Discrete Charges on a Two Dimensional Conductor

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( √ log N /N 2 ) 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.

Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

“…The symmetry breaking phenomenon for the configurations minimizing energy (2.1) on certain planar curves was studied in [4]. Papers [24] and [25] estimate the difference between the potential of the equilibrium distribution on closed smooth Jordan arcs on the plane and the potential of the minimum Riesz energy configurations for s = 0.…”

Exact Solution of the Gauge Symmetric p-Spin Glass Model on a Complete Graph

“…Proposition 1 applies also to other bounded domains, in particular curves! I should emphasize that the logarithmic interactions between charges constrained to (planar) curves can be studied in quite some detail with complex variable techniques, see [Ketal04]. Proposition 1 can easily be generalized to unbounded domains, with lower semi-continuity replaced by another appropriate condition guaranteeing minimizing configurations for all N. A physically important example is Λ = R 3 with U R 3 (q i , q j ) = |q i − q j | −12 − |q i − q j | −6 , which has minimizing N-body configurations for each N, known as LennardJones clusters, see [AtSu03] for a recent survey.…”

Phase transitions in the one-dimensional coulomb medium

“…The problem of finding N point particle configurations on a manifold having minimal energy (or even fixed configurations) was claimed important already long ago [1]. That is why we shall say shortly about the history of this question.…”