Journal of Statistical Physics

, Volume 159, Issue 4, pp 972–986 | Cite as

Existence of Ground States of Nonlocal-Interaction Energies

Article

Abstract

We investigate which nonlocal-interaction energies have a ground state (global minimizer). We consider this question over the space of probability measures and establish a sharp condition for the existence of ground states. We show that this condition is closely related to the notion of stability (i.e. \(H\)-stability) of pairwise interaction potentials. Our approach uses the direct method of the calculus of variations.

Keywords

Ground states Global minimizers H-stability Pair potentials Self-assembly Aggregation 

Mathematics Subject Classification

49J45 82B21 82B05 35R09 45K05 

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)Google Scholar
  2. 2.
    Balagué, D., Carrillo, J.A., Laurent, T., Raoul, G.: Dimensionality of local minimizers of the interaction energy. Arch. Ration. Mech. Anal. 209, 1055–1088 (2013)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Balagué, D., Carrillo, J.A., Laurent, T., Raoul, G.: Nonlocal interactions by repulsive–attractive potentials: radial ins/stability. Phys. D 260, 5–25 (2013)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Balagué, D., Carrillo, J.A., Yao, Y.: Confinement for attractive–repulsive kernels. Discret. Contin. Dyn. S. (to appear)Google Scholar
  5. 5.
    Benedetto, D., Caglioti, E., Pulvirenti, M.: A kinetic equation for granular media. RAIRO Modél. Math. Anal. Numér. 31(5), 615–641 (1997)MATHMathSciNetGoogle Scholar
  6. 6.
    Bernoff, A.J., Topaz, C.M.: A primer of swarm equilibria. SIAM J. Appl. Dyn. Syst. 10(1), 212–250 (2011)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bertozzi, A.L., Carrillo, J.A., Laurent, T.: Blow-up in multidimensional aggregation equations with mildly singular interaction kernels. Nonlinearity 22(3), 683–710 (2009)CrossRefADSMATHMathSciNetGoogle Scholar
  8. 8.
    Bertozzi, A.L., Laurent, T.: Finite-time blow-up of solutions of an aggregation equation in \(\mathbf{R}^n\). Commun. Math. Phys. 274(3), 717–735 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bertozzi, A.L., Laurent, T., Flavien, L.: Aggregation and spreading via the Newtonian potential: the dynamics of patch solutions. Math. Model. Methods Appl. Sci. 22(Supp. 1), 1140005 (2012)CrossRefGoogle Scholar
  10. 10.
    Bertozzi, A.L., Laurent, T., Rosado, J.: \({L}^p\) theory for the multidimensional aggregation equation. Commun. Pur. Appl. Math. 64(1), 45–83 (2011)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Billingsley, P.: Weak Convergence of Measures: Applications in Probability. Society for Industrial and Applied Mathematics, Philadelphia (1971)CrossRefMATHGoogle Scholar
  12. 12.
    J.A. Cañizo, Carrillo, J.A. and Patacchini F.S.: Existence of compactly supported global minimizers for the interaction energy. preprint, 2014Google Scholar
  13. 13.
    Calogero, F., Simonov, YuA: Simple upper bound to the ground-state energy of a many-body system and condition on the two-body potential necessary for its stability. Phys. Rev. 183, 869–872 (1969)CrossRefADSGoogle Scholar
  14. 14.
    Carrillo, J.A., Chipot, M., and Huang Y.: On global minimizers of repulsive-attractive power-law interaction energies. preprint, 2014Google Scholar
  15. 15.
    Carrillo, J.A., Delgadino, M.G., and Mellet, A.: Regularity of local minimizers of the interaction energy via obstacle problems. preprint, 2014Google Scholar
  16. 16.
    Carrillo, J.A., DiFrancesco, M., Figalli, A., Laurent, T., Slepčev, D.: Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations. Duke Math. J. 156(2), 229–271 (2011)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Carrillo, J.A., DiFrancesco, M., Figalli, A., Laurent, T., Slepčev, D.: Confinement in nonlocal interaction equations. Nonlinear Anal. 75(2), 550–558 (2012)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Carrillo, J.A., McCann, R.J., Villani, C.: Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179(2), 217–263 (2006)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Choksi, R., Fetecau, R.C., and Topaloglu, I.: On minimizers of interaction functionals with competing attractive and repulsive potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire.( to appear)Google Scholar
  20. 20.
    Dobrushin, R.: Investigation of conditions for the asymptotic existence of the configuration integral of Gibbs’ distribution. Theory Probab. Appl. 9(4), 566–581 (1964)CrossRefGoogle Scholar
  21. 21.
    Fellner, K., Raoul, G.: Stable stationary states of non-local interaction equations. Math. Model. Methods Appl. Sci. 20(12), 2267–2291 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Fetecau, R.C., Huang, Y.: Equilibria of biological aggregations with nonlocal repulsive–attractive interactions. Phys. D 260, 49–64 (2013)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Fetecau, R.C., Huang, Y., Kolokolnikov, T.: Swarm dynamics and equilibria for a nonlocal aggregation model. Nonlinearity 24(10), 2681–2716 (2011)CrossRefADSMATHMathSciNetGoogle Scholar
  24. 24.
    Fisher, Michael E., Ruelle, David: The stability of many-particle systems. J. Math. Phys. 7, 260–270 (1966)CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Haile, J.M.: Molecular Dynamics Simulation: Elementary Methods. Wiley, New York (1992)Google Scholar
  26. 26.
    Holm, D.D., Putkaradze, V.: Aggregation of finite-size particles with variable mobility. Phys. Rev. Lett. 95, 226106 (2005)CrossRefADSGoogle Scholar
  27. 27.
    Holm, D.D., Putkaradze, V.: Formation of clumps and patches in selfaggregation of finite-size particles. Phys. D. 220(2), 183–196 (2006)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Kolokolnikov, T., Huang, Y., Pavlovski, M.: Singular patterns for an aggregation model with a confining potential. Phys. D 260, 65–76 (2013)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Kolokolnikov, T., Sun, H., Uminsky, D., Bertozzi, A.L.: A theory of complex patterns arising from 2D particle interactions. Phys. Rev. E, Rapid. Phys. Rev. E Rapid Commun. 84, 015203(R) (2011)CrossRefADSGoogle Scholar
  30. 30.
    Laurent, T.: Local and global existence for an aggregation equation. Commun. Partial Differ. Equ. 32(10–12), 1941–1964 (2007)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Leverentz, A.J., Topaz, C.M., Bernoff, A.J.: Asymptotic dynamics of attractive–repulsive swarms. SIAM J. Appl. Dyn. Syst. 8(3), 880–908 (2009)CrossRefADSMATHMathSciNetGoogle Scholar
  32. 32.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire. 1(2):109–145 (1984)Google Scholar
  33. 33.
    Mogilner, A., Edelstein-Keshet, L.: A non-local model for a swarm. J. Math. Biol. 38, 534–570 (1999)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Radin, C.: The ground state for soft disks. J. Stat. Phys. 26(2), 365–373 (1981)CrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Ruelle, D.: Statistical Mechanics: Rigorous Results. W. A. Benjamin Inc., New York-Amsterdam (1969)MATHGoogle Scholar
  36. 36.
    Struwe, M.: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 3rd edn. Springer-Verlag, Berlin (2000)CrossRefGoogle Scholar
  37. 37.
    Sütő, A.: Ground state at high density. Commun. Math. Phys. 305(3), 657–710 (2011)CrossRefADSGoogle Scholar
  38. 38.
    Theil, F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262(1), 209–236 (2006)CrossRefADSMATHMathSciNetGoogle Scholar
  39. 39.
    Toscani, G.: One-dimensional kinetic models of granular flows. M2AN Math. Model. Numer. Anal. 34(6), 1277–1291 (2000)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    van der Vaart, A.W., Wellner, J.: Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. Springer, New York (1996)CrossRefGoogle Scholar
  41. 41.
    von Brecht, J.H., Uminsky, D., Kolokolnikov, T., Bertozzi, A.L.: Predicting pattern formation in particle interactions. Math. Model. Methods Appl. Sci. 22(Suppl. 1), 1140002 (2012)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Robert Simione
    • 1
  • Dejan Slepčev
    • 1
  • Ihsan Topaloglu
    • 2
  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations