The fractal dimension of ionization cascades in the glow discharge

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