t</sup> � � <sup>þ</sup> ð Þ <sup>1</sup> � <sup>D</sup><sup>n</sup> <sup>H</sup><sup>0</sup>

where the solution of the eigenvalue problem for <sup>H</sup><sup>0</sup> is known, <sup>H</sup><sup>0</sup>∣ni ¼ En∣ni. The probabilities to find the system in the state ∣ni are given by the diagonal elements of ρð Þt in this representation,

$$p\_1(n, t) = \langle n|\rho(t)|n\rangle. \tag{74}$$

First, we consider the special case λ ¼ 0, where the von Neumann equation is easily solved:

$$
\rho\_{nm}(t) = \langle n|\rho(t)|m\rangle = e^{-i\alpha\_{nm}(t-t\_0)}\rho\_{nm}(t\_0), \qquad \hbar\alpha\_{nm} = E\_n - E\_m \tag{75}
$$

if ρnmð Þ t<sup>0</sup> is given. The nondiagonal elements ρnmð Þt , n 6¼ m are oscillating. The periodic time dependence of the density matrix that arises in the nondiagonal elements has nothing to do with any time evolution or irreversibility. It expresses the coherences in the system. The diagonal elements

$$
\rho\_{nn}(t) = p\_1(n, t) = \langle n|\rho(t)|n\rangle \tag{76}
$$

do not change with time and can be considered as conserved quantities if λ ¼ 0.

To find the initial distribution, we consider the probabilities as relevant observables that describe the nonequilibrium state at t0. If there are no further information on coherence, the relevant statistical operator is diagonal,

$$
\rho\_{\rm rel}(t\_0) = \sum\_n p\_1(n, t\_0) |n\rangle\langle n| = \sum\_n p\_1(n, t\_0) \mathcal{P}\_n. \tag{77}
$$

We introduced the projection operator P<sup>n</sup> ¼ ∣nihn∣. The solution is ρðÞ¼ t ρrelð Þ t<sup>0</sup> . The case λ ¼ 0 is a trivial case, nothing happens.

Now, we consider a small perturbation as expressed by the parameter λ. As before, we consider the probabilities as relevant observables that describe the system in nonequilibrium. We project the diagonal part of the statistical operator,

$$\rho\_{\text{rel}}(t) = \text{diag}[\rho(t)] = \mathbf{D}\_{\text{n}}\rho(t) = \sum\_{n} \mathcal{P}\_{\text{n}}\rho(t)\mathcal{P}\_{n}.\tag{78}$$

The difference ρirrelðÞ¼ t ρðÞ�t ρrelðÞ¼ t ð Þ 1 � D<sup>n</sup> ρ is the irrelevant part of the full statistical operator.

The problem to obtain the time evolution of the probabilities p1ð Þ n; t is solved if we find an equation of evolution for ρrelð Þt . We use the method of the nonequilibrium statistical operator and start with the extended von Neumann equation (27). For the projection, we obtain (D<sup>n</sup> is linear and commutes with ∂=∂t)

$$\frac{\partial}{\partial t}\rho\_{\text{rel}}(t) = \frac{1}{\mathbf{i}\,\hbar}\mathbf{D}\_{n}[\lambda H', \rho\_{\text{irrel}}(t)].\tag{79}$$

We assumed that <sup>H</sup><sup>0</sup> is diagonal with <sup>ρ</sup>relð Þ<sup>t</sup> so that the commutator vanishes. Furthermore, the diagonal elements of the commutator of a diagonal matrix with an arbitrary matrix disappear. For the irrelevant part we have

$$\frac{\partial}{\partial t}\rho\_{\text{irrel}}(t) + \epsilon \rho\_{\text{irrel}}(t) - \frac{1}{\mathbf{i}\,\hbar}(\mathbf{1} - \mathbf{D}\_{\text{n}})[H, \rho\_{\text{irrel}}(t)] = \frac{\mathbf{1}}{\mathbf{i}\,\hbar}(\mathbf{1} - \mathbf{D}\_{\text{n}})[\lambda H', \rho\_{\text{rel}}(t)].\tag{80}$$

Nonequilibrium Statistical Operator DOI: http://dx.doi.org/10.5772/intechopen.84707

<sup>H</sup> <sup>¼</sup> <sup>H</sup><sup>0</sup> <sup>þ</sup> <sup>λ</sup>H<sup>0</sup> (73)

p1ð Þ¼ n; t h i njρð Þj t n : (74)

�iωnmð Þ <sup>t</sup>�t<sup>0</sup> <sup>ρ</sup>nmð Þ <sup>t</sup><sup>0</sup> , <sup>ℏ</sup>ωnm <sup>¼</sup> En � Em (75)

ρnnðÞ¼ t p1ð Þ¼ n; t h i njρð Þj t n (76)

p1ð Þ n; t<sup>0</sup> Pn: (77)

Pnρð Þt Pn: (78)

; ρirrel ½ � ð Þt : (79)

; ρrel ½ � ð Þt : (80)

n

n

1

<sup>i</sup><sup>ℏ</sup> ð Þ <sup>1</sup> � <sup>D</sup><sup>n</sup> <sup>λ</sup>H<sup>0</sup>

where the solution of the eigenvalue problem for <sup>H</sup><sup>0</sup> is known, <sup>H</sup><sup>0</sup>∣ni ¼ En∣ni.

First, we consider the special case λ ¼ 0, where the von Neumann equation is

if ρnmð Þ t<sup>0</sup> is given. The nondiagonal elements ρnmð Þt , n 6¼ m are oscillating. The periodic time dependence of the density matrix that arises in the nondiagonal elements has nothing to do with any time evolution or irreversibility. It expresses

do not change with time and can be considered as conserved quantities if λ ¼ 0. To find the initial distribution, we consider the probabilities as relevant observables that describe the nonequilibrium state at t0. If there are no further information

p1ð Þ n; t<sup>0</sup> ∣n〉〈n∣ ¼ ∑

We introduced the projection operator P<sup>n</sup> ¼ ∣nihn∣. The solution is ρðÞ¼ t ρrelð Þ t<sup>0</sup> .

The difference ρirrelðÞ¼ t ρðÞ�t ρrelðÞ¼ t ð Þ 1 � D<sup>n</sup> ρ is the irrelevant part of the

The problem to obtain the time evolution of the probabilities p1ð Þ n; t is solved if we find an equation of evolution for ρrelð Þt . We use the method of the nonequilibrium statistical operator and start with the extended von Neumann equation (27). For the projection, we obtain (D<sup>n</sup> is linear and commutes with ∂=∂t)

> 1 iℏ

<sup>i</sup><sup>ℏ</sup> ð Þ <sup>1</sup> � <sup>D</sup><sup>n</sup> <sup>H</sup>; <sup>ρ</sup>irrel ½ �¼ ð Þ<sup>t</sup>

D<sup>n</sup> λH<sup>0</sup>

We assumed that <sup>H</sup><sup>0</sup> is diagonal with <sup>ρ</sup>relð Þ<sup>t</sup> so that the commutator vanishes. Furthermore, the diagonal elements of the commutator of a diagonal matrix with an

Now, we consider a small perturbation as expressed by the parameter λ. As before, we consider the probabilities as relevant observables that describe the system in nonequilibrium. We project the diagonal part of the statistical

ρrelðÞ¼ t diag½ �¼ ρð Þt DnρðÞ¼ t ∑

The probabilities to find the system in the state ∣ni are given by the diagonal

elements of ρð Þt in this representation,

Non-Equilibrium Particle Dynamics

ρnmðÞ¼ t h i njρð Þj t m ¼ e

the coherences in the system. The diagonal elements

on coherence, the relevant statistical operator is diagonal,

n

ρrelð Þ¼ t<sup>0</sup> ∑

∂ ∂t

arbitrary matrix disappear. For the irrelevant part we have

1

ρrelðÞ¼ t

The case λ ¼ 0 is a trivial case, nothing happens.

easily solved:

operator,

∂ ∂t

20

full statistical operator.

ρirrelðÞþt ϵρirrelðÞ�t

On the right-hand side, we can drop the projector Dn. Its action disappears because ρrel is diagonal. It is seen that ρirrelð Þt is of the order λ.

In the remaining projection 1ð Þ � <sup>D</sup><sup>n</sup> <sup>H</sup>0; <sup>ρ</sup>irrelð Þ<sup>t</sup> � � <sup>þ</sup> ð Þ <sup>1</sup> � <sup>D</sup><sup>n</sup> <sup>H</sup><sup>0</sup> ; ρirrel ½ � ð Þt , the second contribution is of second order in λ and will be dropped here because we consider only the lowest order in λ (ρirrelð Þt is also of the order λ). This is denoted as Born approximation. We have

$$\frac{\partial}{\partial t}\rho\_{\text{irrel}}(t) + \varepsilon\rho\_{\text{irrel}}(t) - \frac{\mathbf{1}}{\mathbf{i}\,\hbar} \left[H^0, \rho\_{\text{irrel}}(t)\right] = \frac{\mathbf{1}}{\mathbf{i}\,\hbar} [\lambda H', \rho\_{\text{rel}}(t)].\tag{81}$$

The solution is simple by integration,

$$\rho\_{\rm irrel}(t) = \frac{1}{\mathbf{i}\,\hbar} \int\_{-\infty}^{t} e^{c(t\_1 - t)} e^{\frac{i}{\hbar}H^0(t\_1 - t)} [\lambda H', \rho\_{\rm rel}(t\_1)] e^{-\frac{i}{\hbar}H^0(t\_1 - t)} dt\_1. \tag{82}$$

The proof is given by insertion.

