TY - JOUR

T1 - Discrete charges on a two dimensional conductor

AU - Berkenbuseh, Marko Kleine

AU - Claus, Isabelle

AU - Dunn, Catherine

AU - Kadanoff, Leo P.

AU - Nicewicz, Maciej

AU - Venkataramani, Shankar C.

N1 - Funding Information:
We would like to thank Paul Wiegmann, Thomas Erber, David Nelson, and Michael Brenner for helpful conversations. This research was supported by the University of Chicago MRSEC under NSF grant DMR-0213745. Additional support came from the NSF-DMR under grant 094569.

PY - 2004

Y1 - 2004

N2 - 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.

AB - 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.

KW - Electrostatic equilibrium

KW - Fekete points

KW - Singular domains

KW - Symmetry breaking

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U2 - 10.1023/B:JOSS.0000041741.27244.ac

DO - 10.1023/B:JOSS.0000041741.27244.ac

M3 - Article

AN - SCOPUS:26844480956

SN - 0022-4715

VL - 116

SP - 1301

EP - 1358

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 5-6

ER -