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Discrete minimizers of the interaction energy in collective behavior: a brief numerical and analytic review (2403.00594v2)

Published 1 Mar 2024 in math.NA, cs.NA, math-ph, and math.MP

Abstract: We consider minimizers of the N-particle interaction potential energy and briefly review numerical methods used to calculate them. We consider simple pair potentials which are repulsive at short distances and attractive at long distances, focusing on examples which are sums of two powers. The range of powers we look at includes the well-known case of the Lennard-Jones potential, but we are also interested in less singular potentials which are relevant in collective behavior models. We report on results using the software GMIN developed by Wales and collaborators for problems in chemistry. For all cases, this algorithm gives good candidates for the minimizers for relatively low values of the particle number N. This is well-known for potentials similar to Lennard-Jones, but not for the range which is of interest in collective behavior. Standard minimization procedures have been used in the literature in this range, but they are likely to yield stationary states which are not minimizers. We illustrate numerically some properties of the minimizers in 2D, such as lattice structure, Wulff shapes, and the continuous large-N limit for locally integrable (that is, less singular) potentials.

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References (63)
  1. Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. Calculus of Variations and Partial Differential Equations, 44(1–2):81–100, July 2011. ISSN 1432-0835.
  2. Dimensionality of Local Minimizers of the Interaction Energy. Archive for Rational Mechanics and Analysis, 209(3):1055–1088, September 2013a, arXiv:1210.6795.
  3. Nonlocal interactions by repulsive–attractive potentials: Radial ins/stability. Physica D: Nonlinear Phenomena, 260:5–25, October 2013b. ISSN 0167-2789, arXiv:1109.5258.
  4. Bavaud, F. Equilibrium properties of the Vlasov functional: The generalized Poisson-Boltzmann-Emden equation. Reviews of Modern Physics, 63(1):129+, January 1991.
  5. A numerical investigation of the 2-dimensional crystal problem. Preprint CERMICS, available at http://www.ann.jussieu.fr/publications/2003/R03003.html, 2003.
  6. The crystallization conjecture: a review. EMS Surveys in Mathematical Sciences, 2(2):225–306, 2015. ISSN 2308-2151.
  7. Discrete minimisers are close to continuum minimisers for the interaction energy. Calculus of Variations and Partial Differential Equations, 57(1), jan 2018.
  8. Regularity of local minimizers of the interaction energy via obstacle problems. Communications in Mathematical Physics, 343(3):747–781, March 2016. ISSN 1432-0916, arXiv:1406.4040.
  9. The equilibrium measure for an anisotropic nonlocal energy. Calculus of Variations and Partial Differential Equations, 60(3), 2021. ISSN 0944-2669.
  10. Geometry of minimizers for the interaction energy with mildly repulsive potentials. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 34(5):1299–1308, October 2017. ISSN 1873-1430.
  11. Global minimizers of a large class of anisotropic attractive-repulsive interaction energies in 2D. Communications on Pure and Applied Mathematics, 77(2):1353–1404, September 2023. ISSN 1097-0312.
  12. Explicit equilibrium solutions for the aggregation equation with power-law potentials. Kinetic and Related Models, 10(1):171–192, 2017. ISSN 1937-5077.
  13. From radial symmetry to fractal behavior of aggregation equilibria for repulsive–attractive potentials. Calculus of Variations and Partial Differential Equations, 62(1), November 2021. ISSN 1432-0835.
  14. Minimizers of 3D anisotropic interaction energies. Advances in Calculus of Variations, 0(0), November 2022. ISSN 1864-8266.
  15. On global minimizers of repulsive–attractive power-law interaction energies. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372(2028):20130399, 2014. ISSN 1364-503X.
  16. Existence of compactly supported global minimisers for the interaction energy. Archive for Rational Mechanics and Analysis, 217(3):1197–1217, March 2015. ISSN 1432-0673, arXiv:1405.5428.
  17. The sphere packing problem in dimension 24242424. Annals of Mathematics, 185(3), 2017. ISSN 0003-486X.
  18. Universal optimality of the E8subscript𝐸8E_{8}italic_E start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT and Leech lattices and interpolation formulas. Annals of Mathematics, 196(3), 2022. ISSN 0003-486X.
  19. Classifying minimum energy states for interacting particles: Spherical shells. SIAM Journal on Applied Mathematics, 82(4):1520–1536, August 2021. ISSN 1095-712X.
  20. Classifying minimum energy states for interacting particles: Regular simplices. Communications in Mathematical Physics, 399(2):577–598, November 2022. ISSN 1432-0916.
  21. Structural optimization of Lennard-Jones clusters by a genetic algorithm. Chemical Physics Letters, 256(1-2):195–200, 1996. ISSN 0009-2614.
  22. Conquering the hard cases of Lennard-Jones clusters with simple recipes. Computational and Theoretical Chemistry, 1107:7–13, May 2017. ISSN 2210-271X.
  23. Self-propelled particles with soft-core interactions: Patterns, stability, and collapse. Physical Review Letters, 96(10):104302+, March 2006.
  24. On the crystallization of 2D hexagonal lattices. Communications in Mathematical Physics, 286(3):1099–1140, July 2008. ISSN 1432-0916.
