Electrostatic self-energies of discrete charge distributions on Jordan curves

摘要

The characteristic energy of a smooth Jordan curve of length L, defined as the coefficient in the term proportional to N2/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 φ(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 φ(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.