3.2.1 The Liouville equation

The standard treatment of a classical dynamical system can be given in terms of 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 masses. The 6N degrees of freedom r1; p1…rN; p<sup>N</sup> � � define the microstate of the 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 given by the N-particle distribution function f <sup>N</sup> ri; p<sup>i</sup> ; t � � which is normalized,

$$d\Gamma f\_N(r\_i, \mathbf{p}\_i, t) = \mathbf{1}; \qquad d\Gamma = \frac{\mathbf{d}^N r \mathbf{d}^N \mathbf{p}}{N! h^{3N}} = \frac{\mathbf{d}^{3N} \mathbf{x} \mathbf{d}^{3N} p}{N! h^{3N}}.\tag{92}$$

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

The macroscopic properties can be evaluated as averages of the microscopic quantities a ri; p<sup>i</sup> � � with respect to the distribution function <sup>f</sup> <sup>N</sup> <sup>r</sup>i; <sup>p</sup><sup>i</sup> ; t � �:

$$
\langle A \rangle^t = \int d\Gamma a(\mathbf{r}\_i, \mathbf{p}\_i) f\_N(\mathbf{r}\_i, \mathbf{p}\_i, t) \,. \tag{93}
$$

The equation of motion (98) for the reduced distribution function fs is not closed because on the right-hand side the higher order distribution function fsþ<sup>1</sup> appears. In its turn, fsþ<sup>1</sup> obeys a similar equation that contains fsþ2, etc. This structure of a system of equations is denoted as hierarchy. To obtain a kinetic equation that is a closed equation for the reduced distribution function, one has to truncate the BBGKY hierarchy, expressing the higher order distribution function

In the quantum case, the distribution function f <sup>N</sup> is replaced by the statistical operator ρ that describes the state of the system, and the equation of motion is the von Neumann equation (8). The quantum statistical reduced density matrix is

It is related to correlation functions, the Wigner function, Green functions,

We consider the equations of motion for reduced distribution functions. For the

iℏ

Similar as for the BBGKY hierarchy, we obtain in general a hierarchy of equa-

Like in the classical case, we have to truncate this chain of equations. For example, in the Boltzmann equation for f <sup>1</sup>ð Þt , the higher order distribution function

To evaluate the averages of single-particle properties such as particle current or kinetic energy, only the single-particle distribution must be known. Then, the single-particle distribution contains the relevant information, the higher

We are looking for an equation of motion for the single-particle distribution

<sup>∂</sup><sup>p</sup> <sup>f</sup> <sup>1</sup> <sup>¼</sup> <sup>∂</sup> ∂t

f <sup>1</sup> þ v

∂

<sup>∂</sup><sup>r</sup> <sup>f</sup> <sup>1</sup> <sup>þ</sup> <sup>F</sup> <sup>∂</sup>

<sup>∂</sup><sup>p</sup> <sup>f</sup> <sup>1</sup> <sup>¼</sup> <sup>0</sup>:

function f <sup>1</sup>ð Þ r; p; t , taking into account short range interactions and binary collisions. For the total derivative with respect to time we find, see Eq. (95),

<sup>∂</sup><sup>r</sup> <sup>f</sup> <sup>1</sup> <sup>þ</sup> <sup>p</sup>\_ <sup>∂</sup>

f <sup>2</sup>ð Þt is replaced by a product of single-particle distribution functions f <sup>1</sup>ð Þt .

Tr ρ ψ†

<sup>∂</sup><sup>t</sup> <sup>¼</sup> function of <sup>ρ</sup>sð Þ<sup>t</sup> ; <sup>ρ</sup><sup>s</sup>þ<sup>1</sup>ð Þ<sup>t</sup> : (102)

.

ð Þ r<sup>s</sup> ψ r<sup>0</sup> s …ψ r<sup>0</sup>

: (99)

ð Þ <sup>p</sup> <sup>ψ</sup> <sup>p</sup><sup>0</sup> ð Þ : (100)

ð Þ <sup>p</sup> <sup>ψ</sup> <sup>p</sup><sup>0</sup> ð Þ; <sup>H</sup> : (101)

1

fsþ<sup>1</sup> by the lower order distribution functions <sup>f</sup> <sup>1</sup>; …; fs

defined as average over creation and annihilation operators,

; <sup>t</sup> <sup>¼</sup> Tr <sup>ρ</sup>ð Þ<sup>t</sup> <sup>ψ</sup>†ð Þ <sup>r</sup><sup>1</sup> …ψ†

single-particle density matrix in momentum representation, we have

<sup>ρ</sup><sup>1</sup> <sup>p</sup>; <sup>p</sup><sup>0</sup> ð Þ¼ ; <sup>t</sup> Tr <sup>ρ</sup>ð Þ<sup>t</sup> <sup>ψ</sup>†

ð Þ <sup>p</sup> <sup>ψ</sup> <sup>p</sup><sup>0</sup> ð Þ <sup>¼</sup> <sup>1</sup>

3.2.3 Quantum statistical reduced distributions

Nonequilibrium Statistical Operator

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

ρ<sup>s</sup> r1; …; r<sup>0</sup>

dynamical structure factor, and others.

Derivation with respect to time gives

1 iℏ

3.2.4 Stoßzahlansatz and Boltzmann equation

Tr ½ � <sup>H</sup>; <sup>ρ</sup> <sup>ψ</sup>†

<sup>∂</sup>ρsð Þ<sup>t</sup>

distributions are irrelevant and will be integrated over.

<sup>f</sup> <sup>1</sup> <sup>þ</sup> <sup>r</sup>\_ <sup>∂</sup>

∂ ∂t

tions of the form

ρ<sup>1</sup> p; p<sup>0</sup> ð Þ¼ ; t

df <sup>1</sup> <sup>d</sup><sup>t</sup> <sup>¼</sup> <sup>∂</sup> ∂t

25

s

In addition to these so-called mechanical properties there exist also thermal properties, such as entropy, temperature, and chemical potential. Instead of a dynamical variable, they are related to the distribution function. For example, the equilibrium entropy is given by

$$\mathcal{S}\_{\text{eq}} = -k\_{\text{B}} \int d\Gamma f\_N(r\_i, \mathbf{p}\_i, t) \ln f\_N(r\_i, \mathbf{p}\_i, t) \tag{94}$$

We derive an equation of motion for the distribution function f <sup>N</sup> ri; p<sup>i</sup> ; t � �, the Liouville equation, see [5]:

$$\frac{\mathbf{d}f\_N}{\mathbf{d}t} = \frac{\partial f\_N}{\partial t} + \sum\_{i=1}^N \left[ \frac{\partial f\_N}{\partial \mathbf{r}\_i} \dot{\mathbf{r}}\_i + \frac{\partial f\_N}{\partial \mathbf{p}\_i} \dot{\mathbf{p}}\_i \right] = \mathbf{0}.\tag{95}$$

We shortly remember the quantum case. Instead of the N-particle distribution function f <sup>N</sup>ð Þt , the statistical operator ρð Þt is used to indicate the probability of a microstate in a given macrostate. The equation of motion is the von Neumann equation (8). Both equations are closely related and denoted as Liouville-von Neumann equation.
