A path-integral approach to the collisionless Boltzmann gas

TL;DR

This paper introduces a novel path-integral formalism for the collisionless Boltzmann gas using a solid-angle-average distribution function, overcoming limitations of traditional methods and enabling practical computations.

Contribution

It proposes a new distribution function and a path-integral approach that address the weaknesses of the standard theory of collisionless gases.

Findings

The new formalism overcomes standard theory limitations.

It allows practical computation of gas behavior.

The approach is compatible with current computational tools.

Abstract

On contrary to the customary thought, the well-known ``lemma'' that the distribution function of a collisionless Boltzmann gas keeps invariant along a molecule's path represents not the strength but the weakness of the standard theory. One of its consequences states that the velocity distribution at any point is a condensed ``image'' of all, complex and even discontinuous, structures of the entire spatial space. Admitting the inability to describe the entire space with a microscopic quantity, this paper introduces a new type of distribution function, called the solid-angle-average distribution function. With help of the new distribution function, the dynamical behavior of collisionless Boltzmann gas is formulated in terms of a set of integrals defined by molecular paths. In the new formalism, not only that the difficulties associated with the standard theory are surmounted but also that some of practical gases become calculable in terms of today's computer.