Boltzmann equation and $H$-theorem in the functional formulation of classical mechanics

We propose a procedure for obtaining the Boltzmann equation from the Liouville equation in a non-thermodynamic limit. It is based on the BBGKY hierarchy, the functional formulation of classical mechanics, and the distinguishing between two scales of space-time, i.e., macro- and microscale. According to the functional approach to mechanics, a state of a system of particles is formed from the measurements, which have errors. Hence, one can speak about accuracy of the initial probability density function in the Liouville equation. Let's assume that our measuring instruments can observe the variations of physical values only on the macroscale, which is much greater than the characteristic interaction radius (microscale). Then the corresponfing initial density function cannot be used as initial data for the Liouville equation, because the last one is a description of the microscopic dynamics, and the particle interaction potential (with the characteristic interaction radius) is contained in it explicitly. Nevertheless, for a macroscopic initial density function we can obtain the Boltzmann equation using the BBGKY hierarchy, if we assume that the initial data for the microscopic density functions are assigned by the macroscopic one. The $H$-theorem (entropy growth) is valid for the obtained equation.