3.2.10 Conclusions

In Born approximation (or time-dependent perturbation theory), the transition

� � <sup>¼</sup> <sup>w</sup>pp0; Ep <sup>¼</sup> <sup>p</sup><sup>2</sup>

0 <sup>1</sup> Ep

Insertion of Eq. (138) into the Boltzmann equation Eq. (135) yields with Eq. (139)

where we have neglect terms with higher order of E and have used the fact that

e<sup>E</sup> ¼ E=E. The electric current density Eq. (133) depends only on the deviation

The conductivity σ is the proportionality factor between the current density and

We have derived an analytical expression for the conductivity of a Lorentz plasma in terms of the relaxation time tensor ^τð Þ p . For isotropic systems, ^τij ¼ τδij,

as can be verified by insertion. Now, the conductivity reads with Eq. (137)

0 <sup>1</sup> Ep � � <sup>1</sup> � <sup>f</sup>

ð Þ <sup>2</sup>π<sup>ℏ</sup> <sup>3</sup> <sup>w</sup>pp0ð Þ <sup>1</sup> � cos <sup>ϑ</sup> ( )�<sup>1</sup>

pf 0 <sup>1</sup> Ep � � <sup>1</sup> � <sup>f</sup>

> 0 <sup>1</sup> Ep<sup>0</sup> � � <sup>1</sup> � <sup>f</sup>

� � <sup>1</sup> <sup>þ</sup> <sup>Φ</sup>ð Þ <sup>p</sup> <sup>1</sup> � <sup>f</sup>

=2m: (137)

0 <sup>1</sup> Ep � � � � � � : (138)

� � � � : (139)

� � � � <sup>Φ</sup> <sup>p</sup><sup>0</sup> ½ � ð Þ� <sup>Φ</sup>ð Þ <sup>p</sup> ,

(140)

(144)

0 <sup>1</sup> Ep<sup>0</sup>

0 <sup>1</sup> Ep

� � <sup>e</sup><sup>E</sup> � ^<sup>τ</sup> <sup>p</sup><sup>0</sup> ð Þ� <sup>p</sup><sup>0</sup> ð Þ � ^τð Þ� <sup>p</sup> <sup>p</sup> , (141)

<sup>1</sup> is an even function in p (isotropy). We obtain

0 <sup>1</sup> Ep

> 0 <sup>1</sup> Ep

� � � � : (142)

� � � � : (143)

rate is given by Fermi's Golden rule:

Non-Equilibrium Particle Dynamics

<sup>p</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> ℏ H0 p0 p � � �

� � <sup>þ</sup> <sup>Φ</sup>ð Þ <sup>p</sup>

wpp0f 0 <sup>1</sup> Ep<sup>0</sup> � � <sup>1</sup> � <sup>f</sup>

� � � 2

df 0 <sup>1</sup> Ep � � dEp

0 <sup>1</sup> Ep � � � � <sup>¼</sup>

Φð Þ¼ p e=ð Þ mkBT E � ^τð Þ� p p, the equation reads

ð d<sup>3</sup> p0 Ω ð Þ <sup>2</sup>π<sup>ℏ</sup> <sup>3</sup> <sup>w</sup>pp<sup>0</sup>

δ Ep � Ep<sup>0</sup>

To solve the Boltzmann equation Eq. (135), we make use of the ansatz

kBT ¼ f

For equilibrium distributions, we have the detailed balance condition

ð d<sup>3</sup> p0 Ω ð Þ <sup>2</sup>π<sup>ℏ</sup> <sup>3</sup> <sup>w</sup>pp0<sup>f</sup>

Φð Þ p ∝E. With the definition of the relaxation time tensor ^τð Þ p , according to

f 0 <sup>1</sup> Ep<sup>0</sup> � �

f 0 <sup>1</sup> Ep

0

p m Φð Þ p f 0 <sup>1</sup> Ep � � <sup>1</sup> � <sup>f</sup>

ð Þ <sup>2</sup>π<sup>ℏ</sup> <sup>3</sup> pzð Þ ^τð Þ� <sup>p</sup> <sup>p</sup> <sup>z</sup> <sup>f</sup>

the well-known Ziman formula στ <sup>¼</sup> <sup>τ</sup>ne<sup>2</sup>=<sup>m</sup> for the conductivity results. The solution of Eq. (141) for a momentum-dependent relaxation time is

> ð d<sup>3</sup> p0 Ω

0 <sup>1</sup> Ep � � � � <sup>¼</sup> <sup>w</sup><sup>p</sup><sup>0</sup>

w<sup>p</sup><sup>0</sup>

0 <sup>1</sup> Ep

> 0 <sup>1</sup> Ep � � <sup>1</sup> � <sup>f</sup>

e<sup>E</sup> � p ¼

the effective field E:

