Hierarchy of space-time structures, Boltzmann equation, and functional mechanics

TL;DR

This paper explores the hierarchy of space-time structures and introduces a functional mechanics approach to derive the Boltzmann equation, addressing the fundamental problem of reconciling microscopic reversibility with macroscopic irreversibility.

Contribution

It proposes a novel functional formulation of mechanics using probability densities and develops a procedure to derive the Boltzmann equation from microscopic dynamics within a hierarchical space-time framework.

Findings

Derivation of Boltzmann equation from Liouville equation using the BBGKY hierarchy.

Introduction of a hierarchy of space-time structures incorporating macro- and microscale processes.

Justification of the probability density approach via measurement theory.

Abstract

In this report we discuss the organization of different levels of nature and the corresponding space-time structures by the consideration of a particular problem of time irreversibility. The fundamental time irreversibility problem consists in the following: how to reconcile the time-reversible microscopic dynamics and the irreversible macroscopic one. The recently proposed functional formulation of mechanics is aimed to solve this problem. The basic concept of this formulation is not a material point and a trajectory, like in the traditional formulation of mechanics, but a probability density function. Even if we deal with a single particle (not with an ensemble of particles), we describe its state as a probability density function. We justify this approach using measurement theory. A particular problem in the framework of the irreversibility problem is the derivation of the Boltzmann kinetic equation from the equations of microscopic dynamics. We propose a procedure for obtaining the Boltzmann equation from the Liouville equation based on the BBGKY hierarchy, the recently proposed functional formulation of classical mechanics, and the distinguishing between two scales of space-time, i.e., macro- and microscale. The notion of a space-time structure is introduced. It takes into account not only the space-time itself (i.e., a pseudo-Riemannian manifold), but also a characteristic length and time. The space-time structures form a hierarchy in sense that the initial values for the processes on the microscopic space-time structure (interactions of the particles) are assigned from the processes on the macroscopic one (kinetic phenomena).