Liouville equation in statistical mechanics is not applicable to gases composed of colliding molecules

TL;DR

This paper argues that the Liouville equation, fundamental in statistical mechanics, is not applicable to gases due to the discontinuous changes in molecular momenta caused by collisions, which violate the equation's assumptions.

Contribution

The paper identifies and explains the fundamental flaw in deriving Liouville equation for gases, highlighting the discontinuity in molecular trajectories caused by collisions.

Findings

Liouville equation assumes continuous phase space trajectories.

Collisions cause discontinuous changes in molecular momenta.

Liouville equation is not valid for gases with colliding molecules.

Abstract

Liouville equation is a fundamental one in statistical mechanics. It is rooted in ensemble theory. By ensemble theory, the variation of the system's microscopic state is indicated by the moving of the phase point, and the moving trajectory is believed continuous. Thus, the ensemble density is thought to be a smooth function, and it observes continuity equation. When the Hamiltonian canonical equations of the molecules are applied to the continuity equation, Liouville equation can be obtained. We carefully analyze a gas composed of a great number of molecules colliding with each other. The defects in deriving Liouville equation are found. Due to collision, molecules' momenta changes discontinuously, so that the trajectories of the phase points are actually not continuous. In statistical mechanics, infinitesimals in physics and in mathematics should be distinguished. In continuity equation that the ensemble density satisfies, the derivatives with respect to space and time should be physical infinitesimals, while in Hamiltonian canonical equations that every molecule follows, the derivatives take infinitesimals in mathematics. In the course of deriving Liouville equation, the infinitesimals in physics are unknowingly replaced by those in mathematics. The conclusion is that Liouville equation is not applicable to gases.