  25. Stability of stationary states of non-local equations with singular interaction potentials. Mathematical and Computer Modelling, March 2010a. ISSN 0895-7177.
  26. Stable stationary states of non-local interaction equations. Mathematical Models and Methods in Applied Sciences, 20(12):2267–2291, December 2010b. ISSN 1793-6314.
  27. Frank, R. L. Minimizers for a one-dimensional interaction energy. Nonlinear Analysis, 216:112691, 2022. ISSN 0362-546X.
  28. Frank, R. L. Some minimization problems for mean field models with competing forces, pages 277–294. EMS Press, July 2023. ISBN 9783985475513.
  29. The infinite-volume ground state of the Lennard-Jones potential. Journal of Statistical Physics, 20(6):719–724, June 1979.
  30. Hales, T. A proof of the Kepler conjecture. Annals of Mathematics, 162(3):1065–1185, 2005. ISSN 0003-486X.
  31. Hayes, B. The science of sticky spheres. American Scientist, 100(6):442, 2012. ISSN 1545-2786.
  32. The ground state for sticky disks. Journal of Statistical Physics, 22(3):281–287, 1980. ISSN 0022-4715.
  33. Hoare, M. R. Structure and dynamics of simple microclusters. In Prigogine, I. and Rice, S. A., editors, Advances in Chemical Physics, volume 40, pages 49–135. John Wiley & Sons, 1979.
  34. Physical cluster mechanics: Statics and energy surfaces for monatomic systems. Advances in Chemical Physics, 20:161–196, 1971a. ISSN 0001-8732.
  35. Statics and stability of small cluster nuclei. Nature (Physical Sciences), 230:5–8, 1971b. ISSN 0300-8746.
  36. Geometry and stability of “spherical” f.c.c. microcrystallites. Nature (Physical Sciences), 236:35–37, 1972. ISSN 0300-8746.
  37. Structure of finite sphere packings via exact enumeration: Implications for colloidal crystal nucleation. Physical Review E, 85(5):051403, 2012. ISSN 1539-3755.
  38. Optimization by simulated annealing. Science, 220(4598):671–680, 1983. ISSN 1095-9203.
  39. Monte Carlo-minimization approach to the multiple-minima problem in protein folding. Proceedings of the National Academy of Sciences, 84(19):6611–6615, 1987. ISSN 0027-8424.
  40. Crystallization in two dimensions and a discrete Gauss–Bonnet theorem. Journal of Nonlinear Science, 28(1):69–90, 2018. ISSN 0938-8974.
  41. The free-energy landscape of clusters of attractive hard spheres. Science, 327(5965):560–563, 2010. ISSN 0036-8075.
  42. Minimization of small silicon clusters using the space-fixed modified genetic algorithm method. Chemical Physics Letters, 261(4-5):576–582, 1996. ISSN 0009-2614.
  43. Northby, J. A. Structure and binding of Lennard-Jones clusters: 13≤N≤14713𝑁14713\leq N\leq 14713 ≤ italic_N ≤ 147. Journal of Chemical Physics, 87(10):6166–6177, 1987. ISSN 0021-9606.
  44. Next order asymptotics and renormalized energy for Riesz interactions. Journal of the Institute of Mathematics of Jussieu, 16(3):501–569, 2017. ISSN 1474-7480.
  45. Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture. Proceedings of the American Mathematical Society, 148(7):3047–3057, 2020. ISSN 0002-9939.
  46. Radin, C. The ground state for soft disks. Journal of Statistical Physics, 26(2):365–373, October 1981.
  47. Radin, C. Low temperature and the origin of crystalline symmetry. International Journal of Modern Physics B, 1(5/6):1157–1191, October 1987.
  48. Raoul, G. Nonlocal interaction equations: Stationary states and stability analysis. Differential and Integral Equations, 25(5/6), 2012. ISSN 0893-4983.
  49. Ruelle, D. Statistical mechanics: rigorous results. W.A. Benjamin, 1969. ISBN 9780805383607.
  50. Existence of Ground States of Nonlocal-Interaction Energies. Journal of Statistical Physics, 159(4):972–986, February 2015. ISSN 1572-9613, arXiv:1405.5146.
  51. Theil, F. A Proof of Crystallization in Two Dimensions. Communications in Mathematical Physics, 262(1):209–236, February 2006.
  52. From sticky-hard-sphere to Lennard-Jones-type clusters. Physical Review E, 97(4):043309, 2018. ISSN 2470-0045.
  53. Ventevogel, W. J. Why do crystals exist? Physics Letters A, 64(5):463–464, 1978a. ISSN 0375-9601.
  54. Ventevogel, W. J. On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle. Physica A: Statistical Mechanics and its Applications, 92(3-4):343–361, 1978b. ISSN 0378-4371.
  55. On the configuration of systems of interacting particles with minimum potential energy per particle. Physica A: Statistical Mechanics and its Applications, 99(3):569–580, 1979a. ISSN 0378-4371.
  56. On the configuration of systems of interacting particle with minimum potential energy per particle. Physica A: Statistical Mechanics and its Applications, 98(1-2):274–288, 1979b. ISSN 0378-4371.
  57. Viazovska, M. The sphere packing problem in dimension 8888. Annals of Mathematics, 185(3), 2017. ISSN 0003-486X.
  58. The Cambridge Cluster Database. Online. URL http://www-wales.ch.cam.ac.uk/CCD.html.
  59. Wales, D. J. GMIN optimisation software. Available at https://www-wales.ch.cam.ac.uk/GMIN/.
  60. Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms. The Journal of Physical Chemistry A, 101(28):5111–5116, 1997. ISSN 1089-5639.
  61. Wille, L. T. Searching potential energy surfaces by simulated annealing. Nature, 325:374–374, 1987a. ISSN 0028-0836.
  62. Wille, L. T. Minimum-energy configurations of atomic clusters: new results obtained by simulated annealing. Chemical Physics Letters, 133(5):405–410, January 1987b. ISSN 0009-2614.
  63. Zhang, J. A brief review on results and computational algorithms for minimizing the Lennard-Jones potential. December 2010, arXiv:1101.0039 [physics.comp-ph].
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