Time Irreversibility Problem and Functional Formulation of Classical Mechanics

TL;DR

This paper proposes a new 'functional' approach to classical mechanics that attributes physical meaning to distribution functions rather than individual trajectories, offering a potential solution to the time irreversibility problem by deriving irreversibility from the Liouville equation.

Contribution

It introduces a functional formulation of microscopic dynamics where irreversibility arises naturally from the Liouville equation, contrasting with traditional trajectory-based approaches.

Findings

Solutions of the Liouville equation exhibit delocalization leading to irreversibility.

Newton's equations are derived as approximate descriptions of average particle dynamics.

Corrections to Newton's equations are calculated within this framework.

Abstract

The time irreversibility problem is the dichotomy of the reversible microscopic dynamics and the irreversible macroscopic physics. This problem was considered by Boltzmann, Poincar\'e, Bogolyubov and many other authors and though some researchers claim that the problem is solved, it deserves a further study. In this paper an attempt is performed of the following solution of the irreversibility problem: a formulation of microscopic dynamics is suggested which is irreversible in time. A widely used notion of microscopic state of the system at a given moment of time as a point in the phase space and also a notion of trajectory does not have an immediate physical meaning since arbitrary real numbers are non observable. In the approach presented in this paper the physical meaning is attributed not to an individual trajectory but only to a bunch of trajectories or to the distribution function on the phase space. The fundamental equation of the microscopic dynamics in the proposed "functional" approach is not the Newton equation but the Liouville equation for the distribution function of a single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. It is shown that the Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta. Corrections to the Newton equation are computed.