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## О РАВНОВЕСНЫХ КОНФИГУРАЦИЯХ ЗАРЯЖЕННЫХ ИОНОВ В ПЛАНАРНЫХ СИСТЕМАХ С КРУГОВОЙ СИММЕТРИЕЙ

The problem of finding equilibrium configurations of similarly charged particles (ions) induced by

external electrostatic fields in planar systems is of great interest for fundamental and applied research. In this

paper, the results of numerical analysis of equilibrium configurations of negatively charged particles (electrons)

confined in a circular region by an infinite external potential at its boundary are presented. A hybrid

numerical algorithm is developed to seek stable configurations with minimal energies. The algorithm is based

on interpolation formulas derived using the analysis of equilibrium configurations obtained by means of the

variational principle of the minimum energy for an arbitrary finite number of particles in a circular model.

The solution of nonlinear equations of the given model make it possible to predict structure formation in the

form of rings (shells) filled with electrons, the number of which decreases when passing from the external ring

to the internal one. The number of rings depends on the total number of charged particles. The obtained interpolation

formulas of the distribution of the total number of electrons over the rings are used as initial configurations

for the molecular-dynamics method. Our results demonstrate the significant efficiency of the application

of the classical molecular dynamics (MD) method when using interpolation formulas in comparison

with algorithms based on Monte Carlo methods and global optimization. For an arbitrarily chosen number

of particles in the considered system, the proposed method makes it possible to enhance the speed at which

the stable configuration with the minimum energy is reached by several orders in comparison with the classical-

molecular-dynamics method.