32

<sup>σ</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup> <sup>m</sup><sup>2</sup>kB<sup>T</sup> <sup>2</sup>

of the distribution function since f

by insertion of Eq. (138) into Eq. (133)

j el <sup>¼</sup> <sup>e</sup> Ω 2 ð d<sup>3</sup> pΩ ð Þ <sup>2</sup>π<sup>ℏ</sup> <sup>3</sup>

> ð d<sup>3</sup> p

τ Ep � � <sup>¼</sup>

f <sup>1</sup>ð Þ¼ p f

e mkB<sup>T</sup> <sup>E</sup> � <sup>p</sup> <sup>f</sup>

The method of the nonequilibrium statistical operator gives not only the derivation of the Boltzmann equation (quantal and classical), but indicates also possible improvements such as conservation of total energy, inclusion of bound state formation, hydrodynamic equations, etc.

The solution of the general Boltzmann equation is not simple, in addition to numerical simulations different approximations have been worked out. For the linearized Boltzmann equation, the relaxation time approximation can be used for elastic scattering, but for the general case (inclusion of electron-electron collisions in a plasma), the Kohler variational principle [11] can be applied. Landau-Vlasov equations for mean-field effects as well as Fokker-Planck equations for the collision term have been investigated.

The basic assumption to derive the Boltzmann equation is the selection of the single-particle distribution as relevant observable. Correlations are neglected and have to be built up in higher orders of approximation or extending the set of relevant observables. The most appropriate systems for kinetic theory are dilute gases where the collision time is short compared with the time of free flight. Irreversibility is owing to the Stoßzahlansatz for the intrinsic interaction.

### 3.3 Linear response theory

A third example, which allows the explicit elimination of the Lagrange multipliers to fulfill the self-consistency conditions, is a system near to thermodynamic equilibrium which is under the influence of mechanical (external forces) or thermodynamic (gradients of temperature, pressure, chemical potentials, etc.) perturbations. As response, currents appear in the system. Assuming linearity for small perturbations, transport coefficients are defined. Fluctuations in equilibrium are considered as a nonequilibrium state which relaxes to equilibrium, see Eq. (7).

### 3.3.1 Response to an external field

We consider a system under the influence of external (time dependent) fields acting on the particles, see [4, 11–16],

$$\mathbf{H}^t = \mathbf{H}\_\mathbf{S} + \mathbf{H}\_\mathbf{P}^t \tag{148}$$

where HS denotes the system Hamiltonian, containing all kinetic energies of the particles as well as the full interaction part. The second part H<sup>t</sup> <sup>F</sup> describes the coupling of the system to the external fields hj:

$$\mathbf{H}\_{\rm F}^{t} = -\sum\_{j} h\_{j} \mathbf{e}^{-i\alpha t} \mathbf{A}\_{j}. \tag{149}$$

transformation. Within the linear regime, the superposition of different components of the field gives the superposition of the corresponding responses. The treatment of spatial dependent external forces is also possible. As a specific advantage of the Zubarev method, thermodynamic forces such as gradients of tempera-

The main problem is to eliminate the Lagrange multipliers, the generalized response parameters Fnð Þt . As in the case of kinetic theory, this is also possible explicitly in the case of linear response theory (LRT). With the operator relation

> ð 1

dλ ∑ n

All terms have to be evaluated in such a way, that the total expression rests of

HS <sup>þ</sup> <sup>H</sup><sup>t</sup><sup>1</sup> F � �; <sup>ρ</sup>relð Þ <sup>t</sup><sup>1</sup> � � <sup>þ</sup>

Since HS commutes with ρeq (equilibrium!), the curly bracket is of order Oð Þ h . In particular, we have for the first term the time derivative in the Heisenberg picture,

0

dλ ∑ n

order Oð Þ h . For expressions (25) and (26), we find after integration by parts

i ℏ

> 3 5 ¼ β ð 1

For the second term of the integral in Eq. (156), we use Kubo's identity

ð 1

0

0

exp ið Þ Hτ=ℏ O exp ð Þ �iHτ=ℏ with τ ! iℏβλ replacing in the exponents HS by H ¼ HS � ∑cμcNc. We want to calculate expectation values of macroscopic relevant variables that commute with the particle number operator N<sup>c</sup> so that we can use both H and HS synonymously. (Mention that also the Massieu-Planck functional Φð Þt has to be expanded so that the fluctuations around the equilibrium averages

Here, we made use of the modified-Heisenberg picture Oð Þ¼ τ

dλe<sup>λ</sup>ð Þ <sup>A</sup>þ<sup>B</sup> Beð Þ <sup>1</sup>�<sup>λ</sup> A, we get for the relevant statistical operator (150) up

Fnð Þt Bnð Þ iℏβλ ρeq: (155)

∂ ∂t1

� �U†

ρrelð Þ t<sup>1</sup>

Fnð Þti <sup>B</sup>\_ <sup>n</sup>ð Þ <sup>i</sup>λβ<sup>ℏ</sup> <sup>ρ</sup>eq: (157)

<sup>d</sup><sup>λ</sup> <sup>e</sup><sup>λ</sup><sup>A</sup>½ � <sup>B</sup>; <sup>A</sup> <sup>e</sup>ð Þ <sup>1</sup>�<sup>λ</sup> <sup>A</sup>: (158)

ð Þ t; t<sup>1</sup> :

(156)

ture or chemical potentials can be treated [4, 5, 15, 16].

to first order of the nonequilibrium fluctuations Bf g<sup>n</sup>

ρrelðÞ¼ t ρeq þ β

3.3.2 Elimination of the Lagrange multipliers

eAþ<sup>B</sup> <sup>¼</sup> eA <sup>þ</sup> <sup>Ð</sup>

1 0

Nonequilibrium Statistical Operator

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

<sup>B</sup><sup>n</sup> � h i <sup>B</sup><sup>n</sup> eq n o appear).

ρϵðÞ¼ t ρrelðÞ�t

i <sup>ℏ</sup> HS; <sup>β</sup>

so that

35

2 4

3.3.3 Linearization of the NSO

ðt

<sup>d</sup>t1e<sup>ϵ</sup>ð Þ <sup>t</sup>1�<sup>t</sup> <sup>U</sup>ð Þ <sup>t</sup>; <sup>t</sup><sup>1</sup>

Fnð Þ t<sup>1</sup> Bnð Þ iλβℏ ρeq

<sup>B</sup>; eA � � <sup>¼</sup>

�∞

ð 1

dλ ∑ n

0

We characterize the nonequilibrium state by the set Bf g<sup>n</sup> of relevant observables. In the following, we assume that the equilibrium expectation values of the nonequilibrium fluctuations disappear, Bh i<sup>n</sup> <sup>e</sup><sup>q</sup> ¼ 0 (else, we have to subtract the equilibrium values).

Treating the conserved observables explicitly, we write the relevant statistical operator ρrel in the form (H ¼ HS � ∑cμcNc)

$$\rho\_{\rm rel}(t) = \mathbf{e}^{-\Phi(t) - \beta \left(\mathcal{H} - \sum\_{\pi} F\_{\pi}(t) \mathcal{B}\_{\pi}\right)}, \qquad \Phi(t) = \ln \operatorname{Tr} \left\{ \mathbf{e}^{-\beta \left(\mathcal{H} - \sum\_{\pi} F\_{\pi}(t) \mathcal{B}\_{\pi}\right)} \right\}, \tag{150}$$

where the Lagrange multipliers are divided into the equilibrium parameters β, μ and the generalized response parameters Fnð Þt , coupled to the corresponding observables. All Lagrange parameters are determined by the given mean values of these observables. In particular, we have the self-consistency conditions (18)

$$\left\langle \mathbf{B}\_{\mathfrak{n}} \right\rangle\_{\text{rel}}^{t} = \operatorname{Tr} \left\{ \rho\_{\text{rel}}(t) \mathbf{B}\_{\mathfrak{n}} \right\} = \operatorname{Tr} \left\{ \rho(t) \mathbf{B}\_{\mathfrak{n}} \right\} = \left\langle \mathbf{B}\_{\mathfrak{n}} \right\rangle^{t} \tag{151}$$

or

$$\operatorname{Tr}\left\{\rho\_{\text{irrel}}(t)\mathbf{B}\_{\text{n}}\right\}=\mathbf{0},\qquad\rho\_{\text{irrel}}(t)=\rho(t)-\rho\_{\text{rel}}(t).\tag{152}$$

The corresponding self-consistency condition for N and HS lead to the wellknown equations of state for the temperature 1=β and the chemical potential μ. Φð Þt is the Massieu-Planck functional that normalizes ρrelð Þt .

We consider the limit of weak external fields. Compared with the equilibrium distribution (13), we expect that the changes of the state of the system are also weak. We characterize the nonequilibrium state by the set Bf g<sup>n</sup> of relevant observables and assume that the averages

$$
\langle \mathbf{B}\_n \rangle^t = \operatorname{Tr} \{ \rho(t) \mathbf{B}\_n \} \propto h\_j \mathbf{e}^{-\text{i}at} \tag{153}
$$

are proportional to the external fields (linear response).

The basic assumption of LRT is that the average values Bh i<sup>n</sup> <sup>t</sup> of the additional observables, which characterize the response of the system, are proportional to the external fields. Because these external fields are arbitrarily weak, we expand all quantities with respect to the fields up to first order. If the fluctuations Bh i<sup>n</sup> <sup>t</sup> are proportional to these fields, we have also Fn ∝hj. Below, we derive linear equations that relate the response of the system to the causing external fields.

In the linear regime, we await the response parameters Fnð Þt to exhibit the same time dependence as the external fields:

$$F\_n(t) = F\_n \mathbf{e}^{-i\alpha t}.\tag{154}$$

Here, we have harmonic fields hje�iω<sup>t</sup> , but the formulation rests general as we can always express arbitrary time dependences by means of a Fourier where HS denotes the system Hamiltonian, containing all kinetic energies of the

hje�iω<sup>t</sup>

<sup>F</sup> describes the

, (150)

Aj: (149)

n Fnð Þt B<sup>n</sup> ( ) � �

particles as well as the full interaction part. The second part H<sup>t</sup>

Ht

<sup>F</sup> ¼ � ∑ j

We characterize the nonequilibrium state by the set Bf g<sup>n</sup> of relevant observables. In the following, we assume that the equilibrium expectation values of the nonequilibrium fluctuations disappear, Bh i<sup>n</sup> <sup>e</sup><sup>q</sup> ¼ 0 (else, we have to subtract the

Treating the conserved observables explicitly, we write the relevant statistical

where the Lagrange multipliers are divided into the equilibrium parameters β, μ

The corresponding self-consistency condition for N and HS lead to the wellknown equations of state for the temperature 1=β and the chemical potential μ. Φð Þt

We consider the limit of weak external fields. Compared with the equilibrium distribution (13), we expect that the changes of the state of the system are also weak. We characterize the nonequilibrium state by the set Bf g<sup>n</sup> of relevant observ-

The basic assumption of LRT is that the average values Bh i<sup>n</sup> <sup>t</sup> of the additional observables, which characterize the response of the system, are proportional to the external fields. Because these external fields are arbitrarily weak, we expand all quantities with respect to the fields up to first order. If the fluctuations Bh i<sup>n</sup> <sup>t</sup> are proportional to these fields, we have also Fn ∝hj. Below, we derive linear equations

In the linear regime, we await the response parameters Fnð Þt to exhibit the same

FnðÞ¼ <sup>t</sup> Fne�iω<sup>t</sup>

as we can always express arbitrary time dependences by means of a Fourier

and the generalized response parameters Fnð Þt , coupled to the corresponding observables. All Lagrange parameters are determined by the given mean values of these observables. In particular, we have the self-consistency conditions (18)

, <sup>Φ</sup>ðÞ¼ <sup>t</sup> ln Tr e�<sup>β</sup> <sup>H</sup>�<sup>∑</sup>

rel <sup>¼</sup> Tr <sup>ρ</sup>rel f g ð Þ<sup>t</sup> <sup>B</sup><sup>n</sup> <sup>¼</sup> Trf g <sup>ρ</sup>ð Þ<sup>t</sup> <sup>B</sup><sup>n</sup> <sup>¼</sup> h i <sup>B</sup><sup>n</sup> <sup>t</sup> (151)

h i <sup>B</sup><sup>n</sup> <sup>t</sup> <sup>¼</sup> Trf g <sup>ρ</sup>ð Þ<sup>t</sup> <sup>B</sup><sup>n</sup> <sup>∝</sup>hje�iω<sup>t</sup> (153)

: (154)

, but the formulation rests general

Tr ρirrel f g ð Þt B<sup>n</sup> ¼ 0, ρirrelðÞ¼ t ρðÞ�t ρrelð Þt : (152)

coupling of the system to the external fields hj:

Non-Equilibrium Particle Dynamics

operator ρrel in the form (H ¼ HS � ∑cμcNc)

h i <sup>B</sup><sup>n</sup> <sup>t</sup>

ables and assume that the averages

time dependence as the external fields:

Here, we have harmonic fields hje�iω<sup>t</sup>

is the Massieu-Planck functional that normalizes ρrelð Þt .

are proportional to the external fields (linear response).

that relate the response of the system to the causing external fields.

n Fnð Þt B<sup>n</sup> � �

�Φð Þ�t β H�∑

equilibrium values).

ρrelðÞ¼ t e

or

34

transformation. Within the linear regime, the superposition of different components of the field gives the superposition of the corresponding responses. The treatment of spatial dependent external forces is also possible. As a specific advantage of the Zubarev method, thermodynamic forces such as gradients of temperature or chemical potentials can be treated [4, 5, 15, 16].
