Functional stochastic classical mechanics

Abstract

The time irreversibility problem is the dichotomy of the reversible microscopic dynamics and the irreversible macroscopic physics. This problem was considered by Boltzmann, Poincare, 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. In this way the contradiction between the reversibility of microscopic dynamics and irreversibility of macroscopic dynamics is avoided since both dynamics in the proposed approach are irreversible. 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 and microscopic equation of motion 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” stochastic classical mechanics is not the Newton equation but the Liouville or Fokker-Planck-Kolmogorov equation for the distribution function of the 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 for not too long time intervals. Corrections to the Newton equation are computed.