Boltzmann-Kolmogorov equation

TL;DR

This work casts Boltzmann's kinetic reasoning into a linear Kolmogorov framework by adding a stochastic, energy- and momentum-conserving collision term to Liouville dynamics, yielding a microcanonical Boltzmann-Kolmogorov equation whose stationary solution is the microcanonical Gibbs distribution and whose entropy production is nonnegative. By invoking the molecular-disorder (factorization) ansatz, it derives the nonlinear Boltzmann equation and proves the Boltzmann H-theorem, linking microscopic collisions to macroscopic entropy increase. The paper also extends the formalism to open systems, showing that environment-induced transitions drive the canonical Gibbs state and produce a monotone Helmholtz free energy, as well as nonequilibrium steady states with sustained entropy production governed by heat flows and Clausius-type relations. Overall, it provides a unified, stochastic-thermodynamics-consistent view of kinetic evolution in phase space, bridging isolated and open-system behaviors via transition-rate constructions and highlighting the role of detailed balance (or its violation) in setting stationary properties.

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.