  A derivation of the Boltzmann equation from the Liouville equation by the use
of the Grad limiting procedure in a finite volume is proposed. We introduce two
scales of space-time: macro- and microscale and use the BBGKY hierarchy and the
functional formulation of classical mechanics. According to the functional
approach to mechanics, a state of a system of particles is formed from the
measurements, which are rational numbers. Hence, one can speak about the
accuracy of the initial probability density function in the Liouville equation.
We assume that the initial data for the microscopic density functions are
assigned by the macroscopic one (so, one can say about a kind of hierarchy and
subordination of the microscale to the macroscale) and derive the Boltzmann
equation, which leads to the entropy production.
