Equilibrium and uniform charge distribution of a classical two-dimensional system of point charges with hard-wall confinement

We study the minimum energy equilibrium configurations of a classical two-dimensional system of point charges confined by a triangular, square and disk region with a hard-wall boundary. It is assumed that the point charges interact via a repulsive Coulomb interaction potential. Monte Carlo simulations with the annealing algorithm suggest that the equilibrium configurations of a given system are strongly influenced by the external (isotropic/anisotropic) geometry of the hard-wall boundary. The numerically obtained energies extrapolated in the bulk limit converge to the expected continuum equilibrium values (when known). It is found that the equilibrium charge distribution is non-uniform in the continuum limit for all the hard-wall confining regions considered in this work. Since the continuum equilibrium charge distribution is not known for the case of an equilateral triangle or a square domain we choose to compare the numerically obtained bulk energy results to corresponding values for a uniformly charged system. We calculated exactly the electrostatic energy of various uniformly charged planar objects and used the results to assess the discrepancy between such results and the numerically obtained equilibrium bulk energy values for the cases of equilateral triangle and square hard-wall boundaries. These estimates help us understand how an anisotropic boundary with the shape of an equilateral triangle or square influences the energy of an equilibrium charge distribution. The results indicate that the energy discrepancy between equilibrium and uniform charge distributions in the continuum limit is not very large. It is found that the order of magnitude of the relative deviation of the energy for all three different planar domains considered here is approximately the same