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Dielectric breakdown model at small eta: pole dynamics.

M B Hastings1

  • 1Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. hastings@cnls.lanl.gov

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 22, 2002
PubMed
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The dielectric breakdown model, in a specific limit, simplifies to Sivashinsky's equation. Researchers found a stable pole configuration, analogous to diffusion limited aggregation finger stability, and calculated its eigenvalues.

Area of Science:

  • Complex Systems
  • Mathematical Physics
  • Fluid Dynamics

Background:

  • The dielectric breakdown model is a fundamental concept in understanding pattern formation.
  • Sivashinsky's equation describes instabilities in systems like flame propagation and fluid interfaces.
  • Diffusion limited aggregation (DLA) is a key model for random growth processes.

Purpose of the Study:

  • To analyze the dielectric breakdown model in the eta-->0(+) limit.
  • To establish the connection between the dielectric breakdown model and Sivashinsky's equation.
  • To investigate the linear stability of a specific pole configuration within this model.

Main Methods:

  • Mathematical analysis of the dielectric breakdown model in the specified limit.
  • Derivation of Sivashinsky's equation from the dielectric breakdown model.

Related Experiment Videos

  • Linear stability analysis of pole configurations.
  • Exact computation of eigenvalues for the stability matrix.

    • Main Results:

    • The dielectric breakdown model in the eta-->0(+) limit is shown to reduce to Sivashinsky's equation.
    • A particular configuration of poles is identified as linearly stable.
    • The stability is shown to be analogous to the 1/2 finger stability observed in diffusion limited aggregation.
    • The eigenvalues of the stability matrix for this configuration were computed exactly.

    Conclusions:

    • The reduction to Sivashinsky's equation provides a new perspective on the dielectric breakdown model.
    • The identified stable pole configuration offers insights into pattern formation and stability in related physical systems.
    • The analogy with diffusion limited aggregation highlights universal principles in complex system dynamics.