With this expression for ρirrelð Þt , we find a closed equation for ρrelð Þt ,

$$\frac{\partial}{\partial t}\rho\_{\text{rel}}(t) = -\frac{\dot{\lambda}^2}{\hbar^2}\mathcal{D}\_{\text{n}}\int\_{-\infty}^t e^{\varepsilon(t\_1 - t)} \left[H', e^{\frac{i}{\hbar}H^0(t\_1 - t)}[H', \rho\_{\text{rel}}(t\_1)]e^{-\frac{i}{\hbar}H^0(t\_1 - t)}\right]dt\_1. \tag{83}$$

This result describes a memory effect. The change of ρrelð Þt is determined by the values ρrelð Þ t<sup>1</sup> at all previous times t<sup>1</sup> ≤t. In the Markov approximation, we replace ρrelð Þ t<sup>1</sup> by ρrelð Þt so that memory effects are neglected. This is justified in the limit λ ! 0 because then ρrelð Þt changes only slowly with time. Then

$$\frac{\partial}{\partial t}\rho\_{\text{rel}}(t) = -\frac{\lambda^2}{\hbar^2}\mathbf{D}\_{\text{n}}\int\_{-\infty}^{t} e^{c(t\_1-t)} \left[H', \left[e^{\frac{i}{\hbar}H^0(t\_1-t)}H'e^{-\frac{i}{\hbar}H^0(t\_1-t)}, \rho\_{\text{rel}}(t)\right]\right]dt\_1. \tag{84}$$

This expression has similar structure as the QME (39) an can be treated in the same way. The right-hand side Dρrelð Þt is related to the dissipator after subtracting the Lamb shift contribution.

Explicit expressions for the time evolution of the density matrix are obtained by projection on the basis ∣ni. With the matrix elements njρrel h i ð Þj t m ¼ δn,mp1ð Þ n; t as well as <sup>n</sup>jH<sup>0</sup>j<sup>m</sup> � � <sup>¼</sup> <sup>δ</sup>n,mEn and <sup>n</sup>jH<sup>0</sup> h i <sup>j</sup><sup>m</sup> <sup>¼</sup> <sup>H</sup><sup>0</sup> nm we have

$$\frac{d}{dt}p\_1(n,t) = -\frac{\hbar^2}{\hbar^2} \sum\_m H'\_{nm} H'\_{mn} \left[ p\_1(n,t) - p\_1(m,t) \right]$$

$$\times \int\_{-\infty}^t e^{\imath(t\_1 - t)} \left[ e^{\not\!\equiv (E\_m - E\_a)(t\_1 - t)} + e^{-\not\!\equiv (E\_m - E\_a)(t\_1 - t)} \right] dt\_1. \tag{85}$$

Performing the integral over t1, we find [with the Dirac identity lim<sup>ϵ</sup>!þ<sup>0</sup> 1 <sup>x</sup>þi<sup>ϵ</sup> � <sup>P</sup> <sup>1</sup> <sup>x</sup> � iπδð Þ x ] the Pauli equation

$$\frac{d}{dt}p\_1(n,t) = \sum\_{n' \neq n} \left[ w\_{nn'} p\_1(n',t) - w\_{n'n} p\_1(n,t) \right].\tag{86}$$

The transition rates are given by Fermi's Golden rule,

$$w\_{nm} = \lim\_{\varepsilon \to 0} \frac{\lambda^2}{\hbar^2} \left| H'\_{nm} \right|^2 \left( \frac{1}{i o\_{nm} + \varepsilon} + \frac{1}{-i o\_{nm} + \varepsilon} \right) = \frac{2 \pi \lambda^2}{\hbar} \left| H'\_{nm} \right|^2 \delta(E\_n - E\_m). \tag{87}$$

## 3.1.7 Properties of the Pauli equation

The transition rate wnm obeys the condition of detailed balance, wmn ¼ wnm, the inverse transition has the same rate. This follows because H<sup>0</sup> is hermitean,

$$
\langle n|H'|m\rangle = \langle m|H'^+|n\rangle^\* = \langle m|H'|n\rangle^\*.\tag{88}
$$

rate equation (86) describes natural line width, detailed balance, and thermal

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

Historically, nonequilibrium statistical physics was first developed as the kinetic theory of gases [7] by Boltzmann. We start with classical systems to explain the problem to be solved in kinetic theory. The more general case of quantum systems contains no additional complications, but the concepts become more evident in the classical limit. We give results for both cases, the general quantum case and the classical limit. Reduced distribution functions are considered as the relevant observables. Closed equations of evolution are obtained describing irreversible

