Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

Borodachov, Sergiy V.

doi:10.4153/cjm-2011-038-5

Cited by 10 publications

(19 citation statements)

References 28 publications

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“…The two leading coefficients predicted by the formulas of refs. [38,39] are thus reproduced very accurately by the fits of the numerical data (underlined digits in the expressions of c 0 and c 1 obtained from the fit agree with the exact results of refs. [38,39]).…”

Section: An Exactly Solvable Example: the Unit Circlesupporting

confidence: 80%

“…These results should be compared with the exact results of Refs. [38,39] (Γ is the length of the curve) c (exact) 0…”

Section: An Exactly Solvable Example: the Unit Circlementioning

confidence: 99%

“…The focus of the present paper is on one-dimensional systems, represented by N charges arranged on a plane curve, in a way that the total electrostatic energy is globally minimal. Although asymptotic (N → ∞) estimates for the behavior of the total energy of these systems have been derived [38,39], the study for finite N has been essentially restricted to the case of a straight needle [40,41], and mainly focussed on the continuum limit, N → ∞. Jackson in particular has studied the charge density of a thin straight wire, as the transversal dimension shrinks, and proved that it approaches (slowly) a uniform distribution [42,43] (quite interestingly, in ref.…”

Section: Introductionmentioning

confidence: 99%

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Thomson problem in one dimension: Minimal energy configurations of N charges on a curve

Amore

Jacobo

2019

Physica A: Statistical Mechanics and its Applications

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We have studied the configurations of minimal energy of N charges on a curve on the plane, interacting with a repulsive potential Vij = 1/r s ij , with s ≥ 1 and i, j = 1, . . . , N . Among the examples considered are ellipses of different eccentricity, a straight wire and a cardioid. We have found that, for some geometries, multiple minima are present, as well as points of unstable equilibrium. For the case of the cardioid, we observe that the presence of the cusp has a dramatic effect on the distribution of the charges, in the limit N ≫ 1.

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Section: An Exactly Solvable Example: the Unit Circlesupporting

confidence: 80%

“…These results should be compared with the exact results of Refs. [38,39] (Γ is the length of the curve) c (exact) 0…”

Section: An Exactly Solvable Example: the Unit Circlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Thomson problem in one dimension: Minimal energy configurations of N charges on a curve

Amore

Jacobo

2019

Physica A: Statistical Mechanics and its Applications

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“…If the potential V is a positive linear combination of Gaussians, Montgomery's result implies that the optimal lattice is the hexagonal one, for a fixed unit cell volume. For instance, using the integral formula (32), one recovers the previously mentioned result on zeta functions.…”

Section: Optimal Lattices and Special Functionsmentioning

confidence: 98%

“…Several authors have studied the case of a potential which is not locally integrable, typically [143,120,126,127,172,32] that the corresponding term reads…”

Section: 7mentioning

confidence: 99%

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

142

191

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In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance of the system. This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on R 3N where N is the number of particles, which tends to infinity. We review the existing literature and mention several related open problems, of which many have not been thoroughly studied.

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Complete Minimal Logarithmic Energy Asymptotics for Points in a Compact Interval: A Consequence of the Discriminant of Jacobi Polynomials

Brauchart

2023

Constr Approx

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The electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as $$n \rightarrow \infty $$ n → ∞ of the minimal logarithmic potential energy of n point charges restricted to move in the interval $$[-1,1]$$ [ - 1 , 1 ] in the presence of an external field generated by endpoint charges. By the same methods, we determine the complete asymptotic expansion as $$N \rightarrow \infty $$ N → ∞ of the logarithmic energy $$\sum _{j\ne k} \log (1/| x_j - x_k |)$$ ∑ j ≠ k log ( 1 / | x j - x k | ) of Fekete points, which, by definition, maximize the product of all mutual distances $$\prod _{j\ne k} | x_j - x_k |$$ ∏ j ≠ k | x j - x k | of N points in $$[-1,1]$$ [ - 1 , 1 ] . The results for other compact intervals differ only in the quadratic and linear term of the asymptotics. Explicit formulas and their asymptotics follow from the discriminant, leading coefficient, and special values at $$\pm 1$$ ± 1 of Jacobi polynomials. For all these quantities we derive complete Poincaré-type asymptotics.

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Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

Cited by 10 publications

References 28 publications

Thomson problem in one dimension: Minimal energy configurations of N charges on a curve

Thomson problem in one dimension: Minimal energy configurations of N charges on a curve

The crystallization conjecture: a review

Complete Minimal Logarithmic Energy Asymptotics for Points in a Compact Interval: A Consequence of the Discriminant of Jacobi Polynomials

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