Electrostatic self-energies of discrete charge distributions on Jordan curves

Jerzy Cioslowski,Joanna Albin

doi:10.1007/s10910-014-0395-0

Abstract

The characteristic energy of a smooth Jordan curve of length \(L\), defined as the coefficient in the term proportional to \(N^2/L\) in the large-\(N\) asymptotics of the minimal electrostatic self-energy of \(N\) unit charges located on the curve in question, possesses an expansion involving the function \(\varphi (t)\) that measures the deviation from linearity in the dependence of the tangential angle on the arc length. The leading term in this expansion is given by a functional that is quadratic in \(\varphi (t)\). The explicit expression for this functional can be derived without taking into account the energy lowering due to relaxation of the particle positions that, being of the order of \(N^2 (\ln N)^{-1}\) for large \(N\), does not contribute to the characteristic energy.

Highlights

Confinement of interacting classical particles gives rise to diverse patterns of particle positions at equilibrium geometries
Consider a set of N unit point charges located on a smooth Jordan curve ≡ {x(s), y(s)} conveniently defined by its Whewell representation [19], i.e
In order to estimate the lowering of the electrostatic self-energy due to this relaxation, one has to compute the respective leading asymptotic contributions to the energy gradient g(N ) = {gk(N )} and the Hessian H(N ) = {Hkl (N )}

Summary

Introduction

Confinement of interacting classical particles gives rise to diverse patterns of particle positions at equilibrium geometries. The combination of Coulombic interparticle interactions and two-dimensional confining potentials of cylindrical symmetry usually produces assemblies of particles positioned on either vertices of polygons inscribed on concentric rings or nodes of triangular lattice [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] Formation of such patterns, which is observed both in experimental settings and numerical simulations, occurs in systems ranging from electrons in quantum dots [2,3,4,5,6,7] to ions in dusty plasmas [8,9,10] and triboelectrically charged objects [11]. The validity of the derived expressions is demonstrated for a family of simple test curves

Theory

An example

Conclusions