Boltzmann-Kolmogorov equation

TL;DR

This paper explores a Kolmogorov equation framework for phase space probability distributions, deriving the Boltzmann equation and analyzing entropy behavior in isolated and open systems.

Contribution

It introduces a Kolmogorov equation approach to describe both equilibrium and nonequilibrium dynamics, connecting thermodynamics with kinetic theory.

Findings

The equation conserves energy and predicts entropy increase in agreement with thermodynamics.

Derivation of the Boltzmann equation from the Kolmogorov framework.

Description of stationary states including Gibbs distributions and nonequilibrium steady states.

Abstract

We investigate the properties of a Kolmogorov equation governing the time evolution of the probability distribution defined in phase space. Energy is strictly conserved along a trajectory in phase space, meaning the equation is appropriate to describe an isolated system, and the stationary state is the Gibbs microcanonical distribution. The equation predicts the increase in entropy in agreement with thermodynamics, and in contrast with the Liouville equation, which conserves entropy. Using an approximation in which the distribution is a product of one-particle distributions, we derive the Boltzmann equation of kinetic theory. We also consider a Kolmogorov equation to describe an open system in contact with the external environment. In this case the equation describes not only the situation in which the system is found in thermodynamic equilibrium with a Gibbs canonical distribution in the stationary state, but also the nonequilibrium steady state with a continuous production of entropy.