2.3.1 Liouville's Theorem

The time interval during which a closed subsystem is observed can be divided into a large number of equal intervals separated by times $t_{1},t_{2},t_{3},...$. At these moments the subsystem has corresponding phase space^2.12 points $s_{1},s_{2},s_{3},...$, which are distributed in the phase space with the equilibrium distribution function^2.13 $f(\vec{r},\vec{k})$. The points $s_{1},s_{2},s_{3},...$ determine a statistical ensemble at the initial time $t=0$. The motion of the ensemble's phase space points is governed by the semiclassical equations of motion containing coordinates and quasi-momenta of the subsystems only. At each time $t>0$ all the phase points are distributed according to the same equilibrium distribution function. If the subsystem has $l$ degrees of freedom, the motion can formally be considered as the stationary flow of a "phase space gas" in the $2l$-dimensional phase space. This motion obeys a continuity equation^2.14:

$$\mathrm{div}(f\vec{v})=0.$$ (2.42)

In the case of the $2l$-dimensional phase space the last expression gives:

$$\sum_{i=1}^{l}\biggl[\Dot{r}_{i}\frac{\partial f}{\partial r_{i}}... ...ot{r}_{i}}{\partial r_{i}}+\frac{\partial\Dot{p}_{i}}{\partial p_{i}}\biggr]=0.$$ (2.43)

The second term in this expression is equal to zero due to the semiclassical equations of motion^2.15. The first term the total time derivative of the distribution function. Hence (2.43) is reformulated as:

$$\frac{df}{dt}=0,$$ (2.44)

where the differentiation is performed along the semiclassical phase space trajectories. This expression is called Liouville's theorem. From this theorem it follows that the equilibrium distribution function can only be expressed through such mathematical functions of coordinates and quasi-momenta which are conserved during the motion of the closed subsystem. Such functions are called integrals of motion. Thus the equilibrium distribution function itself represents an integral of motion.

As the equilibrium distribution function of two closed subsystems is equal to a product of the distribution functions of these subsystems (see (2.25)), the logarithm of the distribution function represents an additive integral of motion. It is well known from mechanics that there only exist seven independent integrals of motion. One of these integrals is the energy of a subsystem^2.16. The dependence of the equilibrium distribution on the energy has already been shown for fermions (2.30) and bosons (2.34).

S. Smirnov: