The Dielectric Breakdown Model at Small $\eta$: Pole Dynamics

TL;DR

This paper derives a differential equation for the dielectric breakdown model at very small eta, analyzes pole dynamics, finds stationary solutions, and discusses stability and implications for related models like DLA.

Contribution

It introduces a novel pole-based approach to analyze the dielectric breakdown model at small eta, providing explicit solutions and stability analysis.

Findings

Only one stable configuration per viscosity value

Stable configurations are nonlinearly unstable with small noise

The approach aids understanding of finite eta dynamics and DLA

Abstract

We consider the dielectric breakdown model in the limit $\eta\to 0^+$. A differential equation describing the surface growth is derived; this equation is KPZ plus a term causing linear instability, and includes a short-distance regularization similar to a viscosity. The equation exhibits an interesting dynamics in terms of poles, permitting us to derive a large family of solutions to the equation of motion. In terms of poles, we find all stationary configurations of the surface, and analytically calculate their stability. For each value of the viscosity, only one stable configuration is found, but this configuration is nonlinearly unstable in the presence of an exponentially small amount of noise. The present approach may be useful in understanding the dynamics for finite $\eta$, in particular the DLA model, in terms of perturbations to the infinitesimal $\eta$ problem.