Derivation of the Boltzmann equation and entropy production in functional mechanics

TL;DR

This paper derives the Boltzmann equation from the Liouville equation using the Grad limiting procedure within a finite volume, incorporating a functional mechanics approach that accounts for measurement accuracy and scale hierarchy.

Contribution

It introduces a novel derivation of the Boltzmann equation based on functional mechanics and the hierarchy of scales, emphasizing the role of measurement accuracy.

Findings

Derivation of the Boltzmann equation from the Liouville equation.

Establishment of entropy production within the functional mechanics framework.

Demonstration of the hierarchy and subordination of micro- and macro-scales.

Abstract

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.