Thomson rings in a disk

Cerkaski, M.; Nazmitdinov, R. G.; Puente, A.

doi:10.1103/physreve.91.032312

Cited by 16 publications

(29 citation statements)

References 24 publications

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“…The numerical solution of the system (15), (17) for the hard confinement provides a remarkable agreement with the MD calculations for equilibrium configurations up to n = 105, excluding a few cases (see Table I in Ref. 32). Our MD results agree with those of Ref.…”

Section: Molecular Dynamics and The Circular Modelsupporting

confidence: 70%

Self-organization of charged particles in circular geometry

Nazmitdinov

Puente

Cerkaski³

et al. 2017

Phys. Rev. E

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The basic principles of self-organization of one-component charged particles, confined in disk and circular parabolic potentials, are proposed. A system of equations is derived, which allows us to determine equilibrium configurations for an arbitrary, but finite, number of charged particles that are distributed over several rings. Our approach reduces significantly the computational effort in minimizing the energy of equilibrium configurations and demonstrates a remarkable agreement with the values provided by molecular dynamics calculations. With the increase of particle number n>180 we find a steady formation of a centered hexagonal lattice that smoothly transforms to valence circular rings in the ground-state configurations for both potentials.

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Section: Molecular Dynamics and The Circular Modelsupporting

confidence: 70%

Self-organization of charged particles in circular geometry

Nazmitdinov

Puente

Cerkaski³

et al. 2017

Phys. Rev. E

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“…In the Comment [2] by Amore it was found that the minimum energy configuration of N = 395 charges confined in disk and interacting via the Coulomb potential has a lower energy than the result of our molecular dynamics (MD) calculations [1]. Based only on this result Amore concluded that"...the formation of a hexagonal core and valence circular rings for the centered configurations, predicted by the model of Ref.…”

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confidence: 59%

“…Based only on this result Amore concluded that"...the formation of a hexagonal core and valence circular rings for the centered configurations, predicted by the model of Ref. [1], is not supported by numerical evidence and the configurations obtained with this model cannot be used as a guide for the numerical calculations, as claimed by the authors. In light of this findings, the validity of the model of Ref.…”

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confidence: 88%

“…In light of this findings, the validity of the model of Ref. [1] must be questioned, particular for N > ∼ 200. "…”

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confidence: 98%

“…In our publication [1], in order to obtain our estimate of the MD ground state E MD , we generated only 100 configurations with the boundary ring N p=9 = 147 charges, where the internal charges were randomly distributed. As a result, we have obtained suggests that the effectivity of the CM prediction might be improved if the second ring, the nearest neigbor to the boundary one, should be taken into account.…”

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confidence: 99%

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Reply to “Comment on ‘Thomson rings in a disk' ”

Puente

Nazmitdinov

Cerkaski³

et al. 2017

Phys. Rev. E

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We demonstrate that our model [Phys.Rev. E91, 032312 (2015] serves as a useful tool to trace the evolution of equilibrium configurations of one-component charged particles confined in a disk.Our approach reduces significantly the computational effort in minimizing the energy of equilibrium configurations and demonstrates a remarkable agreement with the values provided by molecular dynamics calculations. We show that the comment misrepresents our paper, and fails to provide plausible arguments against the formation hexagonal structure for n ≥ 200 in molecular dynamics calculations. PACS numbers: 64.70.kp,64.75.Yz,02.20 Hereafter, for the sake of convenience we refer to our model as the circular model (CM).We agree with the author that his possible global MD minimum is better than our estimate for the particular case N = 395. However, this is not enough to conclude that the CM can not help to reduce substantially the computational effort in MD or simulated annealing (SA)calculations for the following reasons.1. From the Monte Carlo and MD calculations, even for a relatively small number of charged particles, it follows that the amount of stable configurations grows very rapidly with the number of particles. Sometimes, metastable states with lower (or higher) symmetry are found with much higher probability than the true ground state. This fact was confirmed by the author who "generated 3001 configurations ..." to get just one instance of the improved E MIN = 110664.44 new tentative ground state, with our prediction for the particle number at the boundary ring: "... Np = 147 charges are disposed on the border of the disk, in agreement with Ref.1". Evidently, in contrast to his claim, Amore has confirmed the usefulness of the CM.Indeed, the particle number on the boundary ring N p is one of the key elements for any calculation, since once it is defined, it is necessary to simulate less various configurations (with a number of charges N − N p ). We recall thatwhere p is a number of rings, and N is a total number of charges.In fact, external ring occupations are extremely well predicted with some occasional ±1 mismatch by means of the expression. It is noteworthy that these expressions are obtained from the systematic CM 2 results.2. In our publication [1], in order to obtain our estimate of the MD ground state E MD , we generated only 100 configurations with the boundary ring N p=9 = 147 charges, where the internal charges were randomly distributed. As a result, we have obtained.44. Note, however, that the disagreement between the author's new result and our model prediction E CM is still less than 3 × 10 −3 % (as we stated in our paper it is 2 × 10 −3 %). Moreover, the occupations for the external (approximately circular) shells are quite accurately predicted within CM for any N . In the case of N = 395 we have obtained (147, 65, 50, 40), while the analysis of the Amore's MD ground state yields (147, 66, 51, 40). This comparison suggests that the effectivity of the CM prediction might be improved if the seco...

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Correspondence between Electrostatics and Contact Mechanics with Further Results in Equilibrium Charge Distributions

Batle,

Vlasiuk,

Ciftja

2023

Annalen der Physik

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It is common in mesoscopic systems to find instances where several charges interact among themselves. These particles are usually confined by an external potential that shapes the symmetry of the equilibrium charge configuration. In the case of classical charges moving on a plane and repelling each other via the Coulomb potential, they possess a ground state à la Thomson or Wigner crystal. As the number of particles N increases, the number of local minima grows exponentially and direct or heuristic optimization methods become prohibitively costly. Therefore the only feasible approximation to the problem is to treat the system in the continuum limit. Since the underlying framework is provided by potential theory, we shall by‐pass the corresponding mathematical formalism and list the most common cases found in the literature. Then we prove a (albeit known) mathematical correspondence that will enable us to re‐discover analytical results in electrostatics. In doing so, we shall provide different methods for finding the equilibrium surface density of charges, analytical and numerical. Additionally, new systems of confined charges in three‐dimensional surfaces will be under scrutiny. Finally, we shall highlight exact results regarding a modified power‐law Coulomb potential in the d‐dimensional ball, thus generalizing the existing literature.

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Thomson rings in a disk

Cited by 16 publications

References 24 publications

Self-organization of charged particles in circular geometry

Self-organization of charged particles in circular geometry

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Correspondence between Electrostatics and Contact Mechanics with Further Results in Equilibrium Charge Distributions

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