The standard treatment of a classical dynamical system can be given in terms of

system. The evolution of a particular system with time is given by a trajectory in the phase space. Depending on the initial conditions different trajectories are taken. Within statistical physics, instead of a special system, an ensemble of identical systems is considered, consisting of the same constituents and described by the same Hamiltonian, but at different initial conditions (microstates), which are compatible with the values of a given set of relevant observables characterizing the macrostate of the system. The probability of the realization of a macrostate by a special microstate, i.e., a point in the 6N-dimensional phase space (Γ-space), is

� � define the microstate of the

; t � � which is normalized,

<sup>N</sup>!h<sup>3</sup><sup>N</sup> : (92)

<sup>N</sup>!h<sup>3</sup><sup>N</sup> <sup>¼</sup> <sup>d</sup><sup>3</sup>Nxd<sup>3</sup>Np

the Hamilton canonical equations. In classical mechanics, we have generalized coordinates and canonic conjugated momenta describing the state of the system, e.g., a point in the 6N-dimensional phase space (Γ-space) in the case of N point

; <sup>t</sup> � � <sup>¼</sup> <sup>1</sup>; d<sup>Γ</sup> <sup>¼</sup> <sup>d</sup><sup>N</sup>rd<sup>N</sup><sup>p</sup>

In nonequilibrium, the N-particle distribution function depends on the time t.

equilibrium as stationary solution.

DOI: http://dx.doi.org/10.5772/intechopen.84707

Nonequilibrium Statistical Operator

3.1.9 Conclusions

system variables.

3.2 Kinetic theory

processes.

23

3.2.1 The Liouville equation

masses. The 6N degrees of freedom r1; p1…rN; p<sup>N</sup>

given by the N-particle distribution function f <sup>N</sup> ri; p<sup>i</sup>

dΓf <sup>N</sup> ri; p<sup>i</sup>

ð

An important property is that it describes irreversible evolution with time. For the relevant entropy SrelðÞ¼� t kB∑np1ð Þ n; t ln p1ð Þ n; t we find

$$\frac{dS\_{\rm rel}(t)}{dt} = -k\_{\rm B} \sum\_{n} \sum\_{m} w\_{nm} \left[ p\_1(m, t) - p\_1(n, t) \right] \ln \left[ p\_1(n, t) \right] - k\_{\rm B} \sum\_{n} \frac{p\_1(n, t)}{p\_1(n, t)} \frac{\partial p\_1(n, t)}{\partial t} \tag{8}$$

$$= \frac{1}{2} k\_{\rm B} \sum\_{n} \sum\_{m} w\_{nm} \left[ p\_1(n, t) - p\_1(m, t) \right] \left[ \ln \left[ p\_1(n, t) \right] - \ln \left[ p\_1(m, t) \right] \right] \ge 0. \tag{8}$$

We used <sup>d</sup> dt <sup>∑</sup>np1ð Þ¼ <sup>n</sup>; <sup>t</sup> <sup>d</sup> dt 1 ¼ 0 and interchanged n with m in the half 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 equilibrium.

### 3.1.8 Example: transition rates

We consider transitions between eigenstates of H<sup>0</sup> owing to interaction. A typical case is the collisions expressed by a† k1 a† k2 ak0 2 ak0 <sup>1</sup> between the (momentum) 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 (Ek <sup>¼</sup> <sup>ℏ</sup><sup>2</sup> k2 =2m,ω<sup>q</sup> ¼ c∣q∣)

$$H\_0 = \sum\_k E\_k a\_\mathbf{k}^\dagger a\_\mathbf{k} + \sum\_q \hbar \alpha\_q b\_\mathbf{q}^\dagger b\_\mathbf{q} \tag{90}$$

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

$$H\_{\rm int} = \sum\_{k,k',q} \nu \left( kk', q \right) a\_{k'}^{\dagger} a\_{k} b\_{q}^{\dagger} + \text{h.c.} \tag{91}$$

The transition rates (87) are calculated between the initial state ∣ni ¼ ∣ki, energy En ¼ Ek, and the final state ∣mi ¼ ∣k<sup>0</sup> , qi, energy Em ¼ Ek<sup>0</sup> þ ℏω<sup>q</sup> for emission in the vacuum state. For absorption, the corresponding process can be given. For free particles ∣ki ¼ ∣k, σi, the matrix element v k; σ; k<sup>0</sup> ; σ<sup>0</sup> ; <sup>q</sup> <sup>∝</sup>δk<sup>0</sup> <sup>þ</sup>q,k must fulfill 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

rate equation (86) describes natural line width, detailed balance, and thermal equilibrium as stationary solution.
