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

Derivation of the Boltzmann equation and entropy production in functional mechanics