3.2.8 The linearized Boltzmann equation

We interchange indices 1 \$ 2, 1<sup>0</sup> \$ 2<sup>0</sup>

part is not conserved. The equilibrium solution f

1 f 0 <sup>1</sup> p<sup>1</sup> � � <sup>∓</sup><sup>1</sup> !

, use the symmetries of w p1p2p<sup>0</sup>

because ln x is a monotonous function of x. We obtain 4 <sup>d</sup>Srel

1 \$ 2<sup>0</sup>

If f 0

quantum gases,

, 2 \$ 1<sup>0</sup>

1 f 0 <sup>1</sup> ð Þ p

(relevant) entropy can increase.

Non-Equilibrium Particle Dynamics

∓1 !

> 1 f 0 <sup>1</sup> ð Þ p

In the classical limit, we have f

order collisions are improbable.

the set of relevant observables.

30

entropy (Φ ¼ ln Z is the Matthieu-Planck function)

Seq ¼ �k<sup>B</sup>

ð

where s denotes the spin of the particle.

3.2.7 Beyond the Boltzmann kinetic equation

; furthermore 1 \$ 1<sup>0</sup>

1p0 2

0

The collision integral guarantees conservation of total momentum, particle number, and kinetic energy. However, the total energy including the interaction

> � <sup>1</sup> f 0 <sup>1</sup> <sup>p</sup><sup>0</sup> ð Þ <sup>∓</sup><sup>1</sup> !

<sup>∓</sup><sup>1</sup> <sup>¼</sup> <sup>e</sup><sup>β</sup>ð Þ <sup>E</sup>p�<sup>μ</sup> , f <sup>0</sup>

0

In deriving the Boltzmann equation, different approximations have been performed: only binary collisions are considered, three-particle, and higher order collisions are neglected. Memory effects and spatial inhomogeneities have been neglected. The single-particle distribution was considered as relevant observable in the Markov approximation. These approximations can be compared with the Born-Markov approximation discussed in context with the quantum master equation. Instead of the Born approximation that is possible for weak interactions, the binary collision approximation is possible in the low-density limit, where three- and higher

In the case of thermal equilibrium, the Boltzmann entropy Srel (127) coincides with the entropy of the ideal (classical or quantum) gas. The equilibrium solution of the Boltzmann equation leads to the entropy of the ideal gas and gives not the correct equation of state for an interacting system that are derived from the Gibbs

see Eq. (13). This deficit of the Boltzmann equation arises because binary collisions are considered where the kinetic energy of the asymptotic states is conserved. Only the single-particle distribution is a relevant observable and is correctly reproduced. It can be improved if the total energy, which is conserved, is considered as a relevant observable. Alternatively, we can also include the two-particle distribution function in the set of relevant observables. An important example is the formation of bound states as a signature of strong correlations in the system. Then, the momentum distribution of bound states has to be included in

<sup>1</sup> ð Þ p depends only on energy, we find the well-known result for ideal

, 2 \$ 2<sup>0</sup>

� � and ð Þ <sup>x</sup><sup>1</sup> � <sup>x</sup><sup>2</sup> ð Þ ln <sup>x</sup><sup>1</sup> � ln <sup>x</sup><sup>2</sup> <sup>≥</sup><sup>0</sup>

<sup>1</sup> ð Þ <sup>p</sup> follows from <sup>d</sup>Srel

1 f 0 <sup>1</sup> p<sup>0</sup> 1 � � <sup>∓</sup><sup>1</sup> !

<sup>1</sup> ð Þ¼ <sup>p</sup> <sup>e</sup><sup>β</sup>ð Þ <sup>E</sup>p�<sup>μ</sup> � <sup>1</sup>

<sup>1</sup> ð Þ¼ <sup>p</sup> <sup>e</sup>�<sup>β</sup>ð Þ <sup>E</sup>p�<sup>μ</sup> with eβμ <sup>¼</sup> <sup>N</sup>

h i�<sup>1</sup>

dΓ Φð Þ þ βH exp ½ � �Φ � βH , (131)

Ω

2πℏ<sup>2</sup> mkBT � �<sup>3</sup>=<sup>2</sup>

; and

<sup>d</sup><sup>t</sup> ¼ 0:

¼ 0: (129)

: (130)

1 ð Þ <sup>2</sup>sþ<sup>1</sup> ,

<sup>d</sup><sup>t</sup> ≥0, the Boltzmann

Different approximations are known to obtain solutions of the Boltzmann equation, see [4, 5]. A serious problem in solving the Boltzmann equation is its nonlinearity as we have terms of the form <sup>f</sup> <sup>1</sup> <sup>p</sup>1; <sup>t</sup> � �<sup>f</sup> <sup>1</sup> <sup>p</sup>2; <sup>t</sup> � �. Special cases that allow for linearization are two-component systems with a large difference in the masses or concentration. Linearization is also possible in the case where the deviation from some equilibrium distribution is small. As an application, we consider the calculation of electrical conductivity in plasmas.

We investigate a plasma of ions and electrons under the influence of an external electric field Eext. For simplicity, we assume Eext to be homogeneous and independent of time (statical conductivity σ). For moderate fields, we await a linear behavior of the plasma following Ohm's Law:

$$
\mathbf{j}\_{\rm el} = \sigma \mathbf{E}.\tag{132}
$$

[Note that in Eq. (132) E is not the external field, but the effective electric field in the medium (the plasma), being the superposition of the external field Eext and the polarization field εP]. j el is the average electric current defined via the singleparticle distribution function f <sup>1</sup>

$$\mathbf{j}\_{el} = \frac{1}{\Omega} \left\langle \sum\_{i}^{N} e\_{i} \boldsymbol{\nu}\_{i} \right\rangle = \sum\_{\boldsymbol{s}} e\_{\boldsymbol{s}} \int \mathbf{d}^{3} \boldsymbol{\nu} \boldsymbol{\nu} f\_{1}(\boldsymbol{\nu}, \boldsymbol{s}) = \sum\_{\boldsymbol{s}} \frac{e\_{\boldsymbol{s}}}{m\_{\boldsymbol{s}}} \left[ \frac{\mathbf{d}^{3} \mathbf{p}}{(2\pi\hbar)^{3}} \,\, \mathbf{p} f\_{1}(\boldsymbol{\nu}, \boldsymbol{s}) . \right. \tag{133}$$

Here, we have kept the index s for the different sorts. In the following, we will skip this index as we only consider electrons being responsible for the electric current.

We recall the Boltzmann equation

$$\frac{\partial}{\partial m}\frac{\partial}{\partial r}f\_{\ 1} + e\mathbf{E}\frac{\partial}{\partial \mathbf{p}'}f\_{\ 1} + \left(\frac{\partial}{\partial t}f\_{\ 1}\right)\_{\text{St}} = \mathbf{0},\tag{134}$$

m is the electron mass and �e the electron charge. The first term in this equation vanishes because of homogeneity of the system. For the collision term, we take the expression Eq. (125) in the generalized form for quantum systems. After the distribution function of the collision partner has been replaced by the equilibrium distribution, we have

$$\left(\frac{\partial}{\partial t}f\_1\right)\_{\text{St}} = \left[\frac{\text{d}^3 \text{p}'\Omega}{\left(2\pi\hbar\right)^3} \{f\_1(\text{p}')w\_{\text{pp}'}\left[1 - f\_1(\text{p})\right] - f\_1(\text{p})w\_{\text{p}'\text{p}}\left[1 - f\_1(\text{p}')\right]\},\tag{135}$$

where wpp<sup>0</sup> is the transition rate from the momentum state p to the state p<sup>0</sup> . The quantum behavior of the collisions is taken into account via the Pauli blocking factors 1 � <sup>f</sup> <sup>1</sup>ð Þ <sup>p</sup> � �.
