The Fractal Dimension of Ionization Cascades in the Glow Discharge

TL;DR

This paper explores the fractal nature of ionization cascades in glow discharges, linking mathematical models to physical phenomena and potential applications in pattern formation and discharge analysis.

Contribution

It introduces a novel approach to characterize ionization avalanches as fractal sets with a specific Hausdorff dimension, connecting branching process models to physical discharge phenomena.

Findings

Ionization cascades can be modeled as branching processes.

The ionization avalanche has a unique Hausdorff fractal dimension.

Fractal dimensions can inform applications based on similarity principles.

Abstract

The glow discharge's main ionization breakdown processes have been understood for about one hundred years. The glow discharge, however, still remains an area of active research in relation to pattern formation and far-from-equilibrium systems. The primary and secondary ionization processes can be mathematically modelled as general branching processes. Not only is the Townsend breakdown criterion obtained but the ionization avalanche can be characterized as a branching set with a unique Hausdorff fractal dimension. These fractal dimensions can be utilized in applications using similarity principles and Paschen's Law.