Note on Divergence of the Chapman-Enskog Expansion for Solving Boltzmann Equation

TL;DR

This paper addresses the divergence issues in the classical Chapman-Enskog expansion for the Boltzmann equation by introducing a modified series with a variable upper limit, achieving convergence and better explanations for previous anomalies.

Contribution

It proposes a novel modified Chapman-Enskog expansion using a M"obius series inversion with a variable upper limit, resolving divergence problems.

Findings

The new expansion converges where the classical one diverges.

It explains previously puzzling scenarios in kinetic theory.

Provides a better mathematical framework for the Boltzmann equation.

Abstract

Within about a year (1916-1917) Chapman and Enskog independently proposed an important expansion for solving the Boltzmann equation. However, the expansion is divergent or indeterminant in the case of relaxation time $\tau \geq 1$. Even since this divergence problem has puzzled this subject for a century. By using a modified M\"obius series inversion formula, this paper proposes a modified Chapman-Enskog expansion with a variable upper limit of the summation. The new expansion can give not only a convergent summation but also provide the best-so-far explanation on some unbelievable scenarios occurred in previous practice.