2.2.1.1 Statistical Distribution

Considering an infinitesimal element of the phase space $d\vec{r}d\vec{k}$ the probability $dw$ that coordinates and quasi-momenta have values within intervals $[\vec{r},\vec{r}+d\vec{r}]$ and $[\vec{k},\vec{k}+d\vec{k}]$ is introduced through the expression:

$$dw = f(\vec{r},\vec{k})\,d\vec{r}d\vec{k},$$ (2.21)

where $f(\vec{r},\vec{k})$ is called a phase space distribution function^2.3. The distribution function must satisfy the normalization condition

$$\int f(\vec{r},\vec{k})\,d\vec{r}d\vec{k}=1,$$ (2.22)

which means that the sum of probabilities for all possible states must be equal to unity. The important property of the distribution function of a subsystem is that it does not depend on the initial state of another subsystem as its influence dies out by other subsystems. It does not depend on its own initial state either because the subsystem passes all its states during a long time interval and each of these states can be chosen as an initial one.

Using the distribution function it is possible to calculate the average of a function $g(\vec{r},\vec{k})$ which depends on the coordinates and quasi-momenta of a subsystem

$$\langle g \rangle=\int g(\vec{r},\vec{k})f(\vec{r},\vec{k})\,d\vec{r}d\vec{k}.$$ (2.23)

This statistical average removes the necessity to follow $g(\vec{r},\vec{k})$ in time in order to make an average. It is completely equivalent to the time average

$$\langle g \rangle=\underset{T\rightarrow\infty}{\lim}\frac{1}{T}\int g(t)\,dt.$$ (2.24)

If the closed system is in a state in which all its parts have their physical values close to their statistical averages, the system is in the statistical or thermodynamic equilibrium. It is clear that a closed system spends the most of its time in the thermodynamic equilibrium. If at some moment it is not in the thermodynamic equilibrium then it will relax to the equilibrium state. The time interval of the transition to the equilibrium state is called a relaxation time.

The fact that different subsystems do not interact with each other^2.4 leads to the possibility to consider them independent in a statistical sense. The statistical independence implies that a state of one subsystem does not influence the state probabilities of other subsystems.

From the mathematical point of view statistical independence means that the probability for a subsystem which consists of two parts to be in the element of its phase space^2.5 $d\vec{r}d\vec{k}=d\vec{r}_{1}d\vec{k}_{1}d\vec{r}_{2}d\vec{k}_{2}$ is equal to the product of the probabilities for each of the two subsystems to have coordinates and quasi-momenta in $d\vec {r}_{1}d\vec {k}_{1}$ and $d\vec {r}_{2}d\vec {k}_{2}$. Using (2.21) this gives

$$f(\vec{r}_{1},\vec{r}_{2},\vec{k}_{1},\vec{k}_{2})=f_{1}(\vec{r}_{1},\vec{k}_{1})f_{2}(\vec{r}_{2},\vec{k}_{2}),$$ (2.25)

where $f(\vec{r}_{1},\vec{r}_{2},\vec{k}_{1},\vec{k}_{2})$ is the distribution function for the constituent subsystem and $f_{1}(\vec{r}_{1},\vec{k}_{1})$, $f_{2}(\vec{r}_{2},\vec{k}_{2})$ are the distribution functions of the two subsystems. It is obvious that the same equality (2.25) is valid for any number of subsystems.

Let two functions $g_{1}(\vec{r}_{1},\vec{k}_{1})$ and $g_{2}(\vec{r}_{2},\vec{k}_{2})$ describe the two subsystems. In this way they describe the subsystem which consists of the two subsystems. From (2.23) and (2.25) it follows that the statistical average of the product $g_{1}(\vec{r}_{1},\vec{k}_{1})g_{2}(\vec{r}_{2},\vec{k}_{2})$ is equal to the product of the statistical averages:

$$\langle g_{1}g_{2} \rangle = \langle g_{1} \rangle\langle g_{2} \rangle.$$ (2.26)

Considering particles as independent subsystems it is possible to determine their equilibrium distribution functions. However, these distributions depend on the wave function which describes the whole system. It can be either symmetric or antisymmetric with respect to the exchange of any two particles of the system. In turn it depends on the spin of particles. Particles with an integer spin are subject to the Bose-Einstein statistics, while particles with fractional spin are subject to the Fermi-Dirac statistics. S. Smirnov: