Classification of Particle Numbers with Unique Heitmann–Radin Minimizer

Abstract

We show that minimizers of the Heitmann–Radin energy (Heitmann and Radin in J Stat Phys 22(3):281–287, 1980) are unique if and only if the particle number N belongs to an infinite sequence whose first thirty-five elements are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120 (see the paper for a closed-form description of this sequence). The proof relies on the discrete differential geometry techniques introduced in De Luca and Friesecke (Crystallization in two dimensions and a discrete Gauss–Bonnet Theorem, 2016).

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References

  1. 1.

    Au Yeung, Y., Friesecke, G., Schmidt, B.: Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. Calc. Var. Partial. Differ. Equ. 44(1–2), 81–100 (2012)

  2. 2.

    Davoli, E., Piovano, P., Stefanelli, U.: Sharp \(N^{3/4}\) law for the minimizers of the edge-isoperimetric problem on the triangular lattice. J. Nonlinear Sci. 27(2), 627–660 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    De Luca, L., Friesecke, G.: Crystallization in two dimensions and a discrete Gauss–Bonnet Theorem, preprint. arXiv:1605.00034 (2016)

  4. 4.

    Dobrushin, R.L., Kotecky, R., Shlosman, S.B.: The Wulff Construction: A Global Shape from Local Interactions. AMS, Providence (1992)

    Google Scholar 

  5. 5.

    Fonseca, I., Müller, S.: A uniqueness proof of the Wulff theorem. Proc. R. Soc. Edinb. Sect. A 119, 125–136 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Heitmann, R.C., Radin, C.: The ground states for sticky discs. J. Stat. Phys. 22(3), 281–287 (1980)

    ADS  Article  Google Scholar 

  7. 7.

    Mermin, N.D.: Crystalline order in two dimensions. Phys. Rev. 176(1), 250–254 (1968)

    ADS  Article  Google Scholar 

  8. 8.

    Polyá, G., Szegó, G.: Problems and Theorems in Analysis I. Springer, Berlin (1991)

    Google Scholar 

  9. 9.

    Richthammer, T.: Translation-invariance of two-dimensional Gibbsian point processes. Commun. Math. Phys. 274(1), 81–122 (2007)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Schmidt, B.: Ground states of the 2D sticky disc model: fine properties and \(N^{3/4}\) law for the deviation from the asymptotic Wulff shape. J. Stat. Phys. 153(4), 727–738 (2013)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Taylor, J.E.: Unique structure of solutions to a class of nonelliptic variational problems. In: Differential Geometry (Proc. Sympos. Pure. Math., Vol. XXVII), Part 1, pp. 419–427. AMS, Providence (1975)

  12. 12.

    The On-line Encyclopedia of Integer Sequences. www.oeis.org

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Acknowledgements

We are indebted to Michael Baake for pointing out to us the website [12] after attending a lecture on the work reported here. The research of LDL was funded, and that of GF partially supported, by the DFG Collaborative Research Center CRC 109 “Discretization in Geometry and Dynamics”.

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Correspondence to Lucia De Luca.

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De Luca, L., Friesecke, G. Classification of Particle Numbers with Unique Heitmann–Radin Minimizer. J Stat Phys 167, 1586–1592 (2017). https://doi.org/10.1007/s10955-017-1781-3

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Keywords

  • Crystallization
  • Wulff shape
  • Heitmann–Radin potential
  • Discrete differential geometry
  • Energy minimization