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We consider the dielectric breakdown model in the limit eta-->0(+). This is shown to lead to Sivashinsky's equation. We show that a particular configuration of poles is linearly stable, in analogy to the stability of the 1/2 finger for diffusion limited aggregation, and compute exactly the eigenvalues of the stability matrix.

physical review. e, statistical physics, plasmas, fluids, and related interdisciplinary topics

Dielectric breakdown model at small<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>η</mml:mi><mml:mo>:</mml:mo></mml:math>Pole dynamics

Dielectric breakdown model at smallη:Pole dynamics