Discrete charges on a two dimensional conductor

Marko Kleine Berkenbuseh, Isabelle Claus, Catherine Dunn, Leo P. Kadanoff, Maciej Nicewicz, Shankar C. Venkataramani

Research output: Contribution to journalArticle

7 Scopus citations

Abstract

We investigate the electrostatic equilibria of N discrete charges of size 1/N on a two dimensional conductor (domain). We study the distribution of the charges on symmetric domains including the ellipse, the hypotrochoid and various regular polygons, with an emphasis on understanding the distributions of the charges, as the shape of the underlying conductor becomes singular. We find that there are two regimes of behavior, a symmetric regime for smooth conductors, and a symmetry broken regime for "singular" domains. For smooth conductors, the locations of the charges can be determined, to within O(√logN/N2) by an integral equation due to Pommerenke [Math. Ann., 179: 212-218, (1969)]. We present a derivation of a related (but different) integral equation, which has the same solutions. We also solve the equation to obtain (asymptotic) solutions which show universal behavior in the distribution of the charges in conductors with somewhat smooth cusps. Conductors with sharp cusps and singularities show qualitatively different behavior, where the symmetry of the problem is broken, and the distribution of the discrete charges does not respect the symmetry of the underlying domain. We investigate the symmetry breaking both theoretically, and numerically, and find good agreement between our theory and the numerics. We also find that the universality in the distribution of the charges near the cusps persists in the symmetry broken regime, although this distribution is very different from the one given by the integral equation.

Original languageEnglish (US)
Pages (from-to)1301-1358
Number of pages58
JournalJournal of Statistical Physics
Volume116
Issue number5-6
DOIs
StatePublished - 2004
Externally publishedYes

Keywords

  • Electrostatic equilibrium
  • Fekete points
  • Singular domains
  • Symmetry breaking

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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  • Cite this

    Berkenbuseh, M. K., Claus, I., Dunn, C., Kadanoff, L. P., Nicewicz, M., & Venkataramani, S. C. (2004). Discrete charges on a two dimensional conductor. Journal of Statistical Physics, 116(5-6), 1301-1358. https://doi.org/10.1023/B:JOSS.0000041741.27